Emanuele Ferragina

Martin Seeleib-Kaiser

Thees Spreckelsen

The Four Worlds of 'Welfare Reality' -Social Risks and Outcomes in Europe

Keywords: Welfare state regimes, poverty, inequality, comparative, social policy, cluster analysis, EU-SILC

After three decades of welfare state crisis, change and transformation can we still speak of welfare state regimes when looking at their outcomes? The analysis of outcomes provides a picture of 'the real worlds of welfare' and is of considerable importance to understanding political legitimacy across countries. We use aggregate longitudinal data for West European countries in order to map welfare outcomes and cluster countries. The cluster results are also assessed for their sensitivity to the choice of different countries, years or indicators. All European welfare states have a significant capacity for reducing poverty and inequality. However, the degree of this reduction varies considerably, especially when examining different social groups, i.e. unemployed people, children, youths or the elderly. Outcomes cluster countries largely in line with previous institutionalist literature, differentiating between conservative, liberal, Mediterranean and social-democratic regimes. As the main exception, we identify Germany, which can no longer be characterised as the prototypical conservative welfare state. When analysing old social risks such as unemployment and old age, Europe appears to be characterised by two groups, i.e. one consisting of liberal and Mediterranean countries and a second made up of social-democratic and conservative countries. New social risks such as child and youth poverty, by contrast, replicate very closely the theoretical four-cluster typology. Our sensitivity analyses reveal that our clusters tend to be stable over time. Welfare regimes continue to serve as a useful analytical tool and relate to outcomes experienced by European citizens.

Worlds of Welfare: institutional configurations and outcomes

Esping-Andersen's (1990,1999) seminal work on 'Worlds of Welfare' has inspired comparative social policy research for the past twenty-five years. Empirically, he identified regimes as clusters based on the concepts of decommodification, social stratification and the state-market relationship, i.e. the specific welfare mix. 'Decommodification refers to the degree to which individuals, or families, can uphold a socially accepted standard of living independently of market participation' (Esping-Andersen, 1990: 37). In contrast, Esping Andersen used the 'social stratification' dimension to assess how welfare state institutions structure classes and consequently social order (Esping- Andersen, 1990: 55).

He operationalised 'decommodification'1 by including indicators for eligibility rules and levels of income replacement for cash benefits insuring against risks from unemployment, sickness2 and old age (Esping-Andersen, 1990: 47), and 'social stratification' through seven indicators that aim at capturing the three worlds of welfare: corporatism; etatism -using indicators of pension segmentation -(conservative index); means-tested poor relief spending; private pension and health care spending as a share of the total respective spending (liberal index); average universalism; and average benefit equality (socialist index). Social policy can be provided through different institutional arrangements and combinations of the market, non-profit organisations and the state.

This operationalisation of decommodification and social stratification across eighteen OECD countries resulted in a parsimonious taxonomy based on different institutional configurations and legal entitlements for the average production worker (APW). Esping-Andersen classified these configurations by their underlying political traditions, i.e. social democracy, Christian democracy and liberalism. His three-fold taxonomy has been widely scrutinised, criticised and updated (for a summary of the debate, see Arts andGelissen, 2002, 2010;[START_REF] Ferragina | The welfare regime debate: past, present, future?[END_REF]; for an in depth theoretical discussion, see [START_REF] Powell | An audit of the welfare modelling business[END_REF].

At the substantive level, this literature on the welfare regime debate has mainly followed four streams of criticism: (1) the identification of additional regimes, (2) the presumed misclassification of specific countries, (3) the application of the regime theory to other policy domains and (4) the taxonomy's limitations resulting from being based on legal entitlements for the APW (for a summary of all four criticisms, see [START_REF] Ferragina | The welfare regime debate: past, present, future?[END_REF]. Our work mainly contributes to the fourth stream.

We analyse welfare regimes on the basis of outcomes for different groups. This contrasts with previous analyses based only on inputs. Outcomes, such as achieved poverty reduction, we argue, are essential to the legitimacy of welfare states. Legitimacy in turn, we assume, is the basis for enduring welfare state arrangements that can be recognised as regimes.

From a theoretical point of view, polities obtain a general political legitimacy either through the democratic institutional process (input legitimacy) or through their problem-solving capacity (outcome legitimacy) [START_REF] Scharpf | Governing in Europe: Effective and Democratic?[END_REF]. For instance, research has demonstrated that support, or the lack thereof, for the European Union is strongly based on EU citizens' perceptions of (welfare-) state outcomes [START_REF] Gabel | Economic conditions, economic perceptions, and public support for European integration[END_REF][START_REF] Haller | Is the European Union legitimate? To what extent?[END_REF]. We argue similarly that welfare states obtain their legitimacy from their long-standing institutional structures, but also from their ability to reach certain societal goals. Legitimacy provides the basis for acceptance of a given structure; therefore, our key hypothesis is that: Welfare regimes persist, not only when looking at institutional configurations, but also when considering main 'desired' outcomes.

Welfare states' 'desired' outcomes can be defined, in general, as the results from the objectives of social policies and welfare states. Depending on their institutional configurations, the literature [START_REF] Goodin | The Real Worlds of Welfare Capitalism[END_REF] often distinguishes three objectives of welfare states. They aim to: (a) provide autonomy to citizens/residents; (b) provide social stability in case of risk; and (c) reduce/ameliorate poverty.

Historically, as well as in Esping-Andersen's original work, these objectives (often implicitly) related to outcomes for one specific societal group, the APWs. These outcomes are the reduction of risks of old age poverty, unemployment rates and poverty of the male breadwinners.

Changing socio-economic conditions, including globalisation, de-industrialisation, demographic change and changed gender roles, as well as welfare state changes, including retrenchment, recommodification and recalibration [START_REF] Pierson | Coping with permanent austerity: welfare state restructuring in affluent democracies[END_REF], are said to have had a major impact on changing risk structures. Hence, to establish a more comprehensive assessment of outcomes for different societal groups, we extend the analyses beyond the APW and old risks, to also include new social risks (NSR) [START_REF] Taylor-Gooby | New Risks, New Welfare[END_REF][START_REF] Bonoli | The politics of the new social policies: providing coverage against new social risks in mature welfare states[END_REF][START_REF] Bonoli | Time matters: postindustrialization, new social risks, and welfare state adaptation in advanced industrial democracies[END_REF]. We deliberately chose the old versus new risks distinction, as it builds on the core aim of welfare states -to insure against social risks -and is less problematic than the normatively and analytically blurred distinction between those policies that primarily focus on social investment and those that primarily deal with compensation [START_REF] Nolan | What use is "social investment"?[END_REF]. Core NSR are: the (in)ability to reconcile work and family life; single parenthood with the increased risk of child poverty; low skills and education; as well as insufficient social security coverage for atypical workers. NSR tend to be most prevalent among younger people, families and women [START_REF] Bonoli | The politics of the new social policies: providing coverage against new social risks in mature welfare states[END_REF]. Countries that effectively address these new risks should have low child and youth poverty rates, a high percentage of youth in education to improve human capital formation (ibid.), as well as high female labour force participation to minimise the risk of poverty and reduce inequality in societies that are increasingly characterised by homogamy [START_REF] Esping-Andersen | The Incomplete Revolution[END_REF]. Hence, on the one hand, we capture the extent of insurance against old social risks, measuring the incidence of poverty among pensioners, unemployed and prime-aged workers, as well as replacement rates for pensioners and the overall unemployment rate; and we assess the incidence of new social risks by considering child and youth poverty rates, percentage of youth in education and female labour force participation.

There have been intense debates, numerous empirical studies of regimes using cluster analysis (for example, [START_REF] Kangas | The politics of social security: on regressions, qualitative comparisons, and cluster analysis[END_REF][START_REF] Gough | Social assistance regimes: a cluster analysis[END_REF], and regimes are in continuous use as analytical tools for comparative social policy analysis. Despite this, few studies have tested whether welfare regimes translate into clusters similar to the original worlds of welfare when one considers welfare state outcomes (but see [START_REF] Goodin | The Real Worlds of Welfare Capitalism[END_REF], and with a redistribution focus, [START_REF] Kammer | Welfare regimes and welfare state outcomes in Europe[END_REF], and particularly outcomes for new social risks.

Such a test needs a longitudinal perspective since, theoretically, regimes are said to be stable over time (Esping-Andersen, 1996, 1999;[START_REF] Pierson | Coping with permanent austerity: welfare state restructuring in affluent democracies[END_REF]. [START_REF] Danforth | Worlds of welfare in time: a historical reassessment of the three-world typology[END_REF] identifies variations in welfare arrangements over a long time period and argues that the three regimes were only established by 1975. Outcomes of welfare state changes will only become manifest over time, with lags, some would suggest, of a decade (i.e. [START_REF] Sabatier | An advocacy coalition framework of policy change and the role of policyoriented learning therein[END_REF], as well as variation due to contextual factors. According to some observers, the 1980s and 1990s were primarily characterised by welfare state retrenchment and recommodification, while more recent reforms are said to have been of amore transformative nature, including recalibration [START_REF] Pierson | Coping with permanent austerity: welfare state restructuring in affluent democracies[END_REF][START_REF] Hemerijck | Changing Welfare States[END_REF]. As these processes (especially since the early 2000s) have also been said to be characterized by 'new politics' [START_REF] Pierson | Coping with permanent austerity: welfare state restructuring in affluent democracies[END_REF], it cannot be assumed that the outcomes follow the logic previously observed.

Esping-Andersen's original taxonomy is based on three regimes. Following [START_REF] Ferrera | Modelli di Solidarieta[END_REF][START_REF] Ferrera | The "southern model" of welfare in social Europe[END_REF], as well as [START_REF] Leibfried | Towards a European welfare state? On integrating poverty regimes into the European community[END_REF], we hypothesise the existence of a fourth Mediterranean regime type different from the original three regimes. In our analyses, we consider empirically four distinct country clusters representing these welfare state regimes.

Methodologically, cluster solutions for a given dataset are necessarily sensitive to the availability and choice of data [START_REF] Aldenderfer | Cluster Analysis[END_REF]. Past cluster analyses of welfare regimes did not account for cluster stability in relation to such factors (for example, [START_REF] Castles | World, families, regimes: country clusters in European and OECD area public policy[END_REF][START_REF] Kammer | Welfare regimes and welfare state outcomes in Europe[END_REF].

In summary, this article, (a) revisits welfare state regime classifications from an outcome perspective, (b) distinguishes analytically and empirically welfare regimes in relation to old and new social risks and (c) for methodological and empirical reasons tests the stability of a four-fold typology across fourteen EU countries during the period 2005 to 2012.

Data and methods

Data

Our analyses are based on aggregate data published by Eurostat. They are mostly generated from the European Union Statistics on Income and Living Conditions (EUSILC), European Union Labour Force Survey (EU-LFS), or, in the case of information on education, the joint UNESCO Institute of Statistics/OECD/Eurostat questionnaires on education statistics (for details on the measures and data sources, see Appendix 1). These data are regarded as a unique source of comparative data on incomes and households relating to welfare states (for example, EU-SILC cf. [START_REF] Iacovou | Using EU-SILC Data for Cross-National Analysis: Strengths, Problems and Recommendations[END_REF] 3 .

Methods and indicators

The empirical analysis is divided into two parts. The first part, based on descriptive statistics, analyses inequality and poverty outcomes, before and after benefits and taxes, alongside the insurance against old and new social risks. The second, based on cluster analyses (for details, see Appendix 2), assesses the similarity between institutional and outcome-based typologies (for an extensive literature on past research, see Appendix 3) in four steps. We cluster countries according to:4 1. all selected measures relating to old and new social risks simultaneously; 2. their outcomes in covering old social risks by including indicators such as poverty levels after social transfers for male workers,5 the elderly,6 and the unemployed; and old-age pension replacement rates;7 3. their outcomes in covering new social risks, including indicators such as poverty rates after social transfers for children and youths, female employment rates, and rates of enrolment in education for people aged between fifteen and twenty-four.

We assess the stability of cluster analyses using:

4. iterative methods [START_REF] Hennig | Cluster-wise assessment of cluster stability[END_REF], dropping in turn indicators, countries and years (for more details, see Appendix 1).

We present analyses for two time periods: (a) we investigate welfare state outcomes in a single cross-section for 2012 to provide a descriptive picture for the outcome measures; (b) we conduct a cluster analysis for the pooled data over the period 2005-12. Finally, based on the literature differentiating between liberal, conservative, social-democratic and Mediterranean regimes [START_REF] Esping-Andersen | The Three Worlds of Welfare Capitalism[END_REF][START_REF] Leibfried | Towards a European welfare state? On integrating poverty regimes into the European community[END_REF][START_REF] Ferrera | Modelli di Solidarieta[END_REF][START_REF] Ferrera | The "southern model" of welfare in social Europe[END_REF][START_REF] Bonoli | Classifying welfare states: a two-dimension approach[END_REF], we include an a priori cut-off of four clusters rather than simply looking for the highest degree of dissimilarity.

Geographical focus

Our research is focused on EU Member States. Europe is the continent most heavily influenced by the three political movements, liberalism, Christian democracy and social democracy, which are said to be the ideological basis for the three worlds of welfare. Thus, we exclude other rich OECD countries from our analysis where these ideological underpinnings are less clearly shared, i.e. Japan (on 'Japanese uniqueness', see [START_REF] Dale | The Myth of Japanese Uniqueness[END_REF][START_REF] Esping-Andersen | Hybrid or unique? The Japanese welfare state between Europe and America[END_REF], the United States (on 'American exceptionalism', see [START_REF] Lipset | American Exceptionalism: A Double-Edge Sword[END_REF], Australia and New Zealand (on the different natures of 'radical welfare states', see Castles andMitchell, 1992, 1993). EU countries have agreed to strive for 'the promotion of a high level of employment, the guarantee of adequate social protection, the fight against social exclusion, and a high level of education, training and protection of human health' (Council of the European Union, 2008: Article 9). Furthermore, the adoption of the Europe 2020 strategy places social policy at the core of EU policy: setting targets for raising the (female) employment rate, reducing early school leaving, increasing the proportion of young people completing tertiary education or equivalent and lifting at least 20 million people out of poverty. These indicators are said to be at the heart of the EU strategy for growth and are very likely to impact on the legitimacy of the EU. Finally, we limited our analyses to countries that had joined the European Union before the enlargement in 2004,8 as central and eastern European countries have not reached 'new social policy equilibria' (Potuček, 2008: 95).9

Results

Describing welfare state outcomes Our first objective is to examine welfare state outcomes amongst the EU 14. Across all countries, the welfare state has a remarkable effect in reducing poverty and inequality levels (Table 1 summarises poverty and inequality data before and after social transfers). Data also suggest a remarkable similarity in poverty reduction levels among EU 14 countries: Austria, Belgium, Denmark, Finland, France, Germany, Ireland, Sweden, and the United Kingdom (nine out of fourteen countries), have a poverty reduction level after social transfers ranging between 65 and 68 per cent.

However, if we look simultaneously at poverty and inequality, a distinctive ranking across countries appears. It is almost as if poverty and inequality reduction are in contrast. This relates to the aforementioned distinction between two objectives of the welfare state [START_REF] Goodin | The Real Worlds of Welfare Capitalism[END_REF]: on the one hand, to provide autonomy (hence, reducing inequality) and, on the other hand, to reduce poverty. Our outcomes-based perspective suggests three different patterns, i.e. (a) Germany and Sweden seem to be committed in similar ways to reducing poverty as well as inequality, (b) France seems committed mainly to poverty reduction, and (c) Greece appears to lack a strong commitment to reduce both (for a review of the debate on the redistributive power of the welfare state, see [START_REF] Esping-Andersen | Economic inequality and the welfare state[END_REF]. Despite the fact that poverty and inequality reduction for the overall population provide good general measures of substantive welfare state outcomes, they do not account for the potential differential outcomes for different societal groups. [START_REF] Bonoli | The politics of the new social policies: providing coverage against new social risks in mature welfare states[END_REF], for example, argued that welfare states are still primarily geared towards covering old social risks, such as unemployment, sickness and old age. Hence, together with the description of overall poverty and inequality reduction, it is important to further distinguish outcomes in relation to old (i.e. unemployment and old-age replacement rates) 10 and new (i.e. child and youth poverty, educational participation and female employment rates) social risks.

In all countries, poverty rates are higher among the unemployed (an old social risk), providing an indication that this societal group is the least protected, if compared to children, youth or pensioners (see Table 2). Moreover, poverty among the elderly is significantly more widespread than poverty among children in Belgium, Denmark, Ireland, Italy, the Netherlands, Spain and the United Kingdom, providing further evidence that in some countries insurance against old social risks would seem less comprehensive than against new social risks. Poverty levels for young people and the elderly seem to be quite high in most nations, and surprisingly so in Scandinavian countries. However, this is partially a statistical artifact due to the high percentage of people living independently rather than being part of a larger household. Since poverty is measured at the household level, poverty tends to be higher in those countries, with smaller households, including youths as well as elderly people who live independently (as in the case of Scandinavian countries). 11 Child poverty seems to be especially high in Mediterranean countries. 12 The percentage of young people in education is high in Nordic countries and the Netherlands, whilst it is particularly low in the United Kingdom, Austria, Italy and France (see Table 3). Female employment rates are the lowest in Greece, Italy, Spain and Ireland, followed by Portugal and France.

10 Measuring social stability. 11 Indeed, calculations on the 2012 EU-SILC data show exactly that Scandinavian countries have a smaller average size of families and higher number of young and old people living independently than the other countries analysed (analyses available from the authors on request). It is worth noting that the relative reduction of poverty across different societal groups is much less consistent across countries, than the levels of poverty reductions observed for the overall population. We find large differences, ranging, for example, from a 15.9 per cent child poverty reduction in Greece to a 64.1 per cent reduction in Finland, highlighting the much greater effectiveness of the Finnish benefit and tax system in reducing child poverty. The data also suggest there might be more variation across countries when looking at new rather than old social risks. This means that despite the aim of institutional convergence amongst European countries towards employment-oriented social investment policies, and some convergence in overall poverty rates (Cantillon and Vandenbroucke, 2014: xiii), many differences continue to persist among European welfare states.

Welfare regimes outcome clusters

Our second objective is to cluster welfare states according to their outcomes. In a first step, we include all outcome indicators (including change in Gini before and after transfers) 13 in our analysis, pooled over the period 2005 to 2012. Where Tables 1 and2 only look at, for example, child poverty in 2012, we now include child poverty for each year between 2005 and 2012, in each country. This provides for more robust results and takes into account variations over time.

We find a set of outcome-based regimes similar to those based on institutional configurations, with the following exceptions (see Figure 1): Germany clusters with the United Kingdom, and Ireland falls into the conservative cluster. The clustering of Ireland within the conservative group might be dependent on achieving lower poverty rates amongst the unemployed and the elderly compared to the UK (data for 2012, see Table 2).

Although these findings are largely in line with the welfare regime literature, we also checked the robustness of clusters proposing a battery of sensitivity analyses. Ideally, we would find similar countries in each cluster, even if we changed some of the indicators or dropped a country or year of observation. In the following sensitivity analyses, Jaccard 14 coefficients represent the effects of such data variations.

We examine the influence of omitting single indicators, by looking at the overall set of clusters and each cluster separately (see Table 4). Jaccard coefficients superior to 0.75 indicate cluster stability, even after dropping from the sample a variable or an observation. Values below 0.75 and superior to 0.65 suggest the existence of patterns in the data; however, in this case clusters are generally not perfectly stable against the variation in the data used. Our findings are rather sensitive to the omission of any indicator, with a coefficient of 0.57 for the overall solution. In contrast to the other clusters, the conservative cluster (Table 4) is robust to the omission of indicators, except for the rate of female employment; similar levels thereof seem to define the conservative regime. By contrast, the sensitivity analysis provides an explanation of the 'surprising' classification of Germany among the liberal cluster: high levels of poverty among the unemployed, rather than female employment rates, are the most defining feature of liberal welfare states (hence, Germany with its high poverty level is fully part of this group). 15 This is in line with previous research focusing on institutional change [START_REF] Seeleib-Kaiser | A dual transformation of the German welfare state?[END_REF][START_REF] Bleses | The Dual Transformation of the GermanWelfare State[END_REF][START_REF] Fleckenstein | The dual transformation of social protection and human capital: comparing Britain and Germany[END_REF]), but at odds with the notion whereby Germany continues to be the proto-typical conservative welfare state rooted in the 'Bismarckian tradition'. 16 13 Before and after social transfers data are not available for the whole period through Eurostat. We discuss the implications in the limitations section. 14 The Jaccard coefficient measures the ratio of observations, which have changed their cluster membership relative to all observations, when comparing two cluster solutions. As [START_REF] Hennig | Cluster-wise assessment of cluster stability[END_REF] points out, the coefficient is advantageous compared to other measures of stability, as it is not sensitive to sample size differences. 15 To a lower extent than for the conservative cluster, female employment rates seem to play a similar role. 16 As it has been pointed out in the literature, the EU-SILC data for Germany prior to 2008 should be treated with caution [START_REF] Hauser | Problems of the German contribution to EU-SILC: a research perspective comparing EU-SILC, microcensus and SOEP[END_REF] as it was based on potentially not representative quota samples [START_REF] Decancq | The evolution of poverty in the European Union: concepts, measurement and data[END_REF], however the high poverty rates amongst the unemployed materialised particularly after 2008. The social-democratic and Mediterranean clusters are quite sensitive to the choice of indicators, without a single indicator reaching the low cut-off point (0.65) established by the Jaccard coefficient to indicate a certain level of stability.

Following the same metric, we investigate the potential random omission of years or countries from our dataset (Table 5).Welfare outcome clusters are more or less stable over time, despite considerable welfare state change at the institutional level; for example, in the policy domain of family policy [START_REF] Ferragina | Unemployment protection and family policy at the turn of the 21st century: a dynamic approach to welfare regime theory[END_REF]. Clusters are, however, more sensitive to the omission of countries. As the liberal cluster consists of only two countries, it is thus more prone to dissolution when randomly dropping one country. Amongst the social-democratic and Mediterranean clusters, one country is always less similar to the others. It is not surprising that the Netherlands do not squarely fit the social-democratic cluster, as they have been characterised as social-democratic, conservative or hybrid welfare regime, depending on the indicators chosen. Also Portugal is institutionally different from the other Mediterranean countries [START_REF] Ferrera | The "southern model" of welfare in social Europe[END_REF].

To summarise: welfare state outcome clusters are largely in line with the previous regime literature based on institutional configurations. Clusters are also stable over time; however, they are quite sensitive (with the exception of the conservative cluster) to the choice of indicators and countries included in the sample. The cluster analysis for old social risk outcomes partitions countries into two large blocks and two smaller groups of outliers, which include only three countries (see Figure 2). Germany stands alone, mostly because of the high rate of poverty among the unemployed, and France and the Netherlands form another cluster due to the considerably low rates of old age poverty. The two large clusters include: Belgium, Denmark, Sweden, Austria and Finland (the first group), with higher levels of inequality reduction compared to those in the second group, Ireland, Greece, Spain, Portugal, Italy and the United Kingdom. The cluster based on new social risk outcomes stands in sharp contrast to the old social risks cluster. New social risk indicators almost reproduce the clustering presented for the overall analysis (Figure 3), with the exceptions of Germany and Austria. Whilst Germany joins the social-democratic cluster, Austria joins the United Kingdom in the liberal world. This liberal cluster, formed only by two countries, should be treated with caution, as it is largely determined by a combination of high levels of female labour market participation and low rates of youth enrolment in education programs. Potential problems of accounting for vocational training in Austria might undermine our findings. As an aside, when disregarding the theory-and literature-driven four-cluster cut-off, the cluster analysis seems to be dominated by two super-clusters containing within each other the conservative and the social-democratic on the one hand, and the Mediterranean and liberal cluster on the other hand.

Different to the general clustering, the sensitivity analysis presented for old and new social risks is more limited. This is due to the small number of indicators used (Table 6 provides a comparison of the stability of the old and new social risk clusters). The old social risk clusters are not very stable to dropping any year, but are more stable to the omission of single countries. This is due to the presence of two large clusters and two small middle groups (as we have mentioned they only include Germany, the Netherlands and France). Also, the cluster analysis based on new social risks is quite sensitive to the choice of indicators and countries included, with the remarkable exceptions of the conservative and liberal worlds.

In sum, our cluster solutions for old and new social risks suggest three interesting points. First, we find a significant difference in regimes, depending on the analysis of old social and new social risk outcomes. Second, the old social risk clustering differs significantly from the original welfare state analysis, whereas new social risk outcome clustering seems to be closer to the more conventional welfare regimes typologies. The stability test, finally, points towards the combined importance of the old and new social risk outcomes for identifying typologies. Neither the old nor the new social risks clusters are as stable as the general cluster solution (Table 4).

Discussion and conclusion

The aim of this work is to assess the potential use of welfare state outcomes to delineate four welfare regimes. In doing so, we also distinguish outcomes according to old and new social risks, going beyond policies focusing solely on the APW. To this end, we selected eleven outcome indicators, largely inspired by [START_REF] Goodin | Reasons for Welfare: The Political Theory of theWelfare State[END_REF] reasons for welfare -autonomy, social stability and poverty alleviation -and distinguished by the type of social risk addressed -old or new. The results presented above show that all European welfare states have a strong effect in reducing poverty and inequality across the overall population. However, the high level of similarity, measured when looking at poverty reduction for the overall population, is not replicated when considering different societal groups. For example, the Finnish and Irish welfare states have a similar capacity in reducing poverty among the overall population, but they radically differ in relation to new social risks, such as child poverty (see Tables 1 and3). The different levels of outcomes related to social protection for societal groups confirm the continued importance of welfare regime typologies, but simultaneously demonstrate the need to carefully consider the outcomes of the welfare state for different societal groups to identify regime variations.

The clustering based on the whole set of outcome indicators does provide a similar picture to previous welfare typologies, with the important exceptions of the classification of Germany and Ireland. In particular, while the classification of Ireland as belonging to the conservative cluster might be the result of achieving lower poverty rates amongst the unemployed and the old compared to the UK (the prototypical European liberal country), the classification of Germany within the Liberal regime is rather 'unexpected'.

When analysing old social risk outcomes, Europe appears to be split into two groups. 'The periphery', which includes Liberal and Mediterranean countries, seems to be characterised by lower replacement rates and higher levels of inequality, than 'the core' countries (the social-democratic and continental clusters). Furthermore, considering old social risk outcomes, Germany stands alone, following a unique pattern compared to the rest of the continent.

For new social risks, the most striking finding is that the clustering replicates very closely the theoretical four-cluster typology. In addition, the clustering of new social risks divides European countries into two groups; on the one hand, the social-democratic and liberal countries, characterised by higher youth education and female employment; and, on the other hand, with opposite characteristics, the conservative and Mediterranean countries.

Our cluster analysis is robust against the choice amongst the indicators; in addition, the clusters are stable over time, but quite sensitive when old and new social risks are considered separately. The conservative cluster seems the most stable, while the others show more erratic patterns determined by the choice of indicators and countries. In sum, our findings carry a clear implication: across the EU 14, several groups of countries show a strong persistence in producing a certain set of social outcomes. This suggests that the contextual changes, for example the 2008 crisis, might have had less impact than one would expect, particularly with regard to insurance against new social risks.

The above findings should be interpreted with caution in the light of three major caveats: the limitation of data availability, the dependence on the context of outcome data and the fact that clusters are data-driven. First, the present study uses data from 2005 until 2012, and thereby is restricted to a rather short period of observation; for most countries, this eight-year period equates to two terms of parliament. Whilst we restricted the country sample on theoretical grounds, the choice of indicators was in part determined by the available data. Ideally, a detailed analysis of new and old social risks using poverty levels for different societal groups should have been complemented with the inequality reduction rates for the same groups. The limited availability of comparable indicators and data over time is a major limitation of this study. Second, welfare state outcomes are more context-dependent than welfare state inputs. Hence, the outcomes investigated above (and mainly ascribed to the welfare states) might vary considerably due to other factors than social policy. Third, our analyses are not a formal statistical test of the outcome regimes approach. They are datadriven and exploratory attempts to demonstrate the substantive validity of the approach.

Considering these limitations, future research should investigate longer time periods and perform analyses on the basis of individual level data (for example, EU-SILC and EULFS). Such research should aim at statistically testing the robustness of the outcome regime clusters. The most fruitful avenue might be to systematically contrast input-and outcome-based welfare state regimes.

In particular, such research should focus on assessing the differences in terms of welfare state efficacy (see a similar analysis by [START_REF] Vandenbroucke | Mapping at-risk-of-poverty rates, household employment, and social spending[END_REF].

Despite the limitations, this article clearly shows that welfare state regimes are 'alive'. They serve as useful analytical tools and also relate to the outcomes experienced by citizens. For them, small social policy changes, that do not radically affect welfare state structures, are highly unlikely to reduce poverty and inequality. Consistently high levels of inequality and poverty are very likely to have a detrimental impact on political legitimacy.

Figure 2 -

 2 Figure 2 -(Colour online) Outcome clusters for old social risks

Figure 3 -

 3 Figure 3 -(Colour online) Outcome clusters new social risks

Table 1 -

 1 Welfare state outcomes: poverty levels and inequality across the EU 14 (pre-2004 EU members) in 2012

	Country	Poverty level (all ages) a		Gini coefficient		Difference in
		Before	After	Change	Before	After	Change	change (Gini%-
								poverty%)
	Austria	44.2	14.4	-67.4	47.1	27.6	-41.4	26.0
	Belgium	42.3	15.0	-64.5	47.7	26.6	-44.2	20.3
	Denmark	41.2	13.1	-68.2	53.4	28.1	-47.4	20.8
	Finland	41.3	13.2	-68.0	46.4	25.9	-44.2	23.9
	France	43.7	14.1	-67.7	49.8	30.5	-38.8	29.0
	Germany	43.3	16.1	-62.8	54.4	28.3	-48.0	14.8
	Greece	49.8	23.1	-53.6	56.9	34.3	-39.7	13.9
	Ireland	50.4	15.7	-68.8	53.5	29.9	-44.1	24.7
	Italy	44.5	19.4	-56.4	47.5	31.9	-32.8	23.6
	Netherlands	36.7	10.1	-72.5	46.1	25.4	-44.9	27.6
	Portugal	45.4	17.9	-60.6	55.9	34.5	-38.3	22.3
	Spain	46.2	22.2	-51.9	52.8	35.0	-33.7	18.2
	Sweden	41.8	14.1	-66.3	52.4	24.8	-52.7	13.6
	United	45.4	16.2	-64.3	57.3	32.8	-42.8	21.6
	Kingdom
	Notes:

a Gross and net household income <60% median household income.

Source: Gini coefficient of equivalised disposable income; Gini coefficient of equivalised disposable income before social transfers (pensions included in social transfers. Eurostat (2014a).

Table 2 -

 2 Detailed welfare state outcome indicators: poverty levels by social group after social transfers (and change compared to before transfer levels)

	Country			New risks				Old risks
		Total poverty	Children (<16)	Youth (16-24)	Male worker a	Old (75+)	Unemployed
		Level	Change	Level	Change	After	Change	Level	Change	Level b	Level b
	Austria	14.4	(-67.4%)	18.3	(-56.8%)	17.3	(-41.9%)	12.5	(-47.2%)	16.6	44.9
	Belgium	15	(-64.5%)	16.6	(-49.5%)	16.7	(-49.2%)	12.5	(-50.6%)	18.4	34.5
	Denmark	13.1	(-68.2%)	10.0	(-59.0%)	39.4	(-27.3%)	12.1	(-49.2%)	22.4	26.9
	Finland	13.2	(-68.0%)	11.2	(-64.1%)	24.9	(-39.7%)	10.7	(-44.8%)	27.6	45.3
	France	14.1	(-67.7%)	18.8	(-46.3%)	23.0	(-41.2%)	12.0	(-47.8%)	11.4	36.2
	Germany	16.1	(-62.8%)	14.9	(-53.0%)	20.7	(-36.7%)	13.7	(-60.9%)	12.6	69.3
	Greece	23.1	(-53.6%)	26.5	(-15.9%)	33.1	(-26.4%)	21.9	(-60.8%)	20.0	45.8
	Ireland	15.7	(-68.8%)	16.7	(-63.4%)	23.3	(-60.2%)	13.9	(-36.4%)	13.0	34.1
	Italy	19.4	(-56.4%)	25.9	(-25.8%)	25.4	(-31.4%)	17.5	(-61.2%)	17.9	44.7
	Netherlands	10.1	(-72.5%)	13.6	(-43.8%)	19.8	(-45.0%)	8.5	(-47.0%)	5.9	34.0
	Portugal	17.9	(-60.6%)	21.1	(-34.7%)	22.2	(-42.2%)	15.3	(-51.2%)	21.5	38.5
	Spain	22.2	(-51.9%)	28.9	(-22.7%)	28.4	(-35.7%)	22.0	(-63.0%)	15.9	46.4
	Sweden	14.1	(-66.3%)	13.9	(-57.9%)	27.4	(-34.3%)	11.6	(-46.0%)	25.3	42.3
	United Kingdom	16.2	(-64.3%)	18.1	(-60.1%)	24.1	(-40.9%)	13.0	(-49.4%)	18.9	52.2

Notes:

a 25-54 years old.

b No before-transfers risk-of-poverty levels available, hence no information on change.

Sources: 'At-risk-of poverty rate by most frequent activity status' Eurostat (2014b) and 'at-risk-of poverty rate by detailed age group' Eurostat (2014c).

Table 4 -

 4 Welfare state output clusters: stability by indicator Numbers in bold indicate variables whose removal results in considerably different cluster solutions, or cluster memberships (for the individual clusters). Cluster stability expressed as average Jaccard coefficient for 1,000 runs of randomly dropping observations/years: Coefficients >0.

					Overall	Conservative	Social-Democratic	Liberal	Mediterranean
	Complete solution			0.57	0.79	0.43	0.58	0.49
	Child poverty a			0.71	0.84	0.52	0.65	0.59
	Youth poverty	a			0.43	0.83	0.33	0.64	0.29
	Female employment		0.38	0.58	0.31	0.40	0.57
	In-education				0.49	0.75	0.30	0.55	0.51
	Male worker poverty a		0.72	0.83	0.51	0.63	0.59
	Old age poverty Unemployment poverty a	a	0.38 0.51	0.76 0.83	0.31 0.51	0.56 0.36	0.31 0.35
	Replacement rate			0.72	0.84	0.52	0.64	0.59
	Inequality reduction (Difference Gini)	a		0.67	0.82	0.49	0.65	0.53
	Notes:

75: Stable cluster solution

; >0.65-<0.75: Patterns in data., Seed: 823910233. a (after transfers).

Table 5 -

 5 Welfare state output clusters: stability a by number of observations/years dropped

			Overall	Conservative	Social-Democratic	Liberal	Mediterranean
	1 year dropped	0.81	0.95	0.76	0.71	0.79
	2 years	0.60	0.85	0.45	0.57	0.53
	1 country dropped	0.63	0.93	0.54	0.65	0.69
	2 countries	0.41	0.78	0.36	0.53	0.44
	Notes:	a Cluster stability expressed as average Jaccard coefficient for 1,000 runs of randomly dropping
	observations/years: Coefficients >0.75: Stable cluster solution; >0.65-<0.75: Patterns in data.

Table 6 -

 6 Old and new social risks clusters: stability a by number of observations/years dropped

			Overall	Conservative	Social-Democratic	Liberal	Mediterranean
	Old risk regimes b
	1 year dropped	0.56	0.62	0.73	0.43	0.38
	2 years	0.53	0.58	0.71	0.40	0.35
	1 country dropped	0.65	0.78	0.87	0.78	0.58
	2 countries	0.46	0.64	0.77	0.62	0.38
	New risk regimes b
	1 year dropped	0.47	0.74	0.41	0.40	0.54
	2 years	0.71	0.88	0.52	0.52	0.84
	1 country dropped	0.46	0.64	0.77	0.62	0.38
	2 countries	0.33	0.64	0.28	0.32	0.40
	Notes:	a Cluster stability expressed as average Jaccard coefficient for 1,000 runs of randomly
	dropping observations/years: Coefficients >0.75: Stable cluster solution; >0.65-<0.75: Patterns in data.
	b Cluster solutions correspond to Figures 2 and 3.
	Random drops of observations/years starting with seed: 823910233, runiform() function in Stata 11.0.

For a critique of the decommodification concept, see[START_REF] Room | Commodification and decommodification: a developmental critique[END_REF].

As argued by[START_REF] Moran | Understanding the welfare state: the case of health care[END_REF], the development of health care systems follow different logics compared to other policies.

A limitation with regard to the data published by Eurostat is the absence of any confidence intervals reflecting the uncertainty of the estimates (cf.[START_REF] Goedemé | How much confidence can we have in EU-SILC? Complex sample designs and the standard error of the Europe 2020 poverty indicators[END_REF]. This seems to be a limitation of comparative welfare literature.

All indicators were clustered using complete linkage with Euclidean distance as the dissimilarity measure.

Males aged between twenty-five and fifty-four years.

In order to measure autonomy, poverty rates for pensioners aged seventy-five plus might be indicative. A low rate of pensioners' poverty is an indicator of the decommodifying potential of the pension system. It can be argued that by this age, most people will have used up their savings. Finally, the replacement ratio for pensioners is a proxy measure for social stability.

Rates for people receiving pensions aged sixty-five to seventy-four, as compared to working people aged fiftyfive to sixty-four.

However, excluding the very small and, in many ways, exceptional Duchy of Luxembourg.

This manifest diversity has let some researchers to suggest that the CEE countries constitute a welfare regime in their own right[START_REF] Aidukaite | Old welfare state theories and new welfare regimes in Eastern Europe: challenges and implications[END_REF]; however unlike the Mediterranean regime and its familial solidarity unpinning[START_REF] Ferrera | Modelli di Solidarieta[END_REF][START_REF] Ferragina | Social Capital in Europe: A Comparative Regional Analysis[END_REF], the 'presumed CEE cluster' does not seem easily linked to common structural features of the welfare state.

The following data tables have been used (accessed 23.06.2014)

Appendix 2 -Cluster stability analyses

The stability analyses are conducted for variables, countries and years. Summary stability coefficients are reported for 10,000 and 1,000 iterations respectively. Instead of assessing the variations between cluster solutions one by one, iterative methods of cluster stability assessments examine all variations in random order repeating the variations a large number of times. In addition to the obvious statistical advantages, these procedures do not depend on a researchers subjective choice as to which variation to investigate. Stability coefficients should be interpreted as the general stability of a cluster (solution) to variations in the data. Large changes indicate a low stability of the overall cluster solution for all countries. This follows the procedure proposed by Henning (2007[START_REF] Henning | Dissolution point and isolation robustness: robustness criteria for general cluster analysis methods[END_REF]. Often cluster stability assessments also investigate random perturbation of the values of the observed indicators, so-called jittering. The choice of the random variation is however very subjective, for this reason such analyses were not undertaken.

Appendix 3 -Existing literature on empirical welfare state clusterings Previously, scholars have used different methods to assess welfare regimes: descriptive statistics (Esping- [START_REF] Esping-Andersen | The Three Worlds of Welfare Capitalism[END_REF][START_REF] Castles | Identifying welfare state regimes: the links between politics, instruments and outcomes[END_REF][START_REF] Bonoli | Classifying welfare states: a two-dimension approach[END_REF][START_REF] Korpi | The paradox of redistribution and strategies of equality: welfare state institutions[END_REF][START_REF] Gallie | Welfare Regimes and the Experience of Unemployment in Europe[END_REF][START_REF] Goodin | Work and welfare: towards a post-productivist welfare regime[END_REF][START_REF] Gal | Decommodification and beyond: a comparative analysis of work-injury programmes[END_REF][START_REF] Scruggs | Welfare state decommodification in 18 OECD countries: a replication and revision[END_REF], qualitative comparative analysis [START_REF] Ragin | A qualitative comparative analysis of pension systems[END_REF]Hudson and Kühner, 2009), cluster analysis [START_REF] Kangas | The politics of social security: on regressions, qualitative comparisons, and cluster analysis[END_REF]Obinger andWagschal, 1998, 2001;[START_REF] Gough | Social assistance regimes: a cluster analysis[END_REF][START_REF] Kautto | Investing in services inWest European welfare states[END_REF][START_REF] Saint-Arnaud | Convergence or resilience? A hierarchical cluster analysis of the welfare regimes in advanced countries[END_REF][START_REF] Powell | Welfare regimes and the welfare mix[END_REF][START_REF] Bambra | Worlds of welfare and the health care discrepancy[END_REF][START_REF] Castles | World, families, regimes: country clusters in European and OECD area public policy[END_REF], principal component analysis [START_REF] Shalev | Introduction[END_REF][START_REF] Shalev | Limits and alternatives to multiple regression in comparative research[END_REF][START_REF] Wildeboer Shut | On Worlds of Welfare: Institutions and Their Effects in Eleven Welfare States, The Hague: Social and Cultural Planning Office[END_REF][START_REF] Soede | Unequal Welfare States: Distributive Consequences of Population Ageing in Six European Countries, The Hague: Social Cultural Plannig Office[END_REF][START_REF] Schröder | Integrating welfare and production typologies: how refinements of the varieties of capitalism approach call for a combination of welfare typologies[END_REF][START_REF] Vrooman | Rules of Relief: Institutions of Social Security, and Their Impact[END_REF], and multiple correspondence analysis [START_REF] Ferragina | Unemployment protection and family policy at the turn of the 21st century: a dynamic approach to welfare regime theory[END_REF].


K V Avramov

BIFURCATION ANALYSIS OF A VIBROPERCUSSION SYSTEM BY THE METHOD OF AMPLITUDE SURFACES

An amplitude surface is introduced to analyze the bifurcation behavior of a system upon change in two characteristic parameters. A bifurcation analysis of a vibropercussion system with changing force amplitude and frequency is performed as an example of using amplitude surfaces. Bifurcations of codimension two are revealed.

Introduction and Description of a Mechanical

System. Amplitude surfaces are used here to analyze the bifurcations of nonlinear mechanical systems upon change in two characteristic parameters. In mechanics, the bifurcation analysis is frequently based on amplitude-frequency characteristics [START_REF] Avramov | Regions of random oscillations of discrete mechanical systems with piecewise-linear elastic characteristics[END_REF][START_REF] Avramov | Resonance in random oscillations of discrete mechanical systems with piecewise-linear elastic characteristics[END_REF][START_REF] Avramov | Features of forced oscillations in nonlinear power trains[END_REF][START_REF] Belomyttsev | An algorithm for solution of a nonlinear boundary-value problem for ordinary differential equations in the nonuniqueness region[END_REF]]. An amplitude surface is a three-dimensional generalization of an amplitude-frequency characteristic. The amplitude surfaces of a vibropercussion system can be constructed by a numerical method. Such a surface makes it possible to reveal various codimension two bifurcations formed by joining tangent-bifurcation lines.

Let us consider a vibropercussion system with two constraints whose reactions are determined by the nonlinear dependence R c n ( ) . ξ ξ ξ = + In terms of ordinary dimensionless variables [START_REF] Babitskii | The Theory of Vibropercussion Systems[END_REF][START_REF] Shaw | A periodically forced piecewise linear oscillator[END_REF], the equations of motion of the vibropercussion system has the form ω αω

2 2 ′′ + ′ + = x x H x f t ( ) sin( ), H x x c x x x x cx n n ( ) ( ) , , , , ( ) ( =
+ + >- - --< <- + + - + + 1 1 1 2 1 1 2 1 1 λ λ 2 1 2 1 λ λ + < - ⎧ ⎨ ⎪ ⎩ ⎪ ) , , n x (1)
where ωis the frequency of perturbation, α is the coefficient of viscous resistance, and 2λis the distance between the constraints. Note that the function H x ( ) = -1 for --< <-2 1 1 λ

x describes motion between the two constraints. Equation (1) accounts for the action of a constant force on a material point. System (1) can be rearranged for the state variables ( , ) ( , ) y y x x 1 2 = ′ , which are used below.

2. The Method of Amplitude Surfaces. Amplitude surfaces are convenient for the geometrical representation of the bifurcation behavior of limit cycles upon change in two characteristic parameters of the system. Let such parameters be ωand f and the amplitude surface be the dependence of the semi-range S of periodic motions on ω and f. Therefore, a function

S G f = ( , )
ω represents the amplitude surface in the parameter space ( , , ) S f R ω ∈ 3 . Each point on the amplitude surface corresponds to a certain limit cycle.

Following [START_REF] Kawakami | Bifurcation of periodic responses in forced dynamic nonlinear circuits: computation of bifurcation values of the system parameters[END_REF], let us group the limit cycles of system (1) according to the values of the multiplicators ( , ) ρ ρ 1 2 as follows: directly unstable cycles ( , ) ρ ρ

1 2 1 0 1 > < < , inversely unstable cycles ( , ) ρ ρ 1 2 1 1 0 < --< < , and stable cycles (| | , , ) ρ i i < = 1 1 2 .
Cycles of one type can transform into cycles of other type through local bifurcations. As is known, system (1) can give rise to codimension one tangent bifurcations and period-doubling bifurcations. Note that ρ 1 = 1 for tangent bifurcations and ρ 1 = -1 for period-doubling bifurcations. Let us find what the geometrical representation of bifurcations is from the standpoint of the amplitude surface. Note that amplitude surfaces contain period-doubling and tangent-bifurcation lines. It is clear that the tangent-bifurcation line is one along which stable cycles of order N join with directly unstable cycles of the same order and the period-doubling line is where three types of cycles join together: stable cycles of order 2N, inversely unstable cycles of the same order, and stable cycles of order 2N. Note that bifurcation lines can join at codimension two bifurcation points [START_REF] Allam | Crossroad area-dissymetrical spring area-symmetrical spring area and double crossroad area-double spring area transitions[END_REF][START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF][START_REF] Mira | The dovetail bifurcation structure and its qualitative changes[END_REF].

Let us now describe the amplitude-surface algorithm. First, the amplitude-frequency characteristic is designed to find bifurcation points. Second, these bifurcation points are used to draw bifurcation lines on the parameter plane ( , )

ω f R ∈ 2 .
Finally, the amplitude surface is constructed based on the amplitude-frequency characteristic and the bifurcation lines. Thus, the amplitude-surface algorithm includes two algorithms: one is intended for designing amplitude-frequency characteristics and the other for drawing bifurcation lines on the plane ( , ) ω f . To plot the amplitude-frequency characteristic, let us take advantage of the algorithm from [START_REF] Belomyttsev | An algorithm for solution of a nonlinear boundary-value problem for ordinary differential equations in the nonuniqueness region[END_REF][START_REF] Seydel | Nonlinear Computation[END_REF]. Omitting the description of this algorithm, let us dwell on the bifurcation-line algorithm for system [START_REF] Avramov | Regions of random oscillations of discrete mechanical systems with piecewise-linear elastic characteristics[END_REF]. The solutions of Eqs. [START_REF] Avramov | Regions of random oscillations of discrete mechanical systems with piecewise-linear elastic characteristics[END_REF]

0 = ∂ ∂ - ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = det (3)
where

∂ ∂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ( , ) ( , ) y y y y 1 2 01 02
is the Jacobian matrix and E is a unit matrix, describe cycles undergoing bifurcations.

A bifurcation line can be represented by a mesh function ( , ), ( , ),... , ω ω

1 1 2 2 f f ( , ). ω M M f
Each point of the mesh function is the solution of the system of equations ( 2), [START_REF] Avramov | Features of forced oscillations in nonlinear power trains[END_REF]. Therefore, to obtain the bifurcation line M, it is necessary to solve system (2), (3). It can be solved numerically as follows. First, system (2), ( 3) is resolved for one of the vectors v 1

0 1 0 2 = ( , , ) ω y y and v 2 0 1 0 2 = ( , ,
). ω y y

If the vector of unknowns is either v 1 or v 2 , then either f or ω, respectively, is specified prior to solving the equations. The variable specified is called a parametrizing variable [START_REF] Seydel | Nonlinear Computation[END_REF]. The parametrizing variable is selected before the computation of each point of the bifurcation line by the algorithm described in [START_REF] Seydel | Nonlinear Computation[END_REF]. Equations ( 2) and (3) are solved by Newton's method. Let us determine the Jacobian matrix appearing in Newton's method. Let π ω 1 = and π 2 = f for brevity; then the Jacobian matrix can be written in the form 01 02

∂ - - ∂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ∂ ∂ ( , , ) ( , , ) ( , )
1 2 0 π 1 02 , ) y E P Q R - ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ∂ ∂ - ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ - ∂ ∂ ∂ ∂
π π 1 0 2 ∂ ∂ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ W y W i π . ( 4
)
Since the finite-difference formulas used to find the derivatives in (4) degrade the convergence of Newton's method, we will take advantage of the following method to determine the elements of the Jacobian matrix.

It is well known that the submatrix

∂ ∂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ( , ) ( , ) y y y y 1 2 01 02
is the solution of the system of differential variational equations

d dt y y y y V t y y y y ∂ ∂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ∂ ∂ ( , ) ( , ) [ ( )] ( , ) ( , 1 2 01 02 1 2 01 02 ) , ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ∂ ∂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⏐ ⏐ ⏐ = = - - = ( , ) ( , ) [ ], [ ( )] y y y y E V t t 1 2 01 02 0 2 2 1 α ω ω dH dy 2 1 0 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ . ( 5
)
To derive the submatrix [P], let us set up two systems of differential equations for the vectors

∂ ∂ ∂ ∂ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ y y i i 1 1
π π i = 1 or 2 by differentiating (1) with respect to π i . Written in a matrix form, these systems look like this:

d dt y y f y y f ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 1 1 2 2 ω ω = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ + + [ ( )] V t y y f y y f y 1 1 2 2 2 1 3 2 2 ω ω α ω ω H y f t t ( ) sin( ) sin( ) . 2 3 2 2 1 0 0 - ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ω ω (6)
The solutions of Eqs. ( 6) with zero initial conditions for t N = 2π are the elements of the submatrix [P] from (4). Let us now derive the submatrices [Q] and [R] appearing in [START_REF] Belomyttsev | An algorithm for solution of a nonlinear boundary-value problem for ordinary differential equations in the nonuniqueness region[END_REF]. Using (4), we find

∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ -
ρ ν 0 02 1 2 ν ν ∂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = y , , . (7)
The formula for ∂ ∂

W i π (i = 1, 2
) is similar. To determine the second derivatives appearing in [START_REF] Allam | Crossroad area-dissymetrical spring area-symmetrical spring area and double crossroad area-double spring area transitions[END_REF], let us set up systems of differential equations for the vectors consisting of these second derivatives by differentiating ( 5) with respect to y 0μ and π i .

Written in a matrix form, these systems look like this:

d dt y y y y f y y y y y y y ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 1 0 0 2 1 0 2 1 0 2 2 0 0 2 μ ν ν ν μ ν ω 2 0 2 2 0 2 1 0 0 ∂ ∂ ∂ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = ∂ ∂ ∂ f y y y V t y y y ν ν μ ν ω [ ( )] ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 1 0 2 1 0 2 2 0 0 2 2 0 2 2 y f y y y y y y y f y y y ν ν μ ν ν ω ω 0ν ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ + - ∂ ∂ ∂ ∂ - ∂ ∂ ∂ ∂ ∂ 1 2
Equations ( 8) have zero initial conditions. Let us solve Eqs. [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF]

= + = - - - = ∑ σ σ δ τ τ δ ( ) ( ), ( ) ( ) | ( )| ( T y T p ν ν ν ) | ( )| , 1 1 2 = ∑ (9)
where q y ( ) 2 is a nonlinear function, δ( ) ⋅ is the delta function, τ j j p ( , ), = 1 1 and T p ν ν ( , ) = 1 2 are values of t found from the nonlinear equations

y y T j 2 2 1 1 2 ( ) , ( ) . τ λ ν = - = -- (10)
The solutions of Eqs. ( 8) are represented as

∂ ∂ ∂ = + ∂ ∂ ∂ = + 2 0 0 2 0 y y y u y f y w p i i i i i i μ ν μν μν ν ν ν ν ( ) ( ) ( ) ( ) , , ∂ ∂ ∂ = + 2 0 y y r s i i i ω ν ν ν ( ) ( ) , i = = = 1 2 1 2 1 2 , , , , , . μ ν (11)
The right-hand sides of relations (1) satisfy the matrix equations [ ]

d dt u w r u w r V t u μν ν ν μν ν ν μ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ν ν ν μν ν ν ( ) ( ) ( ) ( ) ( ) ( ) 1
⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ , (12)
[ ]

d dt p s p s V t v v v μν ν ν μν ν ν μ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ν ν ν μν ν ν ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 2 p s p s v ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + - ∂ ∂ ∂ ∂ - ∂ ∂ ∂ ∂ ∂ q y y y y y q y y f y y y ( ) ( ) 2 2 2 0 2 0 2 2 2 2 0 2 1 2 ω ω α ω ν μ ν ∂ + ∂ ∂ - ∂ ∂ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ y dH dy y y q y y y y 0 3 2 2 0 2 2 2 2 0 2 0 0 0 ν ν ν ω ω ω ( ) ⎥ ⎥ ⎥ . ( 13
)
The linear differential equations ( 12) have semianalytic solutions owing to the delta functions appearing in them. To write the solutions of Eqs. [START_REF] Shaw | A periodically forced piecewise linear oscillator[END_REF], we use the method of variations of constants and the Ostrogradskii-Liouville formula,

u N y y y y T y i t T k p t T i k k k μν ν μ π ( ) ( ) | | 2 2 0 1 2 0 2 2 = ∂ ∂ ∂ ∂ - ∂ = = = ∑ Δ ∂ ∂ ∂ = = = ∑ y y y t j p t i j j j 0 1 2 0 1 ν τ μ τ τ | | ( ), Δ (14) w N y f y y T y f i t T k p t T i k k k ν ν π ( ) ( ) | | | 2 2 1 2 0 2 2 = ∂ ∂ ∂ ∂ - ∂ ∂ = = = ∑ Δ t j p t i j j j y y = = = ∑ ∂ ∂ τ ν τ τ 2 0 1 | ( ), Δ (15)
r N y y y T y i t T k p t T i k k k ν ν π ω ω ( ) ( ) | | | 2 2 1 2 0 2 2 = ∂ ∂ ∂ ∂ - ∂ ∂ = = = ∑ Δ t j p t i j j j y y = = = ∑ ∂ ∂ τ ν τ τ 1 2 0 1 | ( ), Δ (16)
P 4 E 1 E 4 E 3 E 2 (D) P 3 (I) P 2 P 1 P 1 (D) P 2 P 3 L 2 pd2
where

Δ i i t N t z z y z y y y y ( ) exp | ( )| | | = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∂ ∂ ∂ ∂ = 2 2 1 01 2 2 02 α ω ω π = = = - ∂ ∂ ∂ ∂ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ z i t N t z y y y y 02 2 2 01 | | . π (17)
Thus, formulas (4)-(17) allow determining numerically the Jacobian matrix. The solutions of system (2), (3) and the bifurcation line in the form of a mesh function are found by Newton's method. Let us further analyze the bifurcation behavior of a vibropercussion system with the following parameters: c = 0, α = 0.005, and λ = 5. In this case, the system remains essentially nonlinear and approximately describes the oscillations of the power train in an internal-combustion engine [START_REF] Avramov | Features of forced oscillations in nonlinear power trains[END_REF]. The function H x ( )becomes simpler and formulas ( 5)-( 17) remain the same. The software package used to analyze the vibropercussion system allows any value of c.

Codimension Two Bifurcation Points.

Codimension two bifurcation points emerge on the parametric plane ( , ) ω f where the bifurcation lines join [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields[END_REF]. To help visualizing the situation, Fig. 1 and2 show fragments of amplitude surfaces. The bifurcation points shown have been obtained in a numerical analysis of limit cycles of the principal resonance and subharmonic second-order oscillations. These oscillations occur because of period-doubling bifurcations of limit cycles of the first order. As is seen from Figs. 1 and2, the amplitude surface includes joined sheets. A sheet with directly unstable cycles is denoted by D and a sheet with inversely unstable cycles by I.

The points L 1 , L 2 , and L 3 were earlier discovered in mappings [START_REF] Allam | Crossroad area-dissymetrical spring area-symmetrical spring area and double crossroad area-double spring area transitions[END_REF][START_REF] Mira | The dovetail bifurcation structure and its qualitative changes[END_REF] and are called codimension two tangent bifurcation points. The tangent-bifurcation lines at these points are denoted by tg1-tg6 and the period-doubling lines by pd1-pd6. Note that the points L 1 , L 2 , and L 3 differ by the arrangement of the period-doubling lines. According to [START_REF] Allam | Crossroad area-dissymetrical spring area-symmetrical spring area and double crossroad area-double spring area transitions[END_REF], the region on the amplitude surface near L 1 is called a spring area. This area contains a sheet of directly unstable cycles bounded by tangent-bifurcation lines tg1 and tg2. According to [START_REF] Allam | Crossroad area-dissymetrical spring area-symmetrical spring area and double crossroad area-double spring area transitions[END_REF], the regions near L 2 and L 3 are called crossroad and saddle areas, respectively. The amplitude surface near L 4 was earlier observed in a one-dimensional polynomial mapping [START_REF] Mira | The dovetail bifurcation structure and its qualitative changes[END_REF]. The point L 5 was earlier discovered in a mapping of dimension two. This point is where the two tangent-bifurcation lines tg9 and tg10 join. And the point L 6 is where the four tangent-bifurcation lines E 1 , E 2 , E 3 , and E 4 meet. These lines and the corresponding sheets satisfy the following relations: E P P E P P

K 8 ω f D 1 D 2 P 3 P 1 A 7 A 6 D 5 D 4 K 3 (D) K 4 D 6 (I) K 5 D 7 D 8 K 1 K 3 (D) D 1 (D) (D) (D) (D) D 4 D 3 D 2 (I) (I) K 8 P 6 K 7 K 5 K 6 K 2 K 4 K 1 D 8 D 7 K 2 (I)
4. Principal Resonance. Let us consider bifurcations of limit cycles with the period of a perturbation action. A fragment of the amplitude surface representing this bifurcation behavior is shown in Fig. 3. Let us select the parameters so that the cycle appears on the right of the period-doubling line D 6 . As ω decreases, the cycle intersects this line, and stable cycles become inversely unstable. The bifurcations D 5 and D 6 give rise to period-doubling cycles, which are not analyzed here. Because of the tangent bifurcation K 4 , the stable cycles near the period-doubling line D 5 turn into directly unstable cycles. The sheet of these cycles is bounded by the tangent-bifurcation lines K 3 and K 4 , which join at the codimension two bifurcation point P 1 . The stable cycles near the bifurcation line K 3 have the maximum amplitude of oscillations compared with the other cycles shown on the amplitude surface. Crossing the period-doubling line D 4 , these cycles go over into inversely unstable cycles. Analyzing Fig. 3, let us trace the bifurcation behavior of the system. Note that the tangent-bifurcation lines K 1 and K 2 meet at the codimension two bifurcation point P 3 and the lines K 5 and K 6 at the codimension two bifurcation point P 5 . The domain of the amplitude surface containing the bifurcation lines K 5 , K 6 , D 2 , and D 7 and the codimension two bifurcation point P 5 is called a crossroad area. The lines K 5 and K 7 join at the codimension two bifurcation point P 6 .

Conclusions.

A new method has been proposed for the bifurcation analysis of mechanical systems. The method is based on amplitude surfaces, which allow visualizing the bifurcations of mechanical systems. The amplitude-surface algorithm has been detailed and used to analyze the bifurcation behavior of a vibropercussion system, as an example. Owing to the results presented here, various bifurcations of vibropercussion systems have been revealed. They are codimension two bifurcations formed by joined tangent-bifurcation lines on the amplitude surface and corresponding to the principal resonance and subharmonic oscillations of the second order.


P Van Isacker

A O Macchiavelli

P Fallon

S Zerguine

Properties of isoscalar-pair condensates

Keywords: numbers: 21.60.Cs, 21.60.Fw, 03.75.Fi

It is pointed out that the ground state of n neutrons and n protons in a single-j shell, interacting through an isoscalar (T = 0) pairing force, is not paired, J = 0, but rather spin-aligned, J = n. This observation is explained in the context of a model of isoscalar P (J = 1) pairs, which is mapped onto a system of p bosons, leading to an approximate analytic solution of the isoscalar-pairing limit in jj coupling.

I. INTRODUCTION

In 1958, Bohr, Mottelson, and Pines [1] suggested a possible analogy between the excitation spectra of nuclei and those of the superconducting metallic state. Since then, a wealth of experimental data have been accumulated, supporting the important role played by pairing correlations in defining properties of atomic nuclei, such as deformation, moments of inertia, alignments, etc. [START_REF] Brink | Nuclear Superfluidity: Pairing in Finite Systems[END_REF][START_REF]Fifty Years of Nuclear BCS[END_REF]. Today, the study of pairing correlations continues to be a subject of active research in nuclear physics, with an emphasis in exotic nuclei. Of particular interest is the understanding of the role played by the isoscalar (T = 0) and isovector (T = 1) pairing forces [START_REF] Frauendorf | [END_REF] in the structure of N ≈ Z nuclei.

Given the charge independence of the nuclear force, T = 1 pairing is on an equal footing between the T z = 0 neutron-proton (np) and |T z | = 1 neutron-neutron and proton-proton (nn and pp) components. In addition, we have the unique possibility of studying the formation of a condensate of T = 0 np pairs, thus implying the possible co-existence of "Cooper" pairs of isoscalar and isovector type. Although the nuclear force is stronger in the T = 0 channel, it is still not clear how effective the (in-medium) T = 0 correlations are in giving rise to a ground-state isoscalar condensate [START_REF] Frauendorf | [END_REF].

In this paper we consider some interesting properties of the isoscalar condensate in the jj coupling scheme, in particular with regards to its angular momentum. Our motivation starts by studying the numerical results of a shell-model calculation, within the space of single-particle spin-orbit partners, showing that when the isoscalar component is dominant, the ground-state is not paired to J π = 0 + but, rather, it behaves as a state of aligned 1 + quasi-deuterons. To gain further insight into the peculiar structure of these condensates, we develop a boson mapping of the shell model, leading to an approximate analytic solution. Group-theoretical solutions of the pairing problem are known in the isoscalar and isovector limits of LS coupling [5][6][7] and in the isovector limit of jj coupling [8,9] but, to our knowledge, not in the isoscalar limit of jj coupling. Our results therefore provide, for the first time, approximate analytic formulae for the energies of the lowest states in that case. While we are of course aware that this limit is not applicable to real nuclei, these states might exist close to the ground state in specific regions of the N = Z line [10,11] and, more interestingly perhaps, could be realized in atomic traps.

II. SINGLE-j SHELL MODEL

Single-shell models that capture the main ingredients of the problem provide a useful framework to understand the competition of isovector and isoscalar pairing interactions. Here we start by considering the scattering of (L = 0, S = 0, T = 1) nn, np, and pp pairs as well as (L = 0, S = 1, T = 0) np pairs, describing the large spatial overlap of the nucleons' wave function in a relative L = 0 state. In the jj coupling scheme the spin-orbit splitting v s increases the energy required to form the (L = 0, S = 1, T = 0) state, thus favoring a (J = 1, T = 0) quasi-deuteron configuration. It then seems of interest to consider a more realistic case, namely that of a single-j shell that incorporates the jj coupling scheme, more appropriate in heavier nuclei. The difference between these simple LS and jj models has been discussed in terms of the BCS approximation [12].

Our approach to study this problem is to use the shellmodel code OXBASH [13] with an effective two-body force of the form

V (g, x) = -xg VJ=0,T =1 -(1 -x)g VJ=1,T =0 , (1)
FIG. 1: Energies (in units of the pairing strength g) of the lowest two T = 0 states for N = 4 particles in an f 7/2 shell as a function of the relative mixture x of isovector and isoscalar pairing.

with

VJ,T = 1 2 (a † jt × a † jt ) (J,T ) • (ã jt × ãjt ) (J,T ) , (2)
where a † jmj tmt creates a nucleon with angular momentum j and projection m j , isospin t = 1 2 and projection m t , and with ãjmjtmt = (-) j+mj +t+mt a j-mj t-mt . The notation × implies the coupling to angular momentum J and isospin T , and the dot • denotes a scalar product in angular momentum and isospin. The Hamiltonian (1) models the mixture of the two types of competing pairing interactions by the parameter x, with x = 0 corresponding to the isoscalar and x = 1 to the isovector limits respectively. The sign convention in Eq. ( 1) is such that g is positive for an attractive interaction V (g, x). We consider two spin-orbit partners f 7/2 and f 5/2 and study the low-lying spectra obtained as a function of the splitting v s . In the limit v s = 0 we recover the results of the LS coupling scheme. The results for v s V agree with those obtained for a single f 7/2 level only.

The intriguing phenomenon that motivated this study is seen in Fig. 1, showing the evolution of the two lowest states in the N = 4 particle system as a function of x. For an appreciable amount of isoscalar pairing (x < ∼ 0.4) the ground state changes from the expected 0 + to a 2 + state. Moreover, as seen in Fig. 2, the ground state for N = 6 is a 3 + and not 1 + , and so on for more particles. Considering that, for two particles interacting with the force (1), x ∼ 0 favors deuteron-like pairing with angular momentum J = 1, it appears that the ground state of the many-particle system prefers the aligned configuration of the n = N/2 pairs, i.e. the configuration with J = n.

We can trace back the change in the properties of the ground state to the spin-orbit splitting. In Fig. 3 we show FIG. 2: Energies (in units of the pairing strength g) of the lowest two T = 0 states for N = 6 particles in the f 7/2 shell as a function of the relative mixture x of isovector and isoscalar pairing.

the results for the N = 4 system and a pure isoscalar force. The energies of the 0 + and 2 + states are plotted as a function of the spin-orbit splitting v s . The two states cross, with a 0 + ground state in LS coupling, which becomes a 2 + in jj coupling, as we saw above.

To obtain an estimate of the critical value v * s at which the switch occurs, we consider the case of the N = 2 system, also shown in Fig. 3. Taking the limit of large j, to simplify the LS-jj re-coupling coefficients, we have in jj coupling E jj (1 + ) = -g and in LS coupling E LS (1 + ) ≈ -6g.

The 3 S 1 state can be written in terms of the jj-coupled wave functions as [14]

| 3 S 1 ≈ 1 √ 6 |j 2 > + 2 √ 6 |j > j < - 1 √ 6 |j 2 < , (3)
from which we can treat perturbatively the effect of the spin-orbit splitting v s . This gives for an intermediate coupling

E IC (1 + ) ≈ E LS (1 + )+ 1 6 2v s + 4 6 v s = E LS (1 + )+v s . ( 4
)
The critical value is obtained when the energy above equals that of the jj-coupling limit (dashed lines in Fig. 3)

E jj (1 + ) = E LS (1 + ) + v * s ( 5
)
and we find (in the large-j limit)

v * s g ≈ 5. ( 6
)
For the particular case of the f 7/2 -f 5/2 pair (finite-j) we find a value of ∼ 3.5, in agreement with the estimate shown in Fig. 3 (shaded area). To shed further light on the properties of the isoscalar condensate discussed above, we develop in the next section a description based on a mapping to interacting p bosons of angular momentum J = 1 and isospin T = 0. Based on the results above, and on the fermionic nature of the problem, we anticipate that the residual interaction between these bosons favors their aligned coupling.

III. ISOSCALAR PAIRING BETWEEN FERMIONS IN A SINGLE j SHELL

Consider a system of N particles, n neutrons and n protons, in a single-j shell, interacting through an isoscalar pairing interaction with angular momentum J = 1 and isospin T = 0, corresponding to the x = 0 limit of Eq. ( 1), V (g, x = 0) = -g V10 .

As discussed in the previous section, a possible strategy for simplifying the problem starts from the observation that, by definition of the interaction, the dominant pair in the two-particle system has J = 1 and T = 0. We attempt to represent a subset of the 2n-particle eigenstates of this interaction, including hopefully those at lowest energies, in terms of a single state |P ≡ P † |o (with |o the vacuum), which has J = 1 and T = 0,

P † M J ≡ (a † jt × a † jt ) (J=1,T =0) M J ,M T =0 . (7)
The natural framework to test this idea is provided by the nucleon-pair shell model (NPSM), which assumes a basis constructed from nucleon pairs [START_REF] Chen | [END_REF][16][17][18]. In this ap-proximation the full T = 0 shell-model space is truncated to one constructed out of P pairs with basis states |P n J 2 . . . J n-1 J that are proportional to

• • • P † × P † (J2) × P † (J3) × • • • × P † (J) |o . (8)
This 2n-particle state is characterized by the set of intermediate angular momenta {J 2 , . . . , J n-1 }, with J 1 = 1 and J n = J, the total angular momentum of the state.

All pairs have T = 0 and the coupling in isospin need not be considered. In principle, several intermediate couplings {J 2 , . . . , J n-1 } are possible for a given total angular momentum J. Such is the case for arbitrary pairs but not for P pairs since the number of independent states with angular momentum J constructed out of n P pairs cannot exceed the corresponding number constructed out of n p bosons, which is 1 if n-J is non-negative and even, and 0 otherwise. We conclude therefore that, for a given J, at most one state |P n J 2 . . . J n-1 J exists, for which the intermediate angular momenta can be chosen as

J i = i mod 2, 1 ≤ i ≤ n -J, J i = i -n + J, n -J ≤ i ≤ n, ( 9
)
where it is implicitly assumed (as will be from now on) that n -J is non-negative even. We denote normalized states as |P n J , tacitly assuming the intermediate coupling convention (9). In this convention the paired and spin-aligned states of particular interest here correspond to the choice paired :

J i = i mod 2, 1 ≤ i ≤ n, spin-aligned : J i = i, 1 ≤ i ≤ n. (10)
As long as n ≤ (2j + 1)/2 all states (9) exist. This is no longer necessarily true if the shell is more than half filled, in which case it is advantageous to reconsider the problem in terms of holes. We then construct basis states | P 2j+1-n J 2 . . . J n-1 J that are proportional to

• • • P × P (J2) × P (J3) × • • • × P (J) |õ , (11)
where |õ represents a full shell and P annihilates a P pair,

PM J ≡ (ã jt × ãjt ) (J=1,T =0) M J ,M T =0 . ( 12
)
The angular momentum and anti-symmetry considerations concerning the states ( 8) and ( 11) are the same, and consequently the latter lead to the same allowed basis states (9) with n replaced by n ≡ 2j + 1 -n. We denote such states as | P nJ .

In general, |P n J and | P nJ are not the same state,

|P n J = | P nJ , (13)
and it is possible that the state on the left-hand side exists while the one on the right-hand side does not (or vice versa). Only if the shell-model state with a given J and T = 0 is unique, do the particle and hole representations become equivalent, as is the case, for example, for the states

|P 2j+1 J = 0 = |õ , |P 2j J = 1 = | P J = 1 . (14)
The choice |P n J if n ≤ (2j +1)/2 and | P nJ if n ≥ (2j + 1)/2, apart from being computationally simpler, gives the best approximation of shell-model states in terms of P pairs. The summary of the above discussion is that the truncated shell-model basis constructed out of P pairs is spanned by the states |P n J if n ≤ (2j + 1)/2 and by the states | P nJ if n ≥ (2j + 1)/2. These basis states exist (provided n -J or n -J is non-negative even) and are unique for a given n and J, so that no additional labels are needed. Therefore, in the P -pair approximation of the NPSM, the correlation energy due to isoscalar pairing in the state with n neutrons and n protons, coupled to total angular momentum J and isospin T = 0, is1

Ẽf (n, J) ≡ P n J| -g V10 |P n J , (15)
for n ≤ (2j + 1)/2, and by Ẽf (n, J)

≡ P nJ | -g V10 | P nJ , (16)
for n ≥ (2j + 1)/2. The computation of the matrix elements of an arbitrary interaction between nucleon-pair states is possible with the recurrence relation devised by Chen [16]. In the general formulation of the NPSM care should be taken of the over-completeness and nonorthogonality of the pair basis. This is not an issue in the present application since basis states are unique for a given n and J. It should be stressed that Eqs. ( 15) and ( 16) yield an approximation to the exact isoscalarpairing correlation energy.

The energy E f (n, J) of a particle state is calculated with respect to the vacuum |o while that of a hole state, E f (n, J), is with respect to the full shell |õ . The particlehole transformation gives a relation between both quantities, which is exact in the full shell-model space. For our particular case of isoscalar pairing this relation is

E f (n, J) = - 3(2j + 1 -2n) 2j + 1 g + E f (n, J). (17)
We use the same equation to relate the approximate energies Ẽf (n, J) and Ẽf (n, J). In the following absolute energies are quoted with respect to the vacuum |o . For a particle state they are obtained directly while for a hole state they follow from Eq. (17).

A further approximation is to replace the P pairs by p bosons, with single-boson energies and boson-boson interactions derived from the two-particle and four-particle systems, respectively.

With use of the OAI mapping [19] a p-boson Hamiltonian Ĥb is obtained, which can be written as

Ĥb = p p † • p + 1 2 λ=0,2 v b λ (p † × p † ) (λ) • (p × p) (λ) , ( 18
)
where p is the p-boson energy and v b λ are the two-body interaction matrix elements between the p bosons. The definition of the adjoint operator pm ≡ (-) 1-m p -m ensures that pm is an annihilation operator with transformation properties under rotations that are the same as those for the creation operator p † m [START_REF] Iachello | Lie algebras and applications[END_REF]. With the above definitions we have that p † • p = m p † m p m is the number operator np .

The single-boson energy is

p ≡ p| Ĥb |p . = P | -g V10 |P = -g. ( 19
)
where the notation . = is used to indicate that the equality holds by virtue of the mapping procedure. The two-body boson matrix elements with λ = 0, 2 are

v b λ ≡ p 2 λ| Ĥb |p 2 λ -2 p . = -g j 4 [10, 10]λ0| V10 |j 4 [10, 10]λ0 -2 , ( 20
)
where the bra and ket represent normalized, antisymmetric two-pair states,

|j 4 [J 1 T 1 , J 2 T 2 ]JT ∝ A|j 2 (J 1 T 1 )j 2 (J 2 T 2 )JT . (21)
The notation in square brackets [J 1 T 1 , J 2 T 2 ] implies that the state ( 21) is constructed from a parent state with intermediate angular momenta and isospins J 1 T 1 and J 2 T 2 . The anti-symmetrized states |j 4 [J 1 T 1 , J 2 T 2 ]JT can be expanded in terms of the two-pair states |j 2 (J 1 T 1 )j 2 (J 2 T 2 )JT by means of four-to-two-particle coefficients of fractional parentage (CFPs) [START_REF] Talmi | Simple Models of Complex Nuclei. The Shell Model and Interacting Boson Model[END_REF],

[j 2 (J a T a )j 2 (J b T b )JT |}j 4 [J 1 T 1 , J 2 T 2 ]JT ], (22)
which are known in closed form.

From the general expression for the matrix element [START_REF] Iachello | Lie algebras and applications[END_REF] the following results are obtained:

v b 0 /g = -6[j 2 (10)j 2 (10)00|}j 4 [10, 10]00] 2 + 2, v b 2 /g = -6[j 2 (10)j 2 (10)20|}j 4 [10, 10]20] 2 -6[j 2 (30)j 2 (10)20|}j 4 [10, 10]20] 2 + 2, ( 23
)
which, with the help of [j 2 (10)j 2 (10)00|}j 4 [10, 10]00] 2 = 2j 3 -2j + 3 3j(j + 1)(2j + 1) ,

[j 2 (10)j 2 (10)20|}j 4 [10, 10]20] 2 = 10j 3 + 9j 2 -j -3 15j(j + 1)(2j + 1) , [j 2 (30)j 2 (10)20|}j 4 [10, 10]20] 2 = 9(j -1)(j + 2)(2j + 3) 10j(j + 1)(2j + 1)(5j 2 + 7j + 3) , ( 24
)
lead to the following expressions for the p-boson matrix elements:

v b 0 = 6(j 2 + j -1) j(j + 1)(2j + 1) g j→∞ -→ 3 j + O 1 j 2 g, v b 2 = 3(4j 4 + 6j 3 + j 2 + 7j + 12) j(j + 1)(2j + 1)(5j 2 + 7j + 3) g j→∞ -→ 6 5j + O 1 j 2 g. ( 25
)
As anticipated, for an attractive isoscalar pairing interaction the boson-boson matrix elements are repulsive. This is a finite-space effect, due to the Pauli principle, since the matrix elements vanish in the large-j limit. A difference between the λ = 0 and λ = 2 matrix elements also arises due to Pauli effects, and it is seen that v b 2 is less repulsive. This favors the spin-aligned ground state, not only for two but also for more bosons as a result of the following argument.

Since a system of n interacting identical p bosons is solvable by virtue of a U(3) ⊃ SO(3) dynamical symmetry [START_REF] Iachello | Lie algebras and applications[END_REF], the eigenvalues of the Hamiltonian ( 18) are known in closed form,

E b (n, J) = n p + n(n + 1) -J(J + 1) 6 v b 0 + 2n(n -2) + J(J + 1) 6 v b 2 , (26)
where the allowed angular momenta are J = n, n -2, . . . , 1 or 0. The only possible ground states of a pboson system are either paired or spin-aligned [START_REF] Law | [END_REF]22].

The paired state has J = 0 or J = 1 with energies

E b (n, J = 0) = n p + n(n + 1) 6 v b 0 + n(n -2) 3 v b 2 , (27) E b (n, J = 1) = n p + (n -1)(n + 2) 6 v b 0 + (n -1) 2 3 v b 2 ,
depending on whether n is even or odd, respectively. The spin-aligned state has J = n with energy

E b (n, J = n) = n p + n(n -1) 2 v b 2 . (28)
The breaking of the rotational invariance in gauge space [23] leads to the emergence of isoscalar pairing rotational bands, as seen in the quadratic dependence of the energies as a function of the number of pairs n, Eqs. ( 27) and (28).

The difference in energy between the paired and the spin-aligned states can be written as

∆ b (n) = (n -n 2 )(n + 1 + n 2 ) 6 (v b 0 -v b 2 ) ≈ g 3(n -n 2 )(n + 1 + n 2 ) 10j , (29)
where n 2 is 0 for even n and 1 for odd n, n 2 ≡ n mod 2. This shows that for all n the difference in energy between TABLE I: Exact energies E f (n, J) of paired (J = 0 or 1) and aligned (J = n) states with T = 0 of a system of n neutrons and n protons in a single-j shell interacting through an isoscalar pairing force, in units of the strength g, and the corresponding energies Ẽf (n, J) and E b (n, J) obtained in the P -pair and p-boson approximations. A dash -means that a P -pair state does not exist while the absence of an entry indicates that the numerical result could not be obtained. the paired and the spin-aligned states is positive for an attractive pairing interaction, that is, the spin-aligned configuration is the ground state.

E(n, J) j = 7/2 j = 9/2 j = 11/2 j = 13/2 j = 15/2 E f (2, 0) -1.298 -1.424 -1.514 -1.580 -1.631 Ẽf (2, 0) -1.298 -1.424 -1.514 -1.580 -1.631 E b (2, 0) -1.298 -1.424 -1.514 -1.580 -1.631 E f (2, 2) -1.793 -1.825 -1.847 -1.865 -1.879 Ẽf (2, 2) -1.757 -1.799 -1.828 -1.850 -1.866 E b (2, 2) -1.757 -1.799 -1.828 -1.850 -1.866 E f (3, 1) -1.793 -
We recall that the preceding results, valid for an isoscalar pairing interaction in a single-j shell, are derived under the following simplifying assumptions:

1. The full shell-model space is truncated to one constructed out of P pairs. The expectation value of the isoscalar pairing Hamiltonian -g V10 in the (unique) P -pair state takes fully account of the Pauli principle and leads to the approximate correlation energy Ẽf (n, J).

2. The fermionic Hilbert space constructed out of P pairs is mapped onto a corresponding bosonic Hilbert space constructed out of p bosons. The mapping of the Hamiltonian is carried out in the two-and four-nucleon spaces and leads to a boson Hamiltonian with up to two-body interactions.

3. The boson Hamiltonian is used to calculate the energies E b (n, J) of n-boson states.

To gauge the adequacy of the different approximations, we show in Table I the exact energies E f (n, J) (wherever they can be calculated) and the corresponding approximations Ẽf (n, J) and E b (n, J) for 7/2 ≤ j ≤ 15/2. Several comments are in order. First of all, we observe the identity

Ẽf (n = 2, J) = E b (n = 2, J), (30)
that is, the P -pair spectrum of the four-particle system coincides with that obtained for two p bosons. This is a generic property of the mapping and follows from the fact that up to two-body interactions between the bosons are considered. In fact, if up to q-body interactions are considered, the identity (30) remains valid up to the n = q. Secondly, we observe the identity

E f (n = 2, J = 0) = Ẽf (n = 2, J = 0). (31)
This is not a generic property but is valid for the isoscalar pairing interaction, for which |P 2 J = 0 decouples from the rest of the shell-model space. This property of the isoscalar pairing interaction was already pointed out by Fu et al. [24] on the basis of analytic expressions for fournucleon overlaps. Furthermore, we observe from Table I the following hierarchy:

E f (n, J) ≤ Ẽf (n, J) ≤ E b (n, J), (32)
valid for any j, n, and J. The first inequality results from the fact that the lowest eigenvalue of any Hamiltonian in a certain Hilbert space is lower than the lowest eigenvalue of the same Hamiltonian in a truncated subspace.

We remark that an equality E f (n = 4, J) = Ẽf (n = 4, J) can be obtained by constructing effective operators in the truncated space, which is not done in the present application. The second inequality in Eq. ( 32) is a consequence of performing the mapping in the four-particle systems with an unnormalized (i.e., not an effective) Hamiltonian. For a variety of bosonic systems (p, sd, sdp, etc.) we have consistently found that the boson Hamiltonian, as it is derived here from the four-particle system, gives an upper limit for the fermionic interaction energy of the n-particle system.

It is seen from Table I that the quality of the approximation varies with j, n, and J. Two effects are rather obvious: the approximation becomes (i) better with increasing j and (ii) worse with increasing n [as long as n ≤ (2j + 1)/2]. These effects result from the increasing importance of Pauli corrections that are neglected (i.e., beyond two-body interactions between the bosons). A more subtle effect is the dependence on J. It is seen that the approximation for the aligned state J = n is adequate, even close to mid shell, n ≈ (2j + 1)/2, and for low j. On the other hand, it is often rather poor for the paired state with J = 0 or 1. It can be conjectured that this is a generic property of phonon approximations in fermionic systems: while such descriptions are good for high-angular-momentum states, they become highly anharmonic at low angular momenta.

Despite the varying quality of the boson approximation, depending on j, n, and J, the overall conclusion is that the predicted feature of the lower energy of the aligned state as compared to the paired state is confirmed by the exact fermion calculation.

IV. CONCLUSION

We have considered some intriguing properties of a T = 0 isoscalar condensate in single j-shell, in particular with regards to its angular momentum coupling. We developed a description based on a mapping of the shell model to interacting p bosons of angular momentum J = 1 and isospin T = 0, providing for the first time approximate analytic formulae for the energies of the lowest states. Our results show that, due to the Pauli principle, the residual interaction between these bosons favors (a priori unexpected) the aligned configuration of n = N/2 quasi-deuteron pairs, i.e. that with J = n.

While we realize this limit may not be applicable to real nuclei, these states might exist close to the paired ground states in specific regions close to the N = Z line. In fact, it was shown recently in Refs. [10,11], using a phenomenological Hamiltonian within the framework of the Hartree-Fock-Bogoliubov theory, that the spin-triplet phase is favored over the spin-singlet one in the mass region A ≈ 130 with Z ≈ 64 (for example 132 Gd). This is found to depend on the occupation of specific low-j orbitals near the Fermi energy for which the spin-orbit splitting is small. More interestingly perhaps, it is envisioned that these condensates might be realized by tunable spin-orbit coupling in ultracold atomic traps [25], whereby the control parameter x in Eq. ( 1) could be adjusted to drive the system from diamagnetic to magnetic.

A full extension of the present formalism including the effect of the spin-orbit splitting will be the subject of a future publication.

FIG. 3 :

 3 FIG.3: Energies (in units of the pairing strength g) of the T = 0 ground state for N = 2 particles (top) and of the lowest two T = 0 states for N = 4 particles (bottom) as a function of the spin-orbit splitting between the f 7/2 and f 5/2 orbits and for pure isoscalar pairing. The shaded area in the top panel indicates the critical value of the spin-orbit splitting, v * s , at which the jj coupling takes on. (See text for details).

We reserve the notation E f (n, J) and E f (n, J) (i.e., expressions without tilde) for the exact correlation energy of the yrast state with angular momentum J calculated in the full shell-model space.

V. ACKOWLEDGEMENTS This work was supported in part by the FUSTIPEN (French-U.S. Theory Institute for Physics with Ex-otic Nuclei) under U.S. DOE grant No. DE-FG02-10ER41700, and by the U.S. DOE contract No. DE-AC02-05CH11231 (LBNL).