Kirsi Peltonen
email: kirsi.peltonen@uta.fi

Noora Ellonen

Helmer B Larsen

Karin Helweg-Larsen

Parental violence and adolescent mental health 1 Parental violence and adolescent mental health

Keywords: parental violence, adolescence, mental health Parental violence and mental health

Being the target of parental violent acts decreases child adjustment and increases the likelihood of mental health problems in childhood and adolescence. Our study analyzes how different types of parental violence ranging from verbal threats and swearing to hitting and kicking a child, are associated with child adjustment, indicated by SDQ total problem score, internalizing and externalizing problems as well as prosocial behaviour. We also study whether girls and boys and youths in two Nordic countries respond differently to parental violence. The data consists of a large-scale community sample of 15-16 year old Finnish (n=5,762) and Danish (n=3,943) adolescents. The representative data of continental Finland and its Finnish and Swedish speaking 9th graders as well as representative data of Danish 9 th grade pupils was collected by the Police College of Finland and in Denmark by the National Institute of Public Health, University of Southern Denmark. The results show a clear dose-response effect between parental violent behaviour and the adolescent's problems. The more severe forms of parental violence were associated with higher levels of SDQ total difficulties and internalizing and externalizing symptoms. There was also a connection between parental violence and the deterioration of prosocial behaviour. The association was gender and nationality specific. The findings imply a high prevalence of parental violence and adverse mental health among the affected Finnish and Danish adolescents. Though the laws have been set in motion to prevent the use of parental physical violence the challenges remain in several domains of child protection, general health care, prevention and intervention.

abuse [START_REF]Child abuse. Towards a knowledge base[END_REF]. Also, the detailed analyses of the effects of different severity levels of maltreatment on children are practically missing [START_REF] Litrownik | Measuring the severity of child maltreatment[END_REF].

The Developmental Victimization Survey of 2,030 American children [START_REF] Finkelhorn | The victimization of children and youth: A comprehensive, national survey[END_REF] revealed that in the year preceding the survey, child maltreatment (broadly defined) occurred approximately in 1 in 7 of the 2-17 year old children and adolescents. Of the five maltreatment types measured in the study, emotional abuse was the most frequent, with 10% of children experiencing name calling or denigration by an adult. The large scale Finnish study of 7,349 9 th graders conducted in 1988 showed that during the previous 12 months 47% of children experienced verbal aggression, 19% were targets of mild physical violence such as slapping or pushing, and 5% reported an experience of severe physical violence [START_REF] Sariola | The prevalence and context of family violence against children in Finland[END_REF].

Maltreatment is thus a common phenomenon and therefore requires more detailed research. The meaning of different types of maltreatment should be investigated separately, and gender-based differences in reactions to maltreatment studied, in order to provide a better understanding of maltreatment. Also, comparative analyses between countries of the prevalence and meaning of maltreatment is needed to benchmark and identify urgent and important national specific problems or health goals. However, only few such studies have yet been published [START_REF] Gilbert | Burden and consequences of child maltreatment in high-income countries[END_REF].

The Nordic countries have shown to have many similarities in youth victimization studies [START_REF] Enzmann | Self-reported youth delinquency in Europe and beyond: First results of the Second International Self-Report Delinquency Study in the context of police and victimization data[END_REF] but especially related to parental maltreatment, countries also have remarkable differences. Although across the Nordic countries, violence and maltreatment have been rather strictly defined and legislated against, there is a major difference between the two countries included in this study regarding the time when corporal punishment was made a penal act. In the Finnish criminal code, corporal punishment has been a penal act since 1984, whereas in Denmark the same law was not introduced until 1997. This creates an interesting setting for a comparative analysis.

In our study the focus is on the associations between parental violence with its three levels of severity and adolescents' mental health. In addition, comparative analyses of those results will be provided between genders and between the two Nordic countries, Denmark and Finland.

Child physical abuse: Influence on child's mental health and social relations

Previous studies have revealed that children who have been victims of abuse and neglect have more psychosocial problems, and they function less adaptively in several areas of development than do their non abused peers [START_REF] Boney-Mccoy | Psychosocial sequelae of violent victimization in a national youth sample[END_REF][START_REF] Holmes | The role of parental disciplinary practices in the development of depression and alcoholism[END_REF][START_REF] Trickett | The developmental impact of different forms of child abuse and neglect[END_REF]..Concerning physical abuse, some evidence exists that both internalizing and externalizing symptoms seem to occur among children and adolescents exposed to physical maltreatment, and that those who experience more serious physical abuse show more internalizing and externalizing behavioural problems than those who experience less serious abuse [START_REF] Stockhammer | Assessment of the effect of physical child abuse within an ecological framework: Measurement issues[END_REF].

Fantuzzo [START_REF] Fantuzzo | Behavioral treatment of the victims of child abuse and neglect[END_REF] found that physically abused children showed significantly higher levels of aggression than their non-abused peers, even after controlling for several family related factors. Also, clinically significant externalizing symptoms have been reported to be significantly higher among physically abused children compared to the reference group. Ackerman et al. [START_REF] Ackerman | Prevalence of post traumatic stress disorder and other psychiatric diagnoses in three groups of abused children (sexual, physical, and both)[END_REF] found that children with a history of physical abuse have higher rates of oppositional defiant disorder and conduct disorder diagnoses compared to other maltreatment subcategories such as emotional neglect or sexual abuse. Additionally, in their longitudinal birth cohort study, Fergusson & Lynskey [START_REF] Fergusson | Physical punishment/maltreatment during childhood and adjustment in young adulthood[END_REF] found that young people at age 18 reporting exposure to harsh or abusive treatment during childhood had elevated rates of violent offences, suicide attempts and alcohol abuse. Noteworthy, a meta-analysis of corporal punishment revealed similar results. Within 88 reviewed studies, corporal punishment was associated with several negative outcomes in childhood as well as in adulthood, including increased aggression, antisocial behaviour and abusive behaviour towards others [START_REF] Gershoff | Corporal punishment by parents and associated child behaviors and experiences: A metaanalytic and theoretical review[END_REF].

There is also evidence of elevated levels of internalizing symptoms such as depression among physically abused children compared to the reference group [START_REF] Pelcovitz | Psychiatric disorders in adolescents exposed to domestic violence and physical abuse[END_REF]. Finzi et al. [START_REF] Finzi | Depressive symptoms and suicidality in physically abused children[END_REF] substantiated the specific role of parental violence as a source of internalizing problems, reporting that children who were exposed to parental violence had more depressive symptoms and suicidal behaviour compared not only to non maltreated children but also to neglected children. Physical maltreatment has also been found to be associated with problems in dating partners and making close friends [START_REF] Wolfe | Factors associated with abusive relationships among maltreated and nonmaltreated youth[END_REF].

Exposure to parental violence in adolescence

In most studies, the "lifetime" count of maltreatment experiences has been used as a measure of violent situations in home environments and its impact on children's well being and mental health [START_REF] Hanson | Measurement in child abuse research: A survey of researchers[END_REF]. This research tradition, however, fails to show the specific effects of maltreatment experiences at a certain developmental stage, and more systematic research on the association between concurrent exposure to violence and the range of internalizing, externalizing and transitional outcomes has been asked for [START_REF] Barnett | Family violence across the lifespan[END_REF]23]. In our study, we wanted to clarify the specific effects of the acute experiences of parental violence at the age of 15-16. We rely on adolescents self reports, which are regarded as an important and reliable source of information in older children [START_REF] Gilbert | Burden and consequences of child maltreatment in high-income countries[END_REF][START_REF] Fergusson | The stability of child abuse reports: a longitudinal study of the reporting behaviour of young adults[END_REF][START_REF] Finkelhor | Childhood victimization. Violence, crime and abuse in the lives of young people[END_REF].

The extent of violent experiences varies among abused children. Barnett et al. [START_REF] Barnett | Defining Child Maltreatment: The Interface between Policy and Research[END_REF], for example, found that among 11-14 year old children the extent of abusive experiences varied from a single incident of maltreatment to 14 years of recurring incidents of maltreatment. Also, results on the age differences of exposure to child maltreatment are controversial, some studies showing a decline in maltreatment experiences with age and some indicating higher rates of maltreatment experiences in adolescence than at a younger age [START_REF] Belsky | Etiology of child maltreatment: A developmental-ecological analysis[END_REF][START_REF] Cohen | Child abuse and neglect and the development of mental disorders in the general population[END_REF]. In the USA the Developmental Victimization Survey [START_REF] Finkelhorn | The victimization of children and youth: A comprehensive, national survey[END_REF] revealed that physical abuse was highest among the teenage group, boys and girls experiencing similar rates of violence.

In addition to the emphasis on the detrimental nature of physical maltreatment in early childhood [START_REF] Keiley | The timing of child physical maltreatment: A cross-domain growth analysis of impact on adolescent externalizing and internalizing problems[END_REF], the adverse consequences of adolescence-only maltreatment as well as persistent maltreatment are now seen as an important aspect of child mental health and behaviour [START_REF] Thornberry | The importance of timing: The varying impact of childhood and adolescent maltreatment on multiple problem outcomes[END_REF]. So far very few studies apply the experimental comparative developmental approach within maltreatment research, but Stewart, Livingston & Dennison [START_REF] Stewart | Transitions and turning points: Examining the links between child maltreatment and juvenile offending[END_REF] noticed that children whose maltreatment extended into adolescence had higher rates of offences than those with childhood-only maltreatment.

It is noteworthy that the most accurate reports of violent experiences are achieved by using a rather short referent period [START_REF] Czaja | Respondent strategies for recall of crime victimization incidents[END_REF] and a one-year referent has been shown to be relevant [START_REF] Hamby | The victimization of children: Recommendations for assessment and instrument development[END_REF].

Research questions

This study first examines whether different levels of parental violence (during the last 12 months) are differently associated with self-reported mental health, indicated by SDQ total problem score, internalizing and externalizing symptoms as well as prosocial behaviour. We hypothesized a dose-response effect between parental violent behaviour and the adolescent's problems. This means that we expected adolescents exposed to parental violence to have higher levels of mental health problems and lower levels of prosocial behaviour compared to adolescents without experience of parental violence. Further, we hypothesized that adolescents exposed to more severe forms of parental violence had increased levels of these problems compared to adolescents with exposure to milder levels of violence. Second, we examined whether the association between different severity levels of parental violence and mental health was gender or country specific.

Method Participants and procedure of the study

The analysis in this article is based on the Finnish Child Victim Survey (2008) and Danish Youth (2008) study. The surveys covered a wide variety of violence against children, from traditional street violence and violence between peers to sexual abuse, family violence and bullying. In Finland the data was collected among 12-13 (6 th graders) and 15-16 (9 th graders) year old pupils. In this study only the data for 9 th graders is analysed. In Denmark the data was collected among 9 th graders.

In Finland, the survey was conducted by the Police College of Finland and in Denmark by the National Institute of Public Health, University of Southern Denmark in 2008. In both countries the surveys were based on multimedia computer-based self-administered interviews, which the children answered during school hours. The Finnish children accessed the questionnaire via a website which included information about the project as well as about violence in general. The survey was administered by teachers in the schools who were all properly instructed by the research team. The Danish survey was conducted in the school classroom where trained interviewers introduced the survey method, and they remained in the classroom while the students completed the questionnaire.

The data is a representative sample of mainland Finland and its Finnish and Swedish speaking 9 th graders as well as a representative sample of Danish 9 th grade pupils in the mandatory school system. The final Finnish sample consisted of 2,856 girls and 2,906 boys and the final Danish data 1,999 girls and 1,944 boys.

The Finnish sampling was drawn with a stratified cluster sample design based on county, quality of municipality and size of the school. The original sample included 184 schools with children in 9 th grade. Of those 161 participated to the survey (88 %). Data consisted of 5807 responses. The exact response rate can't unfortunately be presented because the variable needed to calculate that (namely the school ID), was vanished during the data collection. However, the representative nature of the final data was checked comparing basic socioeconomic factors to other representative studies (see more 33). Of the total amount of responses 45 were excluded due to lack of consistency in the answers. In the Danish survey, a random sample of 342 schools was selected from lists of all public, non-public, and institutional schools in Denmark. The schools were sampled randomly within six strata corresponding to different classifications of "percent urban" in the Danish census.. Of those, 122 schools participated (35%). No important differences were found between the strata distribution of the participating and nonparticipating schools. There were 5,013 9 th graders in those schools and 4,093 of those participated in the survey. Also in the Danish data, due to lack of consistency in some of the data sets, some responses were excluded from the data.

Measures

Parental violence was measured by Finnish and Danish versions of the Conflict Tactics Scale created by Straus [START_REF] Straus | Measuring intra family conflict and violence: The Conflict Tactics Scale[END_REF]. The scale consists of 14 items beginning with parental aggressive verbalization towards the child and moving to severe violent acts towards the child. Participants responded by indicating whether they had experienced such acts (1 = yes) or not (0= no) during the previous 12 months. The four groups indicating different severity levels of parental violence were formed based on these answers. The "No violence" group included children who had no experiences of verbal aggression or mild or severe physical violence. The "Verbal aggression" group included children who had experienced verbal aggression but no mild or severe physical violence. Acts of verbal aggression included items such as "sulking or refusing to talk, insulting or taunting or swearing, throwing objects and threatening with violence". The "Mild physical violence" group included children who had experienced mild parental violence accompanied or not accompanied by verbal aggression. Acts of mild physical violence included "pushing or shoving or shaking, hair pulling, smacking and whipping". The "Severe physical violence" group included children who had experienced severe parental aggression accompanied or not accompanied by acts of mild physiological violence and/or verbal aggression. Acts of severe physical violence included "battering, hitting with wrist, hitting with object, kicking, threatening with knife or gun and using a knife or gun".

An adolescent was included in one of the three violence groups if she/he had experienced at least one of the acts in a category. That means that there is a variation within the three groups on the frequency of experiences ranging from a onetime experience of one of the included acts, to several experiences of multiple included items. If the information about the experience was missing in some of those individual acts, the case was categorized as no experience before forming the summed variables. That was done to minimize the amount of missing data in the summed variables. SDQ total difficulties, internalizing and externalizing symptoms and prosocial behaviour were measured by the Strengths and Difficulties Scale (SDQ) by Goodman [START_REF] Goodman | The strengths and difficulties questionnaire: A research note[END_REF] The scale consists of 25 items on psychological attributes describing internalizing problems of depression and anxiety, and externalizing problems such as aggression and hyperactivity, plus prosocial behaviour. Participants evaluated how well the description fitted them on a 3-point scale (0 = not true, 1 = somewhat true, 2 = certainly true). Factor analysis (Varimax) was applied to check the validity of dimensionality. The results revealed a somewhat different factor structure than the traditional five scale solution in SDQ and the reliability of the original dimensions of SDQ was low in the data (range .68). The best fitting factor solution in these analyses was a 3-factor model. Based on these factors the averaged sum variables of externalizing, internalizing and pro-social behaviour were formed. The externalizing factor included 7 items ("I get very angry and often lose my temper", "I take things that are not mine from home, school or elsewhere", etc.), the internalizing factor included 8 items ("I have many fears, I am easily scared, I am often unhappy, down-hearted or tearful", etc.) and the prosocial factor 9 items ("I try to be nice to other people. I care about their feelings", "I am helpful if someone is hurt, upset or feeling ill", etc.). The 20 items including questions about emotional symptoms (5items), conduct problems (5items), hyperactivity (5items) and peer relationship problems (5items) were added together to generate a SDQ total difficulties score. The test-retest reliabilities, internal consistency and criterion validity of these scales have been well established [START_REF] Goodman | The strengths and difficulties questionnaire: A research note[END_REF][START_REF] Bourdon | The strengths and difficulties questionnaire: U.S. Normative data and psychometric properties[END_REF]. The reliability of the SDQ for children's self reports was for SDQ total score, for internalizing symptoms, for externalizing symptoms and for pro-social behaviour.

Statistical analysis

To examine the associations between parental violence and child mental health, and their gender and nationality specificity the 4 (violence: no violence, verbal, mild and severe) X 2 (gender) X 2 (nationality) between subjects factorial multivariate analysis of variance (MANOVA) with their two-way interactions were applied to the dependent variables of the SDQ total difficulties score, internalizing symptoms, externalizing symptoms and prosocial behaviour. A Bonferroni correction was used to obtain a more conservative alpha level. The post hoc tests using the Tukey HSD post hoc criterion for significance were conducted to examine the differences between the four severity levels of parental violence. Examinations of the homogeneity of variance-covariance matrices and normality assumptions underlying MANOVA did not reveal any substantial anomalies.

Results

Descriptive results

In the Finnish sample 49% of girls and 74% of boys had no experiences of parental verbal aggression or physical violence during the previous 12 months. Differences between the sexes were clear in all severity levels of parental violence, with girls reporting higher exposure. Being the target of parental verbal aggression (girls 39%, boys 20%), mild physical violence (girls 10%, boys 4%) or severe physical violence (girls 2%, boys 1%) was two times more common among girls compared to boys.

In the Danish sample 58% of girls and 68% of boys reported no experiences of parental verbal aggression or physical violence during the previous 12 months. Being the target of parental verbal aggression (girls 43%, boys 33%) and mild physical violence (girls 9%, boys 4%) was more common among girls compared with boys, whereas the same number of girls and boys reported exposure to severe physical violence (girls 2%, boys 2%).

Insert Table 1 about here

Parental violence and mental health

Table 2 presents the main and interaction effects between exposure to parental violence, gender, country and mental health outcomes. externalizing: M = .46, SD = .35; internalizing: M = .57, SD = .37) were significantly higher than in the no violence condition (SDQ total: M = 10.47, SD = 4.81; externalizing: M = .38, SD = .34; internalizing: M = .44, SD = .33).

Main effect and

Further, the mean scores for exposure to mild violence (SDQ total: M = 14.37, SD = 5.40, Externalizing: M = .61, SD = .39, internalizing: M = .66, SD = .42) were significantly higher than for verbal aggression. The mean scores for exposure to severe violence (SDQ total: M = 15.60, SD = 5.67, externalizing: M = .75, SD = .43, internalizing: M = .73, SD = .40) were significantly higher than for exposure to mild violence concerning the externalizing symptoms but not SDQ total difficulties score or internalizing symptoms. Concerning the prosocial behaviour, however, only the mean scores for exposure to mild parental violence (M = 1.21, SD = .29) was significantly lower than in the no violence condition ( M = 1.27, SD = .33), and there were no other significant differences.

Interactions between exposure to parental violence, gender, nationality and mental health outcomes.

The association between parental violence and internalizing symptoms was gender specific, as evidenced by the significant interaction effect (F(1,9185)=5.24<.001). This means that a high level of parental violence was associated with internalizing symptoms especially among girls. Also, the association between parental violence and mental health was country specific as evidenced by the significant interaction effects of internalizing (F(1,9185)=4.98, p<.01) and externalizing (F(1,9185)=3.67, p<.05) symptoms and total SDQ total difficulties score (F(1,9185)=7.07, p<.01). These interactions indicate that the exposure to parental violence was associated with mental health symptoms especially among Danish adolescents. Generally taken, the symptom scores, however, remained at a lower level in every class of parental violence exposure compared to Finnish adolescents. In other words, there were clearer differences between the classes of exposure to parental violence in symptom scores among Danish than Finnish youths.

Main effects of gender and country

There was a significant main effect of gender on internalizing symptoms (F(1,9185)=9.96,p<.01) and externalizing symptoms (F(1,9185)=7.05,p<.01) indicating that boys have higher levels of symptoms than girls.

There was also a significant main effect of country on internalizing symptoms (F(1,9185)=14. Insert Tables 2 about here

Discussion

Our results showed that about 40% of adolescents in both samples in this study had experienced verbal aggression and/or physical violence from their parents during the previous 12 months prior to the research, revealing a startling reality of negative experiences in the family context. That is much more than the results of the Child Victimization survey [START_REF] Finkelhorn | The victimization of children and youth: A comprehensive, national survey[END_REF] among American children and youth showed, but much less than in the 1988 study among Finnish adolescents [START_REF] Sariola | The prevalence and context of family violence against children in Finland[END_REF]. The American Child Victimization survey, however, excluded episodes of conventional corporal punishment. Exposure to parental violence was greater among girls than boys in this analysis. This was contrary to the findings from American samples showing similar exposure to minor violent acts among girls and boys, but which showed a slightly greater risk of severe physical abuse among boys compared with girls [START_REF] Sedlak | The third national incidence study of child abuse and neglect[END_REF].

In line with earlier research [39,[START_REF] Stockhammer | Assessment of the effect of physical child abuse within an ecological framework: Measurement issues[END_REF], the adolescents who reported more severe forms of parental violence also reported more severe psychological problems. It is important to note that the exposure to verbal aggression was associated with higher levels of total difficulties, internalizing and externalizing compared with the no violence condition. Thus, the verbal threatening and subjugation of a child may pose a risk to the adolescent mental health.

Furthermore, when verbal aggression changes to or is accompanied by physical violence, the level of the adolescent's problems is even higher. This implies that the acts that are traditionally seen as corporal punishment such as slapping or pulling ones hair are acts that are frightening enough to produce agitation. The significance may lie in realizing that a parent, who is supposed to be protective and calm is acting unpredictably. The dose-response effect was not so evident concerning prosocial behaviour, even though adolescents who were exposed to mild parental violence reported poorer prosocial behaviour compared to adolescents with any experiences of parental violence. This trend of prosocial behaviour being less affected by adverse experiences may be seen as a resource for rehabilitation and prevention of mental health problems among exposed adolescents. Intact human relationships and adolescent's ability to act in positive ways outside the family context are of great importance when dealing with mental or behavioural difficulties.

Only the internalization symptoms showed a gender specificity, high levels of parental violence being associated with internalizing symptoms. Girls exposed to parental violence in our study were more likely to react with symptoms of anxiety and depression compared to boys. The national differences found in our study imply that parental violence more strongly increases mental health symptoms among Danish adolescents, although we have to be aware that in general the level of symptoms remained lower compared to Finnish children. Thus, there were clearer differences between the classes of exposure to parental violence in symptom scores among Danish than among Finnish adolescents. The gender specificity in the association between parental violence and internalizing symptoms should make us cautious about drawing general conclusions. However, the adolescent attitudes towards parental violence should be taken into account when explaining the results. The results may also support the widely held view that there are huge individual differences in the way that individuals (both children and adults) respond to stress and adversity [START_REF] Rutter | Understanding and testing risk mechanisms for mental disorders[END_REF].

Although there is no detailed knowledge of the effects of parental violence according to developmental stages, adolescence is generally seen as a vulnerable period of exposure to a harmful home environment [START_REF] Demaris | Violent victimization and women's mental and physical health: Evidence from a national sample[END_REF]. The vulnerability among the affected adolescents in this study manifested itself in elevated levels of SDQ total difficulties and internalizing and externalizing symptoms as well as some deterioration of prosocial behaviour. The fact that both internalizing and externalizing symptoms were associated with the level of parental violence implies that adolescents experience a wide variety of distress when exposed to an adverse home environment. Unfortunately our data does not allow us to take account of the strength of maltreatment in these different subtypes, which would also need further research. One incidence compared to continuous maltreatment may have different kinds of outcomes.

Legislation

In 1988, when the earlier Child Victim Survey was made in Finland [START_REF] Sariola | The prevalence and context of family violence against children in Finland[END_REF], corporal punishment had been illegal for only four years. By the time of the current study twenty-four years have passed since it was prohibited.

Concerning parental violence, the most outstanding difference between these two points of time can be seen in the prevalence of mild physical violence (including the acts that have traditionally been regarded as a form of corporal punishment). The number of children in Finland experiencing slapping and pushing, for example, by their parents in 2008 had dropped to one third of what it was in 1988. The same trend, although not as clear as in mild physical violence, can also be seen in verbal and severe physical violence.

Among Finnish children in the current study sample, corporal punishment has been forbidden by law during their whole lifetime, whereas in Denmark the children in this study sample have spent approximately four of their first years living in communities where slapping, or pulling a child's hair, for example, was still legal. Noticing this difference in the timing of legally abandoning parental corporal punishment, it is interesting to compare the frequencies of parental violent acts between the two countries. Although maltreatment analysed here has occurred during the past 12 months of the survey, when maltreatment was illegal in both countries, it has been shown at least in Finland that attitudes towards physical punishment do not change rapidly [START_REF] Sariola | Älä lyö lasta kampanja muutti asenteita [Don't hit the child -campaign changed attitudes towards physical punishment[END_REF], and therefore the prevalence of maltreatment may vary because of the difference in timing. The results, however, showed that the number of children who had experienced mild or severe physical violence within the previous 12 months was almost exactly the same in Finland and Denmark. The actual impact of the legislation can't be specified but the result is very interesting and reveals that Nordic countries, at least Denmark and Finland, do not differ so remarkably in experiencing maltreatment at home, although actions to prevent it may differ. Although slapping, or pulling a child's hair, for example, have been forbidden by law, there is still a minority of parents in both societies who do so anyway, while most of the parents do not punish their children that way. Discipline practices are closely related to cultural beliefs and, despite the empirical evidence of their negative consequences, they seem difficult to change [START_REF] Westby | Child maltreatment: A global issue[END_REF].

Co-occurring forms of adverse experiences

We have to be aware of the fact that children's experiences within the previous 12 months reflect different kinds of violent history in their homes. When a child is exposed to psychological and physical violence, he/she could at the same time be exposed to other forms of maltreatment, such as sexual abuse or physical/emotional neglect. There is evidence that physical and psychological violence (emotional maltreatment) as well as a lack of general supervision tend to co-occur [39]. Likewise, interparental violence (often referred to as domestic violence)

and parental violence towards children are, at least to some extent, co-occuring phenomena [START_REF] Edleson | The overlap between child maltreatment and woman battering[END_REF][START_REF] Holt | The impact of exposure to domestic violence on children and young people: A review of the literature[END_REF]. For example, mothers who are targets or perpetrators of partner violence are more prone to use physical violence towards their children than mothers who do not have such experiences [START_REF] Ross | Risk of physical abuse to children of spouse abusing parents[END_REF].

This kind of accumulation of victim experiences most definitely also plays a role when associations between maltreatment and adolescents' mental health are studied. Witnessing interparental violence, for example, is linked to adverse mental health outcomes among children [for meta-analysis see for example [START_REF] Evans | Exposure to domestic violence: A meta-analysis of child and adolescent outcomes[END_REF][START_REF] Kitzmann | Child witnesses to domestic violence: A meta-analytic review[END_REF]. Finkelhor and others [START_REF] Finkelhor | Poly-victimization: A Neglected Component in Child Victimization[END_REF] have even suggested that taking so called poly-victimization into account, the impact of individual victimizations on mental health may even disappear. Knowledge of the accumulation of children's violent experiences thus stresses the need for a more comprehensive picture [START_REF] Finkelhorn | The victimization of children and youth: A comprehensive, national survey[END_REF] and should be noted as a consequential factor behind our results, but doesn't diminish the importance to study also the meaning of parental maltreatment.

According to our other research, although experiences of poly-victimization is taken into account and diminishes the impact of some other kinds of individual victimization, it does not make the impact of parental maltreatment on adolescents' mental health disappear [START_REF] Ellonen | Moninaisen väkivallan moninaiset vaikutukset lasten psykososiaaliseen hyvinvointiin[END_REF]. In addition, it has been argued, that child maltreatment (physical abuse or neglect) is the form of victimization that has the strongest independent association with depression and anger/aggression [52].

Limitations of the study:

The Danish sample suffered from a relatively large number of schools who for several reasons did not participate.

Although no important differences were found between the strata distribution of the participating and nonparticipating schools, there may be differences in the available schools from the non-participating schools on characteristics relevant to the study.

Given the cross-sectional nature of the study we have to be aware of the bi-directional nature of our findings. We have shown that parental use of violence (involving swearing, slapping, or even hitting the child) is associated with a child's mental problems. This association can reflect the harmful effect that parents behaviour is causing distress for a child, but it can also be the case that a child's behavioural or emotional problems might result in the parents

  Post hoc comparisons of parental violence. The results show a significant association between exposure to parental violence and SDQ total score (F (3,9185)= 132.84 p <.001), internalizing symptoms (F(3,9185)=75.90, p<.001), externalizing symptoms (F(3,9185)=126.57, p<.001), and prosocial behaviour (F(3,9185)=10.34, p<.001) indicating that children experiencing the more severe forms of parental violence reported poorer mental health and prosocial behaviour. The Post hoc comparisons using the Tukey HSD test indicated a clear dose-response effect. Concerning all symptom measures, the mean scores for exposure to parental verbal aggression (SDQ total: M = 12.19, SD = 5.15;

  being violent towards the child. Gelles & Cornell[53] have suggested that the lack of scientific attention given to parental violence towards adolescents may have reflected the attitude that, because of their own difficult behaviour, adolescents share some complicity when receiving such mistreatment. Based on knowledge of the accumulation of adversity in family contexts it is also difficult to conclude whether the mental health problems found in this study are primary outcomes of physical abuse or of other problems within the family. Further research is needed to illuminate up the complex transactions between different family related factors and their influence on parental violence. This work was started by Jaffee [54], who found a genetically mediated relation between children's antisocial behaviour and their exposure to (nonnormative) physical maltreatment.Additionally, the limitations of the widely used SDQ as a measure should be taken into account. The reliability of the original dimensions of SDQ was low in the data. The earlier applications of this measure were made in Sweden [55] and other Nordic countries[56]. Based on Nordic research the SDQ has shown to be an acceptable screening instrument for large community samples but it needs further evaluation. Its psychometric quality in particular should be improved[57, 56, 58, 55].ConclusionsKnowledge of the prevalence of parental violence and the assessment of its effects should have implications for child protection, general health care, prevention and intervention. The findings of this article point to the high prevalence of parental violence and adverse mental health among the affected Finnish and Danish adolescents. It is noteworthy that among Finish adolescents experiencing severe parental violence the cut-off score for the borderline/abnormal range of SDQ total difficulties score was exceeded. Consequently, the recommendations based on our study could be the following; whenever the knowledge about parental use of severe violence such as hitting or kicking of a child/adolescent is achieved by social worker, nurse, psychologists etc. some psychosocial interventions should be offered. About 40% of Finish and Danish adolescents experience verbal and physical violence at the hand of their parents. The knowledge about its harmful effects could be delivered already in child health clinics and later in parental meetings at the school. The threshold for taking these issues under discussion should be low among professionals working with children and adolescents.Even though the laws have been set in motion to prevent the use of parental physical violence the challenges remain in several domains: 1) controlling people's compliance with these laws, 2) creating conformity in the consequences when these laws are broken, 3) raising the general awareness of the detrimental effects of physical but also of the verbal aggression and subjugation of a child, 4) preventing harmful disciplinary practices and finally, implementing effective mental health interventions for children who have developed symptoms due to these harmful experiences. 52. Turner HA, Finkelhor D, Ormrod R (2006) The effect of lifetime victimization on the mental health of children and adolescents. Social Science & Medicine, 62: 13 53. Gelles RJ, Cornell CP (1990) Intimate violence in families, vol. 2. CA: Sage, Beverly Hills 54. Jaffee SR., Caspi A, Moffitt TE, Polo-Tomas M, Price TS, Taylor A (2004) The limits of child effects: Evidence for genetically mediated child effects on corporal punishment but not on physical maltreatment. Developmental Psychology, 40: 1047 55. Svedin C, Priebe G (2008) The strengths and difficulties questionnaire as a screening instrument in a community sample of high school seniors in Sweden. Nordic Journal of Psychiatry, 63: 225-232 56. Obel C, Heiervang E, Rodriguez A, Heyerdahl S, Smedje H, Sourander A. et al (2004) The strengths and difficulties questionnaire in the Nordic countries. European Child & Adolescent Psychiatry, 13: 32-39 57. Goodman R, Meltzer H, Bailey V (1998) The strengths and difficulties questionnaire: A pilot study on the validity of the self-report version. European Child & Adolescent Psychiatry, 7: 125-130 58. Rønning J, Handegaard B, Sourander A, & Mørch W-T (2004) The strengths and difficulties self-report questionnaire as a screening instrument in Norwegian community samples. European Child & Adolescent Psychiatry, 13, 73-82 Tables

Table 1

 1 Frequencies and percentages of exposure to parental violence according to nationality and genderMain and interaction effects of parental violence, gender and country on mental health outcomes * p< .05 **p<.01 ***p<.001 (the significance levels with Bonferroni correction

	Finland			Denmark
	Total	Girls	Boys	Total	Girls	Boys


Aude Lacourt
email: aude.lacourt@isped.u-bordeaux2.fr

Patrick Rolland

Céline Gramond

Philippe Astoul

Soizick Chamming's

Stéphane Ducamp

Catherine Frenay

Françoise Galateau-Sallé

Anabelle Gilg

Soit Ilg

Ellen Imbernon

Soizick Chamming

Stephane Ducamp

Françoise Galateau-Salle

Nolwenn Le Stang

Jean Claude Pairon

Marcel Goldberg

Yuriko Iwatsubo

Louis-Rachid Salmi

Patrick Brochard

Attributable risk in men in two French case-control studies on mesothelioma and asbestos

Keywords: Asbestos, attributable risk, case-control study, men, mesothelioma, occupational exposure

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers,

INTRODUCTION

Pleural mesothelioma is a primary tumor of the pleura mainly due to asbestos exposure [START_REF] Wagner | Diffuse pleural mesothelioma and asbestos exposure in the North Western Cape Province[END_REF]. Exposureresponse relationships mainly based on cohort studies demonstrate the general pattern of the relationship but with large differences from one study to another due to the type of industry or type of fibers [START_REF] Hodgson | The quantitative risks of mesothelioma and lung cancer in relation to asbestos exposure[END_REF]. As pleural mesothelioma is rare, case-control studies are a well-suited method for analyzing dose-response relationships between occupational asbestos exposure and pleural mesothelioma.

Case-control studies have been performed, but all encountered methodological difficulties in recruiting cases or controls and in assessing occupational asbestos exposure. To study the relationship between pleural mesothelioma and occupational asbestos exposure in France, two large case-control studies have been conducted. A first hospital-based case-control study was conducted between 1987-1993 (hereafter called "study A") [START_REF] Iwatsubo | Pleural mesothelioma: dose-response relation at low levels of asbestos exposure in a French population-based case-control study[END_REF], and a second population-based study nested in the French National Mesothelioma Surveillance Program (PNSM) was performed between 1998-2002 (hereafter called "study B) [START_REF] Goldberg | The French National Mesothelioma Surveillance Program[END_REF]. A substantial difference in the attributable risk in the male population (AR p ) was observed: 44.5% (95% CI: [32.6-56.4]) in study A and 83.2% (95% CI: [76. 8-89.6]) in study B. The results were not comparable due to different methods of asbestos exposure assessment (assessment of individual job periods of subjects in study A and overall assessment of work history of subjects in study B). Furthermore, there was a slight difference in the proportion of exposed subjects (61.2% in study A and 67.5% in study B), maybe due to different population recruitment methods (hospital population in study A and general population in study B). Therefore, the large difference in the AR p might be explained by inadequate exposure assessment in one or other study, that could bias AR p by inadequate exposure classification [START_REF] Hsieh | The effect of non-differential exposure misclassification on estimates of the attributable and prevented fraction[END_REF][START_REF] Wacholder | Attributable risk: advantages of a broad definition of exposure[END_REF], or by the population recruitment. This is especially so for recruitment of controls that could bias prevalence of exposure among controls and thus bias the odds ratio and AR p .

To promote better occupational health and safety strategies, it is important to correctly estimate the AR p related to occupational asbestos exposure. For this reason, it is essential to understand the major difference in AR p between these studies. We made the hypothesis that one way to improve comparisons and to estimate the real difference in the AR p between these two studies was to standardize the exposure assessment across both studies.

The main objectives of this work were to estimate the dose-response relationship between pleural mesothelioma and occupational asbestos exposure among men, to calculate the AR p in the two case-control studies according to a common standardized exposure assessment through the use of a job exposure matrix, and to appreciate the role of subjects" selection.

POPULATION AND METHODS

Studies A [START_REF] Iwatsubo | Pleural mesothelioma: dose-response relation at low levels of asbestos exposure in a French population-based case-control study[END_REF] and B [START_REF] Goldberg | The French National Mesothelioma Surveillance Program[END_REF] were multicenter case-control studies. Study A was a hospital-based casecontrol study where cases were selected in five French administrative regions. Case collection was passive through clinicians from public hospitals (departments of respiratory disease, chest surgery and oncology) and private clinics. Study B was a population-based case-control study nested within the French National Mesothelioma Surveillance Program (PNSM). Since 1998, the PNSM has recorded all incident pleural mesotheliomas in 21 French districts that cover a quarter of the French population. These districts were chosen as being representative of France with regard to the main demographic, occupational and economic characteristics. Recording of cases in study B was active and aimed at exhaustivity. Cases in both studies were alive at the time of interview. Study A and B included 460 and 371 male cases, respectively. Two controls matched for sex, age (+/-5 years) and district of residence were randomly selected from the hospital population in study A and from the general population in study B using the voting rolls of the districts. In both studies, a standardized questionnaire was administered to each subject by a trained interviewer. Experts in industrial hygiene, five in study A and two in study B, assessed occupational asbestos exposure by consensus [START_REF] Fritschi | Validation of expert assessment of occupational exposures[END_REF]. Expertise methods were different: in study A, experts assessed the subjects" jobs sorted by industry, blinded to the full carrier of subjects [START_REF] Iwatsubo | Pleural mesothelioma: dose-response relation at low levels of asbestos exposure in a French population-based case-control study[END_REF], and in study B, experts assessed consecutively all the jobs of each subject [START_REF] Goldberg | The French National Mesothelioma Surveillance Program[END_REF]. The socioeconomic category of the subject was determined by the last occupation and work sector before the interview and was coded using the major groups of the International Standard Classification of Occupations (ISCO edition 1968) [START_REF]International Standard Classification of Occupations[END_REF] and the International Standard Industrial Classification of All Economics Activities (ISIC revision 2) [START_REF]International Standard Industrial Classification of all Economic Activities (Revision 2)[END_REF], respectively. In addition, distribution of the economic status of the general population was obtained from the French national 1999 census in men older than 50 years, restricted to the same districts of studies A and B.

To standardize occupational exposure assessment to asbestos across both studies, we used a job exposure matrix (JEM) that includes a temporal weighting when exposure levels changed within the time-period of interest [START_REF] Goldberg | Job exposure matrices in industry[END_REF]. The JEM used was developed by French industrial hygienists and previously used in several studies [START_REF] Imbernon | Utilisation d'une matrice emplois-exposition pour l'identification des expositions professionnelles à l'amiante[END_REF][START_REF] Orlowski | Retrospective assessment of asbestos exposure--II. At the job level: complementarity of job-specific questionnaire and job exposure matrices[END_REF]. It was designed to estimate asbestos exposure in relevant jobs and includes three exposure parameters for each job: probability, frequency and intensity of exposure. Estimates of these parameters are semi-quantitative and based on the judgment of industrial hygienists. The probability of exposure was categorized as "not exposed", "possible" (where exposure might have occurred) and "definite" (undoubted exposure or highly likely exposure). The frequency of exposure was defined as "sporadic" (less than 5 percent of work time), "irregular" (5-50 percent of work time) and "continuous" (more than 50 percent of work time). The intensity of exposure (expressed in fibers per milliliter (f/ml)) was categorized as "low" (less than 0.1 f/ml), "medium" (01,-1 f/ml), "high" (1-10 f/ml) and "very high" (>10 f/ml). Industries were classified according to the ISIC revision 2 [START_REF]International Standard Industrial Classification of all Economic Activities (Revision 2)[END_REF] and occupations according to the ISCO edition 1968 [START_REF]International Standard Classification of Occupations[END_REF]. In all, the JEM included 10,692 unique exposed industry/occupation combinations. It can be consulted through Internet on the Evalutil web site (Essat 2008) [START_REF] Audignon-Durand | Evalutil : une base de données d'évaluation des expositions professionnelles aux fibres[END_REF].

We linked all subjects" job histories to the JEM. A subject was defined as "occupationally exposed to asbestos" if the probability of exposure for at least one job was different from zero. We also characterized each subject"s history by: the highest probability of any job period during lifetime work history; total duration of exposure (years); time since first exposure (latency period); age at first exposure (year); Cumulative Exposure Index (CEI) expressed in "fibers/ml-years" (f/ml-years) and defined as the sum of the products of probability, frequency, intensity, and duration for each job period. To calculate the CEI, we attributed weighting factors used in a previous study [START_REF] Iwatsubo | Pleural mesothelioma: dose-response relation at low levels of asbestos exposure in a French population-based case-control study[END_REF] to each exposure category of these three parameters. Then, we categorized CEI as continuous variable in four CEI levels.

We estimated odds ratios and 95% confidence intervals (95% CI) by using conditional logistic regression matched for age and French districts. Because the latency period of this disease is long, we did not take into account asbestos exposure during the 20 years before the diagnosis of mesothelioma. Therefore, we calculated AR p according to all CEI levels [START_REF] Bruzzi | Estimating the population attributable risk for multiple risk factors using case-control data[END_REF] and confidence intervals were estimated using the bootstrap method with 500 replicates [START_REF] Cox | Model-based estimation of the attributable risk in case-control and cohort studies[END_REF].

We further compared for socioeconomic status controls from both studies A and B with the general population and cases with cases registered by the PNSM until December 2009. We considered that cases registered by the PNSM were representative of all French cases of mesothelioma, because the PNSM records all incident pleural tumors in 21 French districts that are representative of France regarding the main demographic, occupational and economic characteristics [START_REF] Goldberg | The French National Mesothelioma Surveillance Program[END_REF]. Then, we estimated AR p again according to all CEI levels, after adjusting the two samples by weighting subjects according to distribution of socio-economic status in the general population for controls and according to distribution of socio-economic status of cases registered by the PNSM. We conducted three different simulations to re-estimate the AR p in the two studies. In simulation 1, we weighted only controls and a weight of 1 was given to cases. In simulation 2, we weighted only cases and a weight of 1 was given to controls. In simulation 3, we weighted cases and controls together. SAS 9.1 ® software was used for the analysis.

RESULTS

Study A included 460 male cases and 421 controls and study B included 371 male cases and 732 controls. In each study, cases and controls were age-matched but study A subjects were significantly younger than study B subjects (Table I). Among cases, subjects were on average 63.6 years old in study A and 68.4 years old in study B.

According to the last occupation held, cases and controls differed significantly in both studies with more blue-collars among cases and more professional, technical and related workers among controls.

Overall, cases did not differ between studies. On the contrary, controls had more blue-collars in study A, and more professional, technical and related workers, and administrative and managerial workers, in study B. In addition, with regard to the last work sector, cases and controls differed significantly within each study with more cases in the manufacturing industry. Cases between both studies did not differ but there were more controls in the manufacturing industry in study A and more controls in agriculture, hunting, forestry and fishing in study B.

According to the different exposure parameters, exposure probability did not differ among cases (p=0.47) but did differ among controls between both studies (p=0.01). There were more unexposed controls in study B versus more exposed controls in study A (data not shown). Considering cumulative exposure (Table II), cases between both studies differed significantly (p<0.01), as there were more cases in the highest category of exposure in study B. The distribution was also different for controls within both studies (p<0.01). There were more unexposed controls in study B than in study A.

The exposure-response relationship between mesothelioma and asbestos was confirmed in both studies but was more pronounced in study B (

DISCUSSION

The initial differences between studies still persisted after standardization of exposure assessment.

AR p for occupational asbestos exposure was 36.8% (95% CI: [18.8 -50.8]) in study A and 72.3% (95% CI: [63.8 -78.8]) in study B by using the JEM, so expert methods used primarily to assess occupational exposure to asbestos may not explain the variations in the AR p .

We chose to estimate the exposure-response relationship and AR p according to CEI levels of occupational asbestos exposure. Although this indicator is limited by retrospective exposure assessment [START_REF] Siemiatycki | Invited commentary: Is it possible to investigate the quantitative relation between asbestos and mesothelioma in a community-based study?[END_REF], it was impossible to obtain retrospective quantitative asbestos exposure assessment in the context of case-control studies. We thought that using the CEI to assess semi-quantitative asbestos remained an acceptable approach so we cannot explain the difference observed for AR p values.

Several biases in both studies A and B may explain this difference. The methods of asbestos exposure assessment cannot explain the differences in the findings either, since the assessment was performed with the same JEM in both studies. JEM inevitably produce misclassifications but nondifferential, so odds ratios and AR p were under-estimated similarly in both studies [START_REF] Bouyer | Performance of odds ratios obtained with a job-exposure matrix and individual exposure assessment with special reference to misclassification errors[END_REF]. However, since exposure assessment by a JEM relies on the quality of job histories, we cannot rule out differences with regard to the validity and reliability of the job histories obtained from interviews, nor to their coding quality. Some studies found good validity and reliability of job histories obtained from interviews [START_REF] Baumgarten | Validity of work histories obtained by interview for epidemiologic purposes[END_REF][START_REF] Bond | Validation of work histories obtained from interviews[END_REF][START_REF] Bourbonnais | Validity of self reported work history[END_REF][START_REF] Eskenazi | Validation of a self-administered questionnaire for assessing occupational and environmental exposures of pregnant women[END_REF][START_REF] Pershagen | A validation of questionnaire information on occupational exposure and smoking[END_REF][START_REF] Stewart | The validity of questionnaire-reported work history in live respondents[END_REF], and there is no evidence that the recall of occupations is influenced by disease status [START_REF] Baumgarten | Validity of work histories obtained by interview for epidemiologic purposes[END_REF][START_REF] Bond | Validation of work histories obtained from interviews[END_REF]. A

French study evaluated the quality of coding of job episodes collected by self-questionnaires among retired French men and found that the coding was satisfactory [START_REF] Pilorget | Evaluation of the quality of coding of job episodes collected by self questionnaires among French retired men for use in a job-exposure matrix[END_REF]. Despite potential differences in data collection and coding in both studies, such differences cannot explain such large differences in estimations of AR p.

The most likely explanation is a selection bias due to recruitment of subjects and specially controls.

According to the socioeconomic status, the distribution of the general population and the distribution of the controls in both studies are different. In study B, a complementary analysis was conducted in order to compare participant controls to eligible controls who refused to participate (data not shown). There was a significant difference between participating controls and refusing controls regarding the distribution of their socioeconomic status (p<0.01). There were more blue-collar workers among refusing controls and more white-collar workers among participating controls. Therefore, participating controls may have been less exposed than the general population so the exposure-response relation and AR p may be over-estimated. Conversely, in study A, the controls seemed to be more exposed to asbestos than the general population. Indeed, study A controls were hospital controls, and even if subjects with asbestos-related disease were excluded, there were more blue-collars hospitalized, so the proportion of exposed subjects may be higher in hospital than in the general population. Thus, the exposure-response relation and AR p may be under-estimated.

The cases in both studies seem to be equivalent with regard to probability of occupational asbestos exposure, but study A cases appear less exposed than study B cases considering cumulative occupational asbestos exposure. This can be explained by the geographical recruitment of the subjects. Study B cases were recruited in 21 French districts representative of France regarding the main demographic, occupational and economic characteristics, while study A cases were recruited in five regions with a large proportion in the region of Ile de France, which is a less industrialized region that the rest of France. Therefore, these differences in occupational exposure to asbestos may explain the significant difference observed for AR p values.

The three simulations confirmed this hypothesis. Simulation 1 suggests that, controls seemed to play an important role in the difference observed in the AR p due to the large under-estimation of AR p in study A and the slight over-estimation in study B. In simulation 2, case recruitment seemed to play a smaller role in the difference observed. In simulation 3, the estimated AR p were close to those estimated in simulation 1, thus reflecting more the weight of the controls than the cases. These findings illustrate the potential variations in AR p due to the selection of controls, and may account for the results observed in both studies [START_REF] Ben-Shlomo | Stroke risk from alcohol consumption using different contro l groups[END_REF].

Whatever the simulation, the differences in the AR p initially observed still persisted. Thus, there may have been confounding factors in both studies. Until now, asbestos is the only established risk factor for mesothelioma (except for erionite fibers but these are not present in France). However, in the two studies only occupational asbestos exposure were considered, and domestic or environmental asbestos exposure could have been a possible confounding factor [START_REF] Goldberg | The health impact of nonoccupational exposure to asbestos: what do we know?[END_REF].

To date, several studies have analyzed the exposure-response relation between occupational asbestos exposure and mesothelioma [START_REF] Agudo | Occupation and risk of malignant pleural mesothelioma: A case-control study in Spain[END_REF][START_REF] Howel | Routes of asbestos exposure and the development of mesothelioma in an English region[END_REF][START_REF] Rake | Occupational, domestic and environmental mesothelioma risks in the British population: a case-control study[END_REF][START_REF] Rees | Asbestos exposure and mesothelioma in South Africa[END_REF][START_REF] Rodelsperger | Asbestos and man-made vitreous fibers as risk factors for diffuse malignant mesothelioma: results from a German hospital-based case-control study[END_REF][START_REF] Spirtas | Malignant mesothelioma: attributable risk of asbestos exposure[END_REF], yet few have estimated the AR p . All these studies assessed exposure by expertise, except the work by Spirtas et al. [START_REF] Spirtas | Malignant mesothelioma: attributable risk of asbestos exposure[END_REF] (JEM), and all showed an exposure-response relation, but the magnitude of the relation differed. The main methodological difference between these studies was the mode of recruitment (hospital vs general population).

Table IV summarizes the data of these studies. To be able to compare studies according to AR p and mode of recruitment, we needed to estimate AR p a posteriori from data given in the articles. AR p closer to that in study B were those estimated in studies with the same design as study B, i.e.

population-based case-control studies [START_REF] Howel | Routes of asbestos exposure and the development of mesothelioma in an English region[END_REF][START_REF] Rake | Occupational, domestic and environmental mesothelioma risks in the British population: a case-control study[END_REF][START_REF] Spirtas | Malignant mesothelioma: attributable risk of asbestos exposure[END_REF]. The AR p values were 88% for overall exposure, 69.8%, 85% and 83.2% for occupational exposure in Spirtas et al. [START_REF] Spirtas | Malignant mesothelioma: attributable risk of asbestos exposure[END_REF], Howel et al [START_REF] Howel | Routes of asbestos exposure and the development of mesothelioma in an English region[END_REF], Rake et al.

[29] and our study B, respectively. The Rödelsperger et al. study [START_REF] Rodelsperger | Asbestos and man-made vitreous fibers as risk factors for diffuse malignant mesothelioma: results from a German hospital-based case-control study[END_REF] gave the same results (84.6%) even if the cases were recruited from hospital records. This is in agreement with our findings in simulation 2, showing that hospital case recruitment had a lower impact on AR p estimation. In Agudo et al. [START_REF] Agudo | Occupation and risk of malignant pleural mesothelioma: A case-control study in Spain[END_REF], where controls but not cases came from hospitals, they observed a lower AR p of 62%.

Rees et al [START_REF] Rees | Asbestos exposure and mesothelioma in South Africa[END_REF] used the same design as in our study A (hospital-based case-control study), but they found a higher AR p close to those estimated in population-based case-control studies. This may be due to the geographical recruitment. Indeed, South Africa had mined, transported and used all varieties of asbestos fibers in large amounts. In that study, the proportion of exposed cases and controls was 96% and 60%, respectively. The proportion of exposed controls was close to that in study A and study B (68% and 57%, respectively) but the proportion of exposed cases was higher than in either study A or B (80% and 87%, respectively). This may account for the value of the estimated AR p found in the study by Rees et al [START_REF] Rees | Asbestos exposure and mesothelioma in South Africa[END_REF] despite its design.

Finally, although the true AR p cannot be precisely determinate from our data, results of different simulations give strong arguments to estimate that the AR p should be closer to results obtained in study B than in study A. This is in agreement with majority of recent studies.

CONCLUSION

This work confirms the hypothesis that the difference in ARp in the two French mesothelioma casecontrol studies is probably due to the recruitment of the subjects, and particularly to that of the controls. Despite exposure assessment standardization using a JEM, the initial differences remained, so we were able to focus on the selection of the controls. However, selection bias cannot explain the entire difference and confounding by other type of asbestos exposure or other etiologic factor should be envisaged to explain the entire difference. Owing to the classic biases associated with hospital control selection and according the literature, the results in study B seem to be more reliable than those in study A. The dose-response relation in study A seems to be under-estimated due to overexposure of the controls. This work throws new light on the relation between mesothelioma and asbestos exposure, and confirms the importance of subject selection and recruitment in case control studies.

Table III

 III There was a large difference between the studies with regard to the magnitude of the odds ratios for all exposure parameters. Odds ratios for possible exposure were 1.5 (95% CI: [1.1 -2.1]) in study A and 3.4 (95% CI: [2.4 -4.9]) in study B. For definite exposure, it was 2.9 (95% CI: [2.0 -4.2]) in study A and 10.6 (95% CI: [7.2 -15.6]) in study B. Odds ratios increased in both studies with The variation in AR p initially observed between both studies remained. It was 36.8% (95% CI: [18.8 -50.8]) in study A and 72.3% (95% CI: [63.8 -78.8]) in study B, including all CEI categories (TableIII).The two sets of controls differed from the general population with regard socioeconomic status (p< 0.01). In study A, there were 45.8% blue-collar workers compared to 35.0% in the general population, and 15.2% controls in the "professional, technical and related workers" category compared to 29.5%

	cumulative exposure, but more than two-fold in study B (from 1.8 (95% CI: [1.2 -2.9]) for subjects with
	less than 0.1 "f/ml-years" to 19.6 (95% CI: [11.3 -34.0]) for those with more than 10 "f/ml-years") than
	in study A (from 1.3 (95% CI: [0.9 -2.0]) for the lowest category to 5.3 (95% CI: [2.8 -10.1]) for the
	highest).

). Mesothelioma risk increased with the probability of exposure, duration of job exposure and cumulative exposure in both studies, and with time since first exposure only in study B. A trend was observed only in study B with age at first exposure.

in the general population. In study B, there were 32.1% blue-collar workers in the controls versus 41.1% in the general population. Cases from both studies and cases registered by the PNSM did not differ according to socioeconomic status (p>0.05).

Re-estimated AR p values were 50.1% (95% CI: [36.3 -60.9]) in study A and 70.1% [60.

6-77.3]

in study B according to simulation 1 (weighted controls). When only cases were weighted (simulation 2), AR p values were 43.0% (95% CI: [24.8 -56.7]) in study A and 70.7% [61.1 -77.9] in study B. When cases and controls together were weighted (simulation 3), values were 52.4% (95% CI: [37.9 -63.5]) in study A and 70.2% [60.7 -77.4] in study B.

Table II : Cumulative asbestos exposure of subjects in two French case-control studies of occupational asbestos exposure and pleural mesothelioma.

 II p. p-value comparing cases and controls ; † p ca . p-value comparing cases of both studies; ‡ p co . p-value comparing controls of both studies § SD: Standard deviation; || Min-Max. minimum and maximum, ¶ ISCO ed 1968: international standard classification of occupations ** ISIC rev 2. International standard Industrial Classification of all Economic Activities f/ml-years. Fibers per milliliter years, † p. p-value.

	Results					Cases						Controls
	heading, second paragraph Results heading, third	Cumulative exposure (f/ml-years * ) (JEM)	Study A N=460	Study B N=371	p	†	Study A N=421	Study B N=732	p
	paragraph			n	%	n	%			n	%	n	%
		Not exposed		94	20.4	50	13.5	< 0.01	135	32.1	326	44.5	< 0.01
		> 0 -0.1		82	17.8	42	11.3			93	22.1	154	21.1
		> 0.1 -1		129	28.1	93	25.1			119	28.2	132	18.0
		> 1 -10		100	21.7	116	31.3			58	13.8	94	12.8
		> 10		55	12.0	70	18.8			16	3.8	26	3.6
		4 Sales workers	37	8.0	36	8.6		23	6.2	73 10.0
		5 Service workers	26	5.7	40	9.5		20	5.4	59	8.1
		6 Agricultural, animal
		husbandry and forestry	6	1.3	15	3.6		11	2.9	69	9.4
		workers; fishermen; hunters
		7/8/9 Production and related
		workers, transport equipment operators and	254 55.2	193 45.8		221 59.6	234 32.0
		labourers
		Last held industry
		(ISIC rev. 2**major group)				<0.01					<0.01	0.14 <0.01
		1 Agriculture, hunting, forestry and fishing	6	1.3	11	2.6		13	3.5	69	9.4
		2 Mining and quarrying	7	1.5	6	1.4		1	0.3	2	0.3
		3 Manufacturing	203 44.1	135 32.1		164 44.2	214 29.2
		4 Electricity, gas and water	12	2.6	3	0.7		11	3.0	14	1.9
		5 Construction	66 14.4	38	9.0		51 13.7	62	8.5
		6 Wholesale and retail trade and restaurants and hotels	37	8.1	59 14.0		18	4.8	75 10.2
		7 Transport, storage and communication	38	8.3	48 11.4		37 10.0	64	8.8
		8 Financing, insurance, real estate and business services	25	5.4	31	7.4		20	5.4	57	7.8
		9 Community, social and personal services	64 13.9	89 21.2		56 15.1	175 23.9
		0 Activities not adequately defined	2	0.4	1	0.2		-	-	-	-

* *

Table III : Dose response relation between pleural mesothelioma and occupational asbestos exposure parameters in men, 20 years of laten cy period, two French case-control studies of occupational asbestos exposure and pleural mesothelioma.

 III

						Study A					Study B
	Occupational asbestos exposure parameters (JEM)		Cases (460)	Controls (421)	OR*	95% CI †		Cases (371)	Controls (732)	OR	95% CI
				n	%	n	%			n	%	n	%
	Not exposed		94	20.4	135	32.1	1.0		50	13.5	326	44.5	1.0
	Highest probability of exposure	Possible	205	44.6	202	48.0	1.5	1.0-2.1	146	39.3	289	39.5	3.5	2.4-5.0
			Definite	161	35.0	84	19.9	2.8	1.9-4.2	175	47.2	117	16.0	11.2	7.5-16.7
	Duration of exposed job (years)	1 -10	114	24.8	127	30.1	1.3	0.9-1.9	62	16.7	141	19.3	3.0	1.9-4.6
			>10 -20	100	21.7	64	15.2	2.2	1.4-3.4	66	17.8	96	13.1	4.6	3.0-7.3
			> 20	152	33.1	95	22.6	2.4	1.6-3.6	193	52.0	169	23.1	8.6	5.8-12.6
	Time since first exposure (years)	> 19 -40	106	23.1	92	21.8	1.6	1.0-2.4	55	14.8	111	15.2	3.0	1.9-5.0
			> 40 -50	133	28.9	91	21.6	2.0	1.3-3.0	100	27.0	117	16.0	6.3	4.1-9.6
			> 50	127	27.6	103	24.5	2.0	1.3-3.1	166	44.7	178	24.3	6.8	4.6-10.1
	Age at first exposure (years)	≤ 15	97	21.1	80	19.0	1.8	1.2-2.7	103	27.7	96	13.1	7.6	5.3-11.6
			> 15 -20	161	35.0	115	27.3	2.0	1.3-2.8	109	29.4	131	17.9	5.8	3.9-8.8
			> 20	108	23.5	91	21.6	1.7	1.2-2.6	109	29.4	179	24.5	4.3	2.9-6.3
	Cumulative exposure (f/ml -years ‡ )	> 0 -0.1	82	17.8	93	22.1	1.3	0.9-2.0	42	11.3	154	21.1	1.9	1.2-3.0
			> 0.1 -1	129	28.1	119	28.2	1.5	1.0-2.2	93	25.1	132	18.0	5.0	3.4-7.6
			> 1 -10	100	21.7	58	13.8	2.5	1.6-3.8	116	31.3	94	12.8	9.4	6.1-14.3
			> 10	55	12.0	16	3.8	4.9	2.6-9.3	70	18.8	26	3.6	21.4	12.1-37.7
	AR p	§ [

95% CI] 36.8[18.8-50.8] 72.3 [63.8-78.8]

  OR. Odds ratio matched for age and district, † CI. Confidence interval, ‡ f/ml-years. Fibers per milliliter years, § AR p . Attributable risk in the population

	Results
	heading, fourth
	paragraph

*

ACKNOWLEDGEMENTS

The authors of this study thank the patients and their next of kin, the physicians and pathologists who made these two studies possible. They also thank R.Cooke for his careful reading of the manuscript.


Denis De Carvalho

Braga • Luis

Fernando Mello

Carmen Rocşoreanu
email: carmenr@central.ucv.ro

Mihaela Sterpu
email: mihaelas@central.ucv.ro

Denis De Carvalho Braga

Luis Fernando Mello
email: lfmelo@unifei.edu.br

Control of planar Bautin bifurcation

Keywords: Hopf bifurcation, Bautin bifurcation, stability, limit cycle, control system PACS 02.30.Oz, 05.45.-a

On a versal deformation of the Bautin bifurcation it is possible to find dynamical systems that undergo Hopf or non-hyperbolic limit cycle bifurcations. Our paper concerns a nonlinear control system in the plane whose nominal vector field has a pair of purely imaginary eigenvalues. We find conditions to control the Hopf and Bautin bifurcation using the symmetric multilinear vector functions that appear in the Taylor expansion of the vector field around the equilibrium. The control law designed by us depends on two bifurcation parameters and four control parameters, which establish the stability of the equilibrium point and the orientation and stability of the limit cycles. Two examples are given.

Introduction

During the recent years many publications in the field of dynamical systems focus on bifurcations control. Different control systems were designed in order to create different types of bifurcation and to manipulate the bifurcation characteristics such as the stability and orientation of limit cycles or the stability of equilibria [START_REF] Carrillo | Control of the planar Takens-Bogdanov bifurcation with applications[END_REF].

Many authors developed control laws for linear or nonlinear systems with one or more uncontrollable modes.

Although there exist many papers in the literature concerning the control of the codimension one Hopf bifurcation [START_REF] Verduzco | Hopf bifurcation control for affine systems[END_REF], [START_REF] Hamzi | Analysis and control of Hopf bifurcation[END_REF], [START_REF] Wen | Control algorithm for creation of Hopf bifurcations in continuous-time systems of arbitrary dimension[END_REF], there are few results concerning the control of the generalized Hopf bifurcation of codimension two (Bautin) [START_REF] Braga | Contollable Hopf bifurcations of codimension one and two in linear control systems[END_REF].

In this paper we study a nonlinear planar control system with two uncontrollable modes on the imaginary axis. We design a control law such that the resulting control system undergoes controllable Hopf or Bautin bifurcation. Our analysis is based on the normal form theory [START_REF] Yu | Elements of applied bifurcation theory[END_REF], [START_REF] Guckenheimer | Nonlinear oscillations, dynamical systems and bifurcations of vector fields[END_REF]. We prove that our control law determines the orientation and stability of periodic solutions for both Hopf and Bautin bifurcations.

Consider the planar nonlinear control system ẋ = F (x) + G (x) u, [START_REF] Braga | Contollable Hopf bifurcations of codimension one and two in linear control systems[END_REF] where x = (x 1 , x 2 ) T ∈ R 2 is the state vector, F, G : R 2 → R 2 are smooth vector fields, u = u (x; μ, β) , μ ∈ R 2 , β ∈ R 3 , and G (x) u is the control input. Assume that F (0) = 0, G (0) = 0 and J = dF (0) has purely imaginary eigenvalues λ 1,2 = ±ω 0 i, ω 0 ≥ 0, and

M = dG (0) = ∂G1 ∂x1 ∂G1 ∂x2 ∂G2 ∂x1 ∂G2 ∂x2 |x=0 = (m ij ) i,j=1,2 .
In the following μ = (μ 1 , μ 2 ) ∈ R 2 is the Bautin bifurcation parameter, β = (β 1 , β 2 , β 3 ) ∈ R 3 and tr (M ) are the control parameters. Here β 1 determines the stability of the equilibrium point, β 2 determines the orientation and the stability of the limit cycle emerging at a Hopf bifurcation, whilst β 3 and tr (M ) establish the orientation and the stability of the limit cycles emerging at the Bautin bifurcation.

Consider the following expression for the scalar function u:

u (x, μ, β) = β 1 μ 1 +(β 2 + μ 2 ) x 2 1 + x 2 2 +β 3 x 2 1 + x 2 2 2 .
(

) 2
The aim of the paper is to find values of the control parameters β 1 , β 2 , β 3 and tr (M ) such that system (1) undergoes controllable Hopf or Bautin bifurcations.

In order to do this, two well known theorems from the bifurcation theory [START_REF] Yu | Elements of applied bifurcation theory[END_REF] are used.

Theorem (Hopf ) [START_REF] Yu | Elements of applied bifurcation theory[END_REF]. Suppose the planar system

ẋ = f (x, α) , x ∈ R 2 , α ∈ R, (3)
with smooth f, has for all sufficiently small |α| the equilibrium x = 0 with eigenvalues

λ 1,2 (α) = μ (α) ± iω (α) , ( 4
)
where μ (0) = 0, ω (0) = ω 0 > 0. Let the following conditions be satisfied: (H.1) l 1 (0) = 0, where l 1 (α) is the first Lyapunov coefficient;

(H.2) μ (0) = 0. Then, there are invertible coordinate and parameter changes and a time reparametrization transforming (3) into the complex form

ż = (b 1 + i) z + sz |z| 2 + O |z| 4 , ( 5
)
where s = sign (l 1 (0)) = ±1.

Theorem (Bautin) [START_REF] Yu | Elements of applied bifurcation theory[END_REF]. Suppose the planar system

ẋ = f (x, α) , x ∈ R 2 , α ∈ R 2 , ( 6
)
with smooth f, has the equilibrium x = 0 with eigenvalues

λ 1,2 (α) = μ (α) ± iω (α) , (7)
for all α sufficiently small, where ω (0) = ω 0 > 0. For α = 0, let the Bautin bifurcation conditions hold:

μ (0) = 0, l 1 (0) = 0, (8)
where l 1 (α) is the first Lyapunov coefficient. Assume that the following general conditions are satisfied: (B.1) l 2 (0) = 0, where l 2 (α) is the second Lyapunov coefficient;

(B.2) the map α → (μ (α) , l 1 (α)) T is regular at α = 0.

Then, by the introduction of a complex variable, applying smooth invertible coordinate transformation that depend smoothly on the parameters, and performing smooth parameter and time changes, the system can be reduced to the complex form

ż = (b 1 + i) z + b 2 z |z| 2 + sz z 4 + O |z| 6 , ( 9
)
where s = sign (l 2 (0)) = ±1.

Control of the Hopf bifurcation

In this section we find sufficient conditions such that system (1) undergoes a Hopf bifurcation that can be controlled. To this aim we shall use a complex variable form of a two-dimensional system, following the lines in [START_REF] Yu | Elements of applied bifurcation theory[END_REF].

Expanding system (1) around x = 0 we get

ẋ = Ax + F (x) (10)
where

A = J + β 1 μ 1 M, F (x) = 1 2 B (x, x) + 1 3! C (x, x, x) + 1 4! D (x, x, x, x) + 1 5! E (x, x, x, x, x) + O ||x| | 6
and B, C, D, E are multilinear symmetric forms, with

B (x, x) = d 2 F (0) (x, x) + β 1 μ 1 d 2 G (0) (x, x) , C (x, x, x) = d 3 F (0) (x, x, x) + β 1 μ 1 d 3 G (0) (x, x, x) +6 (β 2 + μ 2 ) x 2 1 + x 2 2 M x, D (x, x, x, x) = d 4 F (0) (x, x, x, x) + β 1 μ 1 d 4 G (0) (x, x, x, x) +12 (β 2 + μ 2 ) x 2 1 + x 2 2 d 2 G (0) (x, x) , E (x, x, x, x, x) = d 5 F (0) (x, x, x, x, x) +β 1 μ 1 d 5 G (0) (x, x, x, x, x) +20 (β 2 + μ 2 ) x 2 1 + x 2 2 d 3 G (0) (x, x, x) +120β 3 x 2 1 + x 2 2 2 M x. ( 11
)
In the following we find sufficient conditions for the parameters β 1 , β 2 such that the conditions of the Hopf bifurcation theorem are fulfilled for system [START_REF] Braga | Contollable Hopf bifurcations of codimension one and two in linear control systems[END_REF].

Condition (H.1)

At μ 1 = 0, in [START_REF] Wen | Control algorithm for creation of Hopf bifurcations in continuous-time systems of arbitrary dimension[END_REF] we have

A = J, B (x, x) = d 2 F (0) (x, x) , (12) C (x, x, x) = d 3 F (0) (x, x, x) + 6 (β 2 + μ 2 ) H (x) ,
where

H (x) = x 2 1 + x 2 2 M x. ( 13
)
In the previous assumptions, the Jacobian matrix A of system (1) at x = 0 and μ 1 = 0, possesses a pair of purely imaginary eigenvalues λ 1,2 = ±iω 0 . Let q ∈ C 2 be an eigenvector of A associated with the eigenvalue λ 1 = iω 0 and p ∈ C 2 an eigenvector of A T associated with λ1 , such that p, q = 1. Using the complex variable z = p, x , we get x = zq + z q, and equation [START_REF] Wen | Control algorithm for creation of Hopf bifurcations in continuous-time systems of arbitrary dimension[END_REF] at μ 1 = 0 reads:

ż = λ 1 z + g (z, z) ,
where g (z, z) = p, F (zq + z q) . Function g can be written as a formal Taylor series in two complex variables as:

g (z, z) = k+l≥2 1 k!l! g kl z k zl ,
where

g kl = ∂ k+l ∂ k z∂ l z g (z, z) |z=0 = ∂ k+l ∂ k z∂ l z p, F (zq + z q) |z=0 .
The first Lyapunov coefficient at the bifurcation point can be expressed as [START_REF] Yu | Elements of applied bifurcation theory[END_REF], p. 99,

l 1 (0) = 1 2ω 2 0 Re (ig 20 g 11 + ω 0 g 21 ) , (14)
where [START_REF] Yu | Elements of applied bifurcation theory[END_REF], p. 93-94,

g 20 = p, B (q, q) , g 11 = p, B (q, q) , ( 15
)
g 21 = p, C (q, q, q) .
For system (1) we have

g 20 = p, d 2 F (0) (q, q) , g 11 = p, d 2 F (0) (q, q) , ( 16
)
g 21 = p, d 3 F (0) (q, q, q) + 6 (β 2 + μ 2 ) p, H (q, q, q) .
Taking into account the expression of the quadratic form H given by (13), we get

H (x, x, y) = 1 3 x 2 1 + x 2 2 M y + 2 3 (x 1 y 1 + x 2 y 2 ) M x, (17)
where

x = (x 1 , x 2 ) T , y = (y 1 , y 2 ) T ∈ C 2 .
Thus, at μ 1 = 0, we obtain:

l 1 (0, μ 2 ) = δ + 3 ω 0 (β 2 + μ 2 ) Re ( p, H (q, q, q) ) , (18) with δ = 1 2ω 2 0 Re i p, d 2 F (0) (q, q) p, d 2 F (0) (q, q) +ω 0 p, d 3 F (0) (q, q, q) (19)
depending only on F. Remark that δ is the first Lyapunov coefficient corresponding to the vector field F . For every μ 2 ∈ R, there exists a unique value

β 2crit (μ 2 ) = -δω 0 3Re ( p, H (q, q, q) ) -μ 2 (20) of β 2 such that l 1 (0, μ 2 ) = 0, provided that Re ( p, H (q, q, q) ) = 0. Considering β 2 = β 2crit (μ 2 ), condition (H.1) is satisfied.
Remark. If the matrix J has the particular form

J = 0 -ω 0 ω 0 0 we can choose the eigenvectors q = (1, -i) T , p = 1 2 , -i 2 T , thus H (q, q, q) = 4 3 M q.
and the first Lyapunov coefficient in (18) reads

l 1 (0, μ 2 ) = δ + 2 ω 0 (β 2 + μ 2 ) tr (M ) . ( 21
)
In this particular case we get

β 2crit (μ 2 ) = - δω 0 2tr (M ) -μ 2 , ( 22
)
provided that tr (M ) = 0.

Condition (H.2) Since for μ 1 = 0 we have Reλ 1,2 (μ 1 ) = 1 2 β 1 μ 1 tr (M ) , it follows that dReλ 1,2 dμ 1 | μ1=0 = 1 2 β 1 tr (M ) , ( 23
)
thus dReλ dμ1 | μ1=0 = 0 iff β 1 = 0, provided tr (M ) = 0. We have proved the following result.

Theorem 1. Consider the planar nonlinear control system

ẋ = F (x) + G (x) u (x, μ, β) , ( 24
)
where x ∈ R 2 , F, G : R 2 → R 2 are smooth vector fields,

F (0) = 0, G (0) = 0, J = dF (0) = 0 -ω 0 ω 0 0
, and tr (dG (0)) = 0. Then there exist β 1 , β 2 ∈ R such that with the control law G (x) u (x, μ, β) , where

u (x, μ, β) = β 1 μ 1 +(β 2 + μ 2 ) x 2 1 + x 2 2 +β 3 x 2 1 + x 2 2 2 , ( 25
)
system (24) undergoes a Hopf bifurcation at μ 1 = 0, provided that β 2 = β 2crit (μ 2 ) . In addition it is possible to control the stability and the direction of the limit cycles emerging near the origin by selecting positive or negative β 1 and

β 2 > β 2crit (μ 2 ) or β 2 < β 2crit (μ 2 ),
where β 2crit (μ 2 ) is given by ( 22).

Remark. According to (21) and ( 22), it follows that

l 1 (0, μ 2 ) > 0 ⇔ ⎧ ⎨ ⎩ β 2 > β 2crit , if tr (M ) > 0, or β 2 < β 2crit , if tr (M ) < 0, while l 1 (0, μ 2 ) < 0 ⇔ ⎧ ⎨ ⎩ β 2 < β 2crit , if tr (M ) > 0, or β 2 > β 2crit , if tr (M ) < 0.
Using Theorem 1, and selecting appropriate values for the control parameters β 1 , β 2 , all the possible Hopf bifurcation diagrams, considering μ 1 as the bifurcation parameter, can be obtained. See Figure 1. In this figure the cases (i) and (iii) correspond to l 1 < 0, thus to a supercritical Hopf bifurcation, while the cases (ii) and (iv) correspond to l 1 > 0, thus to a subcritical Hopf bifurcation.

Control of the Bautin bifurcation

In this section we find the critical value of the control parameter β 3 such that system (1) undergoes all kind of Bautin bifurcation.

Assume that the first Lyapunov coefficient is zero at μ 2 = 0, that is β 2 = β 2crit (0).

Condition (B.1)

In order to compute the second Lyapunov coefficient at (μ 1 , μ 2 ) = (0, 0) , we use the following formula [START_REF] Yu | Elements of applied bifurcation theory[END_REF], p. 310:

12l 2 (0, 0) = 1 ω 0 Reg 32 + 1 ω 2 0 Im[g 20 ḡ31 -g 11 (4g 31 + 3ḡ 22 ) -1 3 g 02 (g 40 + ḡ13 ) -g 30 g 12 ] + 1 ω 3 0 {Re[g 20 (ḡ 11 (3g 12 -ḡ30 )
+g 02 (ḡ 12 -1 3 g 30 ) + The coefficients g kl used in the above formula can be written in terms of the functions C, D, E. Thus, from

1 3! p, C (zq + z q, zq + z q, zq + z q) = k+l=3 1 k!l! g kl z k zl ,
we obtain: g 30 = p, C (q, q, q) , g 21 = p, C (q, q, q) , g 12 = p, C (q, q, q) , g 03 = p, C (q, q, q) .

From 1 4! p, D (zq + z q, zq + z q, zq + z q, zq + z q) = k+l=4 1 k!l! g kl z k zl , we obtain: g 40 = p, D (q, q, q, q) , g 31 = p, D (q, q, q, q) , g 22 = p, D (q, q, q, q) , g 13 = p, D (q, q, q, q) . Similarly we get g 32 = p, E (q, q, q, q, q) . From (11) we have:

D = d 4 F (0) + 12β 2 S, E = d 5 F (0) + 20β 2 K + 120β 3 R,
where S (x, y, z, t) , K (x, y, z, t, u) and R (x, y, z, t, u) are multilinear symmetric forms such that:

S (x, x, x, x) = x 2 1 + x 2 2 d 2 G (0) , K (x, x, x, x, x) = x 2 1 + x 2 2 d 3 G (0) , R (x, x, x, x, x) = x 2 1 + x 2 2 2 M x,
respectively. Applying the above formula, the coefficients g kl can be split as: g 30 = γ 30 + 6β 2 ρ 30 , where γ 30 = p, d 3 F (0) (q, q, q) , ρ 30 = p, H (q, q, q) ; g 21 = γ 21 + 6β 2 ρ 21 , where γ 21 = p, d 3 F (0) (q, q, q) , ρ 21 = p, H (q, q, q) ; g 12 = γ 12 + 6β 2 ρ 12 , where γ 12 = p, d 3 F (0) (q, q, q) , ρ 12 = p, H (q, q, q) ; g 03 = γ 03 + 6β 2 ρ 03 , where γ 03 = p, d 3 F (0) (q, q, q) , ρ 03 = p, H (q, q, q) ; g 13 = γ 13 + 12β 2 ρ 13 , where γ 13 = p, d 4 F (0) (q, q, q, q) , ρ 13 = p, S (q, q, q, q) ; g 40 = γ 40 + 12β 2 ρ 40 , where γ 40 = p, d 4 F (0) (q, q, q, q) , ρ 40 = p, S (q, q, q, q) ; g 22 = γ 22 + 12β 2 ρ 22 , where γ 22 = p, d 4 F (0) (q, q, q, q) , ρ 22 = p, S (q, q, q, q) ; g 31 = γ 31 + 12β 2 ρ 31 , where γ 31 = p, d 4 F (0) (q, q, q, q) , ρ 31 = p, S (q, q, q, q) ; g 32 = γ 32 + 20β 2 ρ 32 + 120β 3 τ 32 , where γ 32 = p, d 5 F (0) (q, q, q, q, q) , ρ 32 = p, K (q, q, q, q, q) , and τ 32 = p, R (q, q, q, q, q) . Substituting into (26) the expressions of the coefficients g kl we get

l 2 (0, 0) = δ + 1 12ω 0 Re (120β 3 τ 32 ) , ( 27
)
where δ depends only on F and G. Hence

l 2 (0, 0) = δ + 10 ω 0 β 3 Re (τ 32 ) . ( 28
)
Consequently, there exists a unique value

β 3crit = - δ ω 0 10Re (τ 32 ) ( 29
)
of β 3 such that l 2 (0, 0) = 0, provided that Re (τ 32 ) = 0.

Considering Proof. It follows from direct computations.

β 3 = β 3crit , condition (B.1) is satisfied. Lemma 1. If Q : R 2 5 → R is a multilinear sym- metric form with Q (x, x, x, x, x) = 5 j=0 c 5-j,j x 5-j 1 x j 2 , then Q (x,
Remark. In the particular case when

J = dF (0) = 0 -ω 0 ω 0 0 , we can choose q = (1, -i) T , p = 1 2 , -i 2 T .
Thus from ( 22) we have

β 2crit (0) = - ω 0 δ 2tr (M ) . ( 30
)
Using Lemma 1, we get R (q, q, q, q, q) = and the second Lyapunov coefficient in (28) reads:

l 2 (0, 0) = δ + 8 ω 0 β 3 tr (M ) . ( 32
)
In this case we have

β 3crit = - δ ω 0 8tr (M ) , ( 33
)
if tr (M ) = 0.

Condition (B.2)

The map

(μ 1 , μ 2 ) → (Reλ (μ 1 , μ 2 ) , l 1 (μ 1 , μ 2 )) is regular at (μ 1 , μ 2 ) = (0, 0) provided Δ = ∂ ∂μ1 Reλ (μ 1 , μ 2 ) ∂ ∂μ2 Reλ (μ 1 , μ 2 ) ∂ ∂μ1 l 1 (μ 1 , μ 2 ) ∂ ∂μ2 l 1 (μ 1 , μ 2 ) = 0 (34)
β 1 < 0, β 3 < β 3crit , tr(M ) < 0. at (μ 1 , μ 2 ) = (0, 0) . Since Reλ (μ 1 , μ 2 ) = 1 2 β 1 μ 1 tr (M ) , ( 35
)
we have ∂ ∂μ2 Reλ (μ 1 , μ 2 ) = 0, hence only the term ∂ ∂μ2 l 1 (μ 1 , μ 2 ) have to be evaluated. The first Lyapunov coefficient corresponding to the Hopf bifurcation in (1) can be written in terms of the derivatives of the functions in the right hand side in (1) as [START_REF] Guckenheimer | Nonlinear oscillations, dynamical systems and bifurcations of vector fields[END_REF]

l 1 (μ 1 , μ 2 ) = R 1 16ω 2 0 + R 2 16ω 0 , ( 36
)
where R 1 contains only terms from the second order derivatives of the vector field F (x) + G (x) u, and

R 2 = ∂ 3 F 1 ∂x 3 1 + ∂ 3 F 1 ∂x 1 ∂x 2 2 + ∂ 3 F 2 ∂x 2 1 ∂x 2 + ∂ 3 F 2 ∂x 3 2 , ( 37
)
where by F 1 , F 2 we denote the components of the two dimensional vector field F (x) + G (x) u. Taking into account (11) we obtain ∂R1 ∂μ2 = 0, and

R 2 = 8 (β 2 + μ 2 ) tr (M ) , (38)
thus ∂R2 ∂μ2 = 8tr (M ) . It follows NODY9779_source.tex; 16/07/2010; 10:46 p. 9

Δ = 1 4ω 0 β 1 (tr (M )) 2 . ( 39
)
Consequently the transversality condition (B.2) is fulfilled, provided

β 1 tr (M ) = 0. ( 40
)
We have proved the following result.

Theorem 2. Consider the planar nonlinear control system

ẋ = F (x) + G (x) u (x, μ, β) , ( 41
)
where x ∈ R 2 , F, G : R 2 → R 2 are smooth vector fields, F (0) = 0, G (0) = 0, J = dF (0) having purely imaginary eigenvalues, M = dG (0) and tr(M ) = 0. Then there exist β 1 , β 2 , β 3 ∈ R, where β 2 = β 2crit (0) is given by ( 30), such that with the control law G (x) u (x, μ, β) , where

u (x, μ, β) = β 1 μ 1 +(β 2 + μ 2 ) x 2 1 + x 2 2 +β 3 x 2 1 + x 2 2 2 , ( 42
)
system (41) undergoes a Bautin bifurcation at μ 1 = μ 2 = 0, for β 3 = β 3crit . In addition it is possible to control the stability and the direction of the limit cycles emerging near the origin by selecting positive or negative β 1 , tr (M ) and β 3 > β 3crit or β 3 < β 3crit , where β 3crit is given by (33).

Remark. According to (32) and (33), it follows that

l 2 (0, 0) > 0 ⇔ ⎧ ⎨ ⎩ β 3 > β 3crit , if tr (M ) > 0, or β 3 < β 3crit , if tr (M ) < 0, while l 2 (0, 0) < 0 ⇔ ⎧ ⎨ ⎩ β 3 < β 3crit , if tr (M ) > 0, or β 3 > β 3crit , if tr (M ) < 0.
Using Theorem 3, and selecting appropriate values for the control parameters β 1 , β 3 and tr (M ), all the possible Bautin bifurcation diagrams, considering (μ 1 , μ 2 ) as the bifurcation parameter, can be obtained. See Figures 23456789. In these figures, the curve C NH contains values of parameters corresponding to non-hyperbolic limit cycle bifurcation. Remark that the curve C NH is tangent to the μ 2 semi-axis, corresponding to l 1 (0, μ 2 ) > 0 as l 2 (0, 0) < 0 or to the μ 2 semi-axis, corresponding to l 1 (0, μ 2 ) < 0 as l 2 (0, 0) > 0. In addition, as μ 1 = μ 2 = 0, the first Lyapunov coefficient is zero, for all Figures 23456789we have β 2 = β 2crit (0). Remark that in Figures 2,4, 6, 8 we have l 2 > 0, while in Figures 3,5 where

F (x) = -x 2 -x 1 x 2 2 x 1 .
It possesses the nonhyperbolic equilibrium x = 0, and

J = dF (0) = 0 -1 1 0 .
Consider the control u (x, μ, β) G (x) , with u (x, μ, β) given by ( 2), and G (x) = sx, s = 0. We have G (0) = 0, M = dG (0) = s 0 0 s and tr (M ) = 2s = 0. Thus the hypothesis of Theorems 1 and 2 are fulfilled for the nonlinear control system (1), which in our case is written as

ẋ1 = -x 2 -x 1 x 2 2 + sx 1 u (x, μ, β) ẋ2 = x 1 + sx 2 u (x, μ, β) (44)
with bifurcation parameter μ = (μ 1 , μ 2 ). Following the lines in Section 2, the first Lyapunov coefficient at μ 1 = 0 has the expression:

l 1 (0, μ 2 ) = - 1 2 + 4s (β 2 + μ 2 ) , (45)
and we obtain the following critical value for β 2 :

β 2crit (μ 2 ) = 1 8s -μ 2 . ( 46
)
For β 2 = β 2crit (0) = 1 8s , applying the formula in Section 3, we find l 2 (0, 0) = 16sβ 3 and β 3crit = 0.

According to our results concerning the Hopf bifurcation control in Section 2, by choosing different values for β 1 , β 2 , s, we can manipulate the behavior of the system (44) in order to obtain all the diagrams from Figure 1. For instance, if In this figure we plotted, using [START_REF] Ermentrout | XPPAUT[END_REF], [START_REF] Ermentrout | Simulating, analyzing and animating dynamical systems: a guide to xppaut for researchers and students[END_REF], the trajectory through the initial point (0.2, 0.25), for three different values of μ 1 , showing the supercritical Hopf bifurcation. Our study is a local one, thus the Hopf bifurcation is emphasized only in a neighborhood of the point (x 1 , x 2 , μ 1 ) = (0, 0, 0) . For parameter values of μ 1 far of the bifurcation value μ 1 = 0, the limit cycle does not exist any longer, since it disappeared by another type of bifurcation (Figure 10 (iv)).

β 1 = 1, β 2 = -1, s = 1, β 3 = 0.
where

F (x) = -x 2 + x 2 1 + 1 3 x 3 1 x 1 ,
which possesses the nonhyperbolic equilibrium x = 0, and J = dF (0) = 0 -1 1 0 .

Consider the control u (x, μ, β) G (x) , with u (x, μ, β) given by ( 2), and

G (x) = sx 1 + x 1 x 2 2 sx 2 + x 3 2 , s = 0.
We have G (0) = 0, M = dG (0) = s 0 0 s and tr (M ) = 2s = 0. Thus the hypothesis of Theorems 1 and 2 are fulfilled for the nonlinear control system ẋ = F (x) + u (x, μ, β) G (x) , (

with bifurcation parameter μ = (μ 1 , μ 2 ) . The first Lyapunov coefficient at μ 1 = 0 has the expression:

l 1 (0, μ 2 ) = 1 2 + 4s (β 2 + μ 2 ) , ( 49
)
and the critical value for β 2 is

β 2crit (μ 2 ) = - 1 8s -μ 2 . ( 50
)
For β 2 = β 2crit (0) = -1 8s , we find 11 The trajectory through the initial point (x 1 , x 2 ) = (0.5, 0.5) and parameters (i) (μ 1 , μ 2 ) = (-0.2, -1), region 1 in Figure 3; (ii) (μ 1 , μ 2 ) = (0, -1), region 3; (iii) (μ 1 , μ 2 ) = (0.5, 1), region 3; (iv) (μ 1 , μ 2 ) = (0, 1), region 4; (v) (μ 1 , μ 2 ) = (-0.2, 1), region 5; (vi) (μ 1 , μ 2 ) = (-0.5, 1), region 1, close to the nonhyperbolic limit cycle bifurcation curve C NH .

Conclusions

In this paper we consider a nonlinear planar system with a non-hyperbolic equilibrium at the origin, which possesses a pair of imaginary uncontrollable modes. We design a control law such that the resulting control system undergoes controllable Hopf or Bautin bifurcation. By two control parameters (β 1 , β 2 ) the stability and orientation of the limit cycle in the Hopf case can be controlled by feedback, obtaining four possible cases. Similarly, the characteristics of the limit cycles in the Bautin case, as β 2 = β 2crit , can be controlled by three control parameters (β 1 , β 3 and tr (M )), obtaining all the eight possible cases. Two examples emphasizing the validity of our theoretical results are given.

Fig. 1

 1 Fig. 1 Hopf bifurcation diagrams:(i) β 1 > 0, β 2 < β 2crit (μ 2 ), tr(M ) > 0 or β 1 < 0, β 2 > β 2crit (μ 2 ), tr(M ) < 0; (ii) β 1 > 0, β 2 > β 2crit (μ 2 ), tr(M ) > 0 or β 1 < 0, β 2 < β 2crit (μ 2 ), tr(M ) < 0; (iii) β 1 < 0, β 2 < β 2crit (μ 2 ), tr(M ) > 0 or β 1 > 0, β 2 > β 2crit (μ 2 ), tr(M ) < 0; (iv) β 1 < 0, β 2 > β 2crit (μ 2 ), tr(M ) > 0 or β 1 > 0, β 2 < β 2crit (μ 2 ), tr(M ) < 0.

8 5 (m 11 -im 12 ) 8 5

 88 (m 21im 22 )

Fig. 2 Fig. 3 Fig. 4 7 Fig. 5 Fig. 6

 234756 Fig. 2 Bautin bifurcation diagram: β 1 > 0, β 3 > β 3crit , tr(M ) > 0.

Fig. 7 Fig. 8 Fig. 9

 789 Fig. 7 Bautin bifurcation diagram: β 1 < 0, β 3 > β 3crit , tr(M ) < 0.

  , 7, 9 we have l 2 < 0.

4 ApplicationsExample 1 .

 41 Consider the planar system ẋ = F (x) , (43)

  3, μ 2 = 0.5, three characteristic phase portraits similar to those from Figure 1(i) are obtained in Figure 10(i)-(iii).

Fig. 10 2 .

 102 Fig. 10 The trajectory through the initial point (x 1 , x 2 ) = (0.2, 0.25) and parameters (i) μ 1 = -0.1; (ii) μ 1 = 0; (iii) μ 1 = 0.1; (iv) μ 1 = 0.5.

l 2 ( 3 .

 23 Choosing appropriate values for the control parameters β 1 , β 2 , β 3 , s =1 2 tr (M ) , all the four types of Hopf and all eight types of Bautin bifurcation diagrams can be obtained.

Fig.

  Fig.11The trajectory through the initial point (x 1 , x 2 ) = (0.5, 0.5) and parameters (i) (μ 1 , μ 2 ) = (-0.2, -1), region 1 in Figure3;(ii) (μ 1 , μ 2 ) = (0, -1), region 3; (iii) (μ 1 , μ 2 ) = (0.5, 1), region 3; (iv) (μ 1 , μ 2 ) = (0, 1), region 4; (v) (μ 1 , μ 2 ) = (-0.2, 1), region 5; (vi) (μ 1 , μ 2 ) = (-0.5, 1), region 1, close to the nonhyperbolic limit cycle bifurcation curve C NH .

Acknowledgments. The first author is partially supported by CAPES. The second author is partially supported by CNPq grant number 476672/2009-0.