Vinh Le

Stéphanie Thai

Marie Jean-Daubias
email: stephanie.jean-daubias@liris.cnrs.fr

Blandine Lefevre
email: blandine.ginon@liris.cnrs.fr

Ginon

Model of Articulation between aLDEAS Assistance Rules

INTRODUCTION

More and more applications are used in different contexts: professional, personal and educational. However, because of technical difficulties of handling or use of these applications, their users can abandon or under-exploit them and lose motivation [START_REF] Gapenne | Defining categories of the human/technology coupling: theroretical and methodological issues[END_REF]. In the educational context, learners use various applications (pedagogical or non-pedagogical applications) to acquire knowledge (Ginon et al., 2014c). These applications must not only resolve technical difficulties as applications used in other interactive environments, but also give learners the pedagogical feedback and guidance which meet the pedagogical goals of teachers. For example, a given hint as a pedagogical feedback helps learners to do an exercise when they meet difficulties. Or, a pedagogical guidance guides learners to choose activities to work. However, these pedagogical feedback and guidance are not always supported by applications, especially nonpedagogical ones used in the educational context. Therefore, adding an assistance system is considered as a solution for both technical and pedagogical problems of an existing application. In details, pedagogical assistance systems can meet both technical (handling, use of applications) and pedagogical assistance needs (hint, explanation, guidance, etc.). However, the existing pedagogical assistance is varied and complex. It can be a complex pedagogical guidance to propose learning activities suitable to learners [START_REF] Antoniadis | NLP-based scripting for CALL activities[END_REF]. For example, the remediation activities are proposed depending on the progression of learners. In a pedagogical activity, the pedagogical assistance can have different modes to sequence of assistance events [START_REF] Melis | ActiveMath: A Generic and Adaptive Web-Based Learning Environment[END_REF], [START_REF] Winke | Hot Potatoes version 5[END_REF]. Theses modes describe the articulation between assistance elements. For instance, a successive assistance gives one message after another in order to guide learners. As part of my thesis, we identified two research issues: "How to help teachers to define the pedagogical guidance?" and "How to help teachers to define the articulation between assistance elements?" In this paper, we present our answer to second issue.

The AGATE project proposes the SEPIA system (Ginon et al., 2014a) that allows assistance designers (teachers) to add an assistance system in the existing ILE (Interactive Learning Enviroment) by creating and executing the aLDEAS rules (Ginon et al., 2014b). SEPIA supports various types of applications (windows, java, web and MacOS applications), assistance techniques (textual, vocal, enhancing, automatic actions, etc.) and it is the independent of application domains. SEPIA is a full solution to create rich assistance systems. However, definition of the articulation between assistance elements is still implicit and difficult. So, this paper presents the evolution made to SEPIA to overcome these limitations.

In this paper, first we present the AGATE project and its major results: the SEPIA system and the aLDEAS language. We also show that expression of the articulation between assistance elements of an assistance system in ILEs is a complex task. Then, we present the different existing modes of articulation through examples of assistance (section 3.1). We too confront these modes to tools that aim at the definition of assistance as well as modes of articulation (section 3.2). These studies allow us to propose a model of articulation between aLDEAS assistance rules (section 4). Then, the implementation of this model is presented (section 5). To validate our approach, we present some results from our evaluation (section 6). Finally, we give the general conclusions and the issues that motivate the future works (section 7).

SEPIA SYSTEM

The AGATE (Approach for Genericity in Assistance To complEx tasks) [START_REF] Agate | [END_REF] project aims at proposing generic models and unified tools to enable the setup of assistance systems in various existing applications, that we call targetapplications, by applying a generic and epiphytic approach. Epiphytic application is the application that is able to perform actions in another application without requiring any change to it. Thus, the functioning of an epiphytic assistance system added in the target-application doesn't disturb the functioning of this application [START_REF] Paquette | Epitalk, a generic tool for the development of advisor systems[END_REF]. The models and tools proposed are specific neither to an application nor to a domain. For that reason, we previously proposed an adjunction process of epi-assistance systems to a given targetapplication (Ginon et al., 2014b). This process (Figure 1) consists of two phases: the assistance specification and the assistance execution in an epiphytic way.

The assistance specification is performed by an expert of the target-application, called the assistance designer. This preparatory phase enables the designer to specify the assistance that he wishes for a given target-application. The assistance execution concerns end-users of the target-application. It is the execution of the assistance designed by the designer; it occurs at any use of the target-application by an end-user. The epi-detectors make possible the monitoring of the target-application. They detect the events related to interactions between the user and the target-application by exploiting the accessibility libraries compatible with a type of applications (windows, java, web applications…). Finally, the epi-assistants handle the elaboration of the answer to provide assistance to the end-user by pop-ups windows, speech or animated agent as well as highlighting a component on interfaces (for instance, by colouring a component).

Figure 1: Adjunction process of epi-assistance systems (Ginon et al., 2014b).

The aLDEAS language (a Language to Define Epi-Assistance Systems) (Ginon et al., 2014b) is proposed in order to connect the two phases of this process. aLDEAS consists of three principal elements: event wait (click on a button…), consultation (of profile, of states of application,…), assistance action (message, enhancing,…). This language is completed by a rules pattern (Figure 2) and also by other patterns facilitating the definition of assistance actions (for example, step by step pattern). A rule begins with event wait called trigger event. When this event occurs, the launch of assistance actions is immediate (upper path in Figure 2), or is constrained by a condition (lower path in Figure 2). This condition takes the form of a consultation with the alternatives each associated with one of these actions. Finally, the rule can be terminated by end event that ends all elementary actions launched by this rule. For example, the example in Figure 2 shows a rule among many rules which define an assistance system. This rule waits a click on the button 'help' in order to verify the answer of the learner and to provide an error message when this answer is not correct (text written by the learner is not equal to 1). This message is closed after 10 secs.

aLDEAS and its patterns are implemented in the SEPIA system (Ginon et al., 2014a) that consists of two tools: an assistance editor and an assistance engine. The assistance editor operationalizes the assistance specification phase (upper part in Figure 1). It provides an interface that allows the assistance designer to define an assistance system by creating a set of aLDEAS assistance rules. The assistance engine operationalizes the assistance execution phase (lower part in Figure 1). It executes the assistance system created in the previous phase by executing its set of aLDEAS assistance rules. SEPIA and aLDEAS allow creating useful assistance systems in various domains, among which ILE. However, the creation of assistance systems which can be found in ILEs is complex. Thus, the principal objective of target-applications in ILE is learning. Assistance systems for these applications must be effective, suited to learners, to their tasks in the target-application and to their progression. Additionally, they must meet the pedagogical objectives and strategies of teachers. In SEPIA, such assistance systems require the definition of a lot of rules which need to be articulated with different modes. For instance, an assistance system provides the learner with progressive assistance to solve an exercise when he asks for assistance. For the first time, this system gives an explanation, then a hint and finally a solution. An explanation can be given by showing the messages step by step. So the rules defining this assistance system are articulated in both progressive and successive modes. However, aLDEAS and SEPIA have not yet provided an explicit and easy way of definition of modes of articulation between rules. To tackle this problematic, we firstly made a state of the art in order to identify existing modes of articulation between assistance elements. We sum up this work through the examples (section 3.1) and present the support of these modes in some tools (section 3.2). Then, in order to allow aLDEAS and SEPIA to support these modes, we proposed a model of articulation between rules (section 4) and its implementation (section 5).We conclude this paper by the presentation of the evaluations that we defined for this research.

STATE OF THE ART

Modes of Articulation between Assistance Elements

Currently, pedagogical assistance is found in some applications. This assistance can be executed according to different modes to sequence of assistance events. These modes describe articulation between assistance elements. In many applications, an assistance element is given independently from another. There are not constraints between assistance elements. We can easily find this mode in most applications with the tooltips. A tooltip appears when the user hovers a component on application interface. So, each tooltip is independently showed. We take a concrete example of IXL learning (IXL Learning, 2015) that provides comprehensive, curriculum-aligned mathematics and English content for preschool to grade 12. It shows overviews of course through a sequence of independent pop-ups (A in Figure 3) when learners hovers links of course. We call this mode of articulation independent mode.

The tutorials integrated in some applications provide step by step assistance in order to guide users. For instance, Connectify (B in Figure 3) (Connectify, 2015) provides the messages one after the others which allow user to learn the use of this application. Such messages may also explain step by step user errors. Each assistance element is constraint by the end of the previous assistance element. We call this mode of articulation successive mode.

Hot Potatoes [START_REF] Winke | Hot Potatoes version 5[END_REF]) allows teachers to create different types of exercises. It is especially useful for creating online, interactive language learning exercises and for providing pedagogical assistance such as diagnosis to verify the answers of the learner. The diagnosis can be given for a several parts at the same time. For instance, an exercise created by EOLF in Franche-Comté university (EOLF, 2016) (C in Figure 3), shows correct or incorrect answers for an English exercise at the same time. We can get the assistance in the forms frequently offered on web where the diagnosis on different user inputs can be simultaneously displayed. These examples show that all assistance elements can be simultaneously given. We call this mode of articulation simultaneous mode. ActiveMaths [START_REF] Melis | ActiveMath: A Generic and Adaptive Web-Based Learning Environment[END_REF] offers to learners a mathematical web application with pedagogical assistance. The learner can ask for assistance in progressive way (D in Figure 3). At the first request, a hint is given to the learner. Then, at the next request, the given hint is more detailed. At the last request, the solution is given in order to avoid blocking the learner in working on problems. The assistance is more and more detailed and concrete. We call this mode of articulation progressive mode.

Some applications consult information sources to provide suitable assistance such as the application state, the user profile or the user's choice. Thus, both ActiveMaths and Hot Potatoes allow consulting the state of the application such as a text filled by the learner. For instance, ActiveMaths (E in Figure 3) verifies answers of the learner and shows the result (syntax error, incorrect/correct answer). Consultations are therefore essential to provide a suitable assistance. We call this mode of articulation interactive mode.

These modes of articulation can also coexist or be combined in assistance systems. For instance, tutorials provide step by step assistance, but in one step, a textual message and an enhancing can be simultaneously performed. This is a combination of successive and simultaneous modes.

Related Works

The modes of articulation presented in the previous section exist in many applications. However, some tools allow also defining these modes explicitly or implicitly. So, we confronted these modes to tools related to our approach.

In Marco advisor systems [START_REF] Richard | Enhancing the adaptivity of an existing website with an epiphyte recommender system[END_REF], the advices are represented as a graph. The subset of graph represents the assisted website. There is no explicit articulation between advices because each advice is represented independently. However, the articulation between the elements of a same advice can be conditioned by the navigation history of the user. This is a case of interactive articulation of assistance but implicitly represented. In the Astus platform [START_REF] Paquette | Designing a Knowledge Representation Approach for the Generation of Pedagogical Interventions by MTTs[END_REF], educational interventions are represented as rules organized in a graph. The conditions of the rule make explicit interactive articulation between interventions. In the same way, the Epitalk system [START_REF] Paquette | Epitalk, a generic tool for the development of advisor systems[END_REF] explicitly represents the advices through a tasks graph. These works concentrate on the definition of assistance but do not explicitly address the aspect of articulation. Therefore, the articulation may be defined explicitly or implicitly by their tools and we can't find the presence of progressive and successive modes of articulation.

In another way, the Grafcet graphical language [START_REF] David | Grafcet: A powerful tool for specification of logic controllers[END_REF] is proposed in order to represent the sequential automation in systems decomposable into steps. Although Grafcet is not specific to assistance systems, the expression of sequence of steps can inspire our work. In Grafcet, a step can be an active step, initial step, macro-step, etc. Actions are associated with a step. The transition between two steps is done through a transition. A transition is one or more logical condition (boolean). With Grafcet, we can describe explicitly the independent, successive, interactive, simultaneous modes but only implicitly the progressive mode. In addition, some elements of Grafcet are not suitable with the ones of aLDEAS. For instance, Grafcet doesn't distinguish events from conditions as aLDEAS does. It contains also useless information in our context to the phase of specification of an assistance system such as active state on step.

To overcome these limitations in the literature, we proposed a model of articulation between aLDEAS rules and implemented in SEPIA. This model and its implementation are presented in the following.

MODEL OF ARTICULATION BETWEEN ALDEAS RULES

If aLDEAS and its implementation in SEPIA already allow the definition of the articulation between assistance elements such as those presented in the section 3 with aLDEAS rules, the expression of the articulation between the rules is implicit and can be complex to define for the assistance designer. To more effectively operationalize these modes of articulation in our propositions, it is necessary to allow defining explicitly the articulation between aLDEAS rules. Thus, an assistance system is currently defined in the AGATE project by a set of aLDEAS rules always at the same level. In the aLDEAS rules pattern (Fig. 2), the trigger event, the end event and the trigger condition are central elements to form the articulation between rules. For instance, we defined two rules R1 and R2 which describe two successive steps in the tutorial of Connectify. So, these rules are articulated in successive mode. It means that R2 is launched at the end of R1. For this order of launch, the trigger event of R2 must be the event "end of R1".

On the one hand, we must carefully define elements in the rules in order to ensure correct articulation between them. On the other hand, we must examine them in order to understand which mode of articulation to choose. Therefore, this articulation between rules is implicitly expressed and is complexly defined with aLDEAS.

For these reasons, we propose to complete our language by a model of articulation between assistance rules. To simplify the representation of the model, we note that rules between which we want to make an articulation are named Ri with i [1, n], (n ≥ 2). The representation of our model is given in Figure 4. It gives an overview of the five modes of articulation that we identified from a study of existing works: independent, successive, simultaneous, progressive and interactive.

In each mode of articulation, there are constraints that rules must respect to ensure the correct articulation between them (for instance, for successive mode, each rule should be launched by the end of the previous rule). The constraints of each mode of articulation are shown in the next section with examples of assistance. These examples of assistance are inspired by examples presented in section 3. To simplify, we describe only three rules articulated for each example.

Independent Mode of Articulation

In the independent mode of articulation (Figure 4) the rules Ri are launched by their own trigger events. This mode doesn't impose any constraint. The definition of rules articulated in independent mode reflects the classical definition with aLDEAS. Obviously, the other modes presented thereafter are specific cases of this mode with specific constraints on rules.

Example A in Figure 5 is a case of assistance to IXL Learning (section 3.1). Here, we present a similar but simpler assistance written in aLDEAS which takes only the three first overviews corresponding to the three first courses. This assistance is created with 3 rules articulated in independent mode. The rules R1, R2, R3 are respectively corresponding to the three courses "Counting review -0 to 10", "Count to fill a ten frame", "Counting reviewup to 20". Each rule begins with its own trigger event "hover on link of course" in order to show a message which presents overview of this course.

Successive Mode of Articulation:

In the successive mode of articulation (Figure 4), the rules are launched one after the other, it means that at the end of the rule Ri, the rule Ri+1 is launched.

In the detailed definition of this mode of articulation (Figure 6), we can see that the rule Ri is forced to have at least an end event and Ri + 1 is forced to have a trigger event "end of Ri." This constraint is applied to all rules except the first and last ones. The first rule R1 can begin with any trigger event(s) and the last rule Rn may end with none, one or several end events. In this mode of articulation, the rule R1 is an entry point of the rules Ri. So, the trigger events of the rule R1 launch this set of rules in successive mode.

However, a rule Ri + 1 cannot be launched until the end of its preceding rule Ri. Consequently, if Ri has a trigger condition that is not validated at the time of the assistance execution, the rule and its assistance actions will not be executed until its end events, and the following rule will therefore not be launched. So, the whole sequence of rules is interrupted. This requires a rule Ri to contain a condition to have an alternative "else". This alternative ensures that the condition is always valid. Let's take the example of the assistance to Connectify (section 3.1). Here, we present an assistance similar but simpler which takes only the three first steps of the tutorial of Connectify. This assistance is created with three rules articulated in successive mode. These three rules (defined in more detail in Figure 7) must respect the constraints of the successive mode (Figure 6). Thus, the first rule R1 waits until a user's click on bouton "Tutorial" in order to show a message of welcome and closes this message after 10 seconds. Then, the rule R2 that waits until the end of R1 shows a message of internet connection check and closes this message after 10 seconds. Finally, the rule R3 that waits until the end of R2 shows a message of choice of an internet connection as well as highlight the related combo box. These message and highlight is also closed after 10 seconds.

Simultaneous Mode of Articulation

The simultaneous mode of articulation (Figure 4) allows executing several assistance rules simultaneously.

In the simultaneous mode of articulation, the rule Ri must begin the same trigger events as ones defined for this mode. When these events occur, all rules are launched at the same time.

Let's take the example of the assistance to an exercise created with Hot Potatoes that shows simultaneously all errors of answers. Here, we present an assistance similar but simpler which takes only the three first answers corresponding to three first user inputs. This assistance is created with three rules R1, R2 and R3 articulated in simultaneous mode (Figure 5). These three rules must begin with trigger event "click on bouton Check". When the learner clicks on this button, they are launched. They verify learner's answers with the consultation of the user inputs on the application and shows the result (correct or incorrect) by adding a text (OK if correct, X else) near to these user inputs.

Progressive Mode of Articulation

In the progressive mode of articulation (Figure 4), the launch of assistance rules depends on the number of times that the learner is in a same situation. In particular, this mode allows to provide the user with the assistance more and more detailed and concrete to meet a repeated request of assistance.

In the successive mode of articulation, the rule R1 is the entry point of assistance to successively start the rules Ri. In the simultaneous mode, all rules Ri start with same trigger events. However, in this progressive mode, there must be an additional rule as an entry point to constraint the launch of the rules Ri. We call this rule R'. R' launches one rule among the rules Ri, according to the number of launches of the rule R'. In the rule R', each rule Ri is associated with an interval [lefti, righti]. It means that Ri is launched for one or several times between lefti and righti. For the first times [left1, right1], R' launches R1 and for the next times [left2, right2] R' launches R2. In this mode of articulation, the rules Ri must begin with a trigger event "launch by a rule (R')".

Let's take the example of the assistance to ActiveMaths (section 3.1) that gives at first a hint, then a more detailed hint and finally the solution when the user repeatedly asks for assistance. The assistance is created with three rules articulated in progressive mode (Figure 5). R' begins with trigger event "click on bouton Help" and launches the rules R1, R2, R3 which show respectively a hint, another more detailed hint and the solution. This launch is constrained by the number of launches of R'. It means that the number of clicks on button "Help" is counted. To be launched, these three rules R1, R2 and R3 must begin with a trigger event "launch by a rule (R')". So, R1 is launched by R' for the first click on bouton "Help", R2 for the second click and R3 for the third click.

Interactive Mode of Articulation

In the interactive mode of articulation (Figure 4), one of the rules Ri is launched according to a consultation of the user profile, of the application state, of the history of the assistance, of the trace and / or of the user.

Again, a rule R' is used as an entry point for the launch of the assistance. Each rule Ri is associated with an alternative of the trigger condition of R'. Ri must begin with the trigger event "launch by a rule (R')." The progressive mode of articulation (see section 5.3) is a special case of the interactive mode, frequently encountered in the existing assistance systems and in which the trigger condition of R' is exclusively a number of launches of R'.

Let's take the example of the assistance to ActiveMath (section 3.1) which shows the diagnosis by consulting the learner's user. This assistance is created with three rules articulated in the interactive mode. The representation of the additional rule R' is the same as the representation of interactive articulation. R' begins with the trigger event "click on bouton Check" and launches the rules R1, R2 and R3 which show respectively a syntax error, a calculation error and success. This launch is constrained by the learner's answer: the value of the text box entered by the learner. To be launched, these three rules R1, R2 and R3 must begin with a trigger event "launch by a rule (R')". When the learner clicks on bouton "Check", one rule among the three rules is launched by R'. If the entered value of the text box does not belong to float type, R1 is launched, if this value is equal to 1, R3 is launched and elsewhere, R2 is launched.

IMPLEMENTATION OF OUR MODEL OF ARTICULATION BETWEEN RULES

We implemented this model of articulation between rules in SEPIA that haven't supported the explicit expression of the articulation until now (Figure 8). More concretely, we enriched the SEPIA assistance editor to support designers to define explicitly the five modes of articulation as well as to facilitate their definition. In order to facilitate the comprehension of designers, we adopt the notion of bloc that regroups the rules articulated in a given mode among these five modes.

Figure 8: SEPIA completed with the model of articulation between rules Thus, SEPIA allows assistance designers to define and view graphically blocs of rules articulated in the wished mode. Each mode has constraints on rules to ensure a correct articulation. The automatic application of these constraints facilitates the definition of assistance systems. It allows automatically generating or modifying aLDEAS rules as well as eventually verifying designer's definition. For instance, to define a bloc of rules articulated in the successive mode, the next rules must have a trigger event "end of previous rule". The definition of the bloc will add automatically this event in these rules. Otherwise, the previous rules must have at least an end event which must be determined by the designers. Therefore, SEPIA will check this constraint and show a message if it is not satisfied.

An assistance system is now represented in SEPIA by rules not only defined by designers but also generated and modified by the application of constraints thanks to blocs (Figure 8). This allows keeping the current operation of the SEPIA engine which doesn't need to change because of the implementation of model of articulation. This implementation allowed us to create and executes with SEPIA examples of assistance similar to assistance presented in the section 3. Figure 9 shows the execution of these examples.

EVALUATION

The aLDEAS language and its implementation in the SEPIA system were previously evaluated. The usability of aLDEAS and the usability of the SEPIA were evaluated in (Ginon et al., 2014b). The execution of assistance systems with the SEPIA in ILEs was evaluated in (Ginon et al., 2014c).

In this paper, we propose a model of articulation between assistance rules. Therefore, in this section, we focus on the evaluation of this model of articulation. Regarding the feasibility of model, it is demonstrated by the implementation of our model in SEPIA. The examples in Figure 9 show the possibility of our model. Thus, the model of articulation allows us to define the assistance systems similar to ones presented in section 3. In addition, we made an experiment of our model with few users and will make another with more users in spring 2016. The objective of these experiments is to evaluate: (C1) the capacity of comprehension of an assistance system created by using blocs, (C2) the capacity of use of blocs, (C3) the benefices of use of blocs, (C4) the coverage of 5 modes relative to expectation of designers. The designers must work with three ways of definition of assistance systems: definition without bloc by using the textual interface, definition without bloc by using graphical interface and definition with bloc by using the graphical interface. With each way, they will execute 3 steps: comprehension, modification, and completion of an assistance system. Next, they must create an assistance system with a preferred way of definition. Each user uses one mode of articulation.

We started the first experiment with students in France in order to observe and improve the future experiments. In this experiment, there are 5 master students. We only observed their tasks without considering the result. However, the majority of them answered that they understood how an assistance system operates and define an assistance system with bloc. Then, we improved the documents for the experiment with 6 Vietnamese students in the course HCI (Human Computer Interaction) in Vietnam. We summarized results, which is presented through the above figures. Figure 10 shows the number of users who succeeded the comprehension, modification and completion by three ways (for evaluation of C1, C2). There are no major difference between them. However, most users (5 out of 6 users) prefer to use the bloc in order to define an asked assistance system (Figure 11). In more detail, these users indicate that the comprehension and the definition of an assistance system with bloc are easier than others (Figure 12) (for evaluation of C3). The five modes are indicated enough for the definition of an assistance system because the users didn't give any other mode existing in other applications or in reality (for evaluation of C4). Through this experiment, we can think that our model of articulation facilitates the comprehension, the definition of an assistance system. The modes of articulation deducted from bibliographical studies (cf. section 3) can define various assistance systems.

However, the number of users who participated in the above experiment is low. Therefore, we will make another experiment in spring 2016 with 30 French master students with the same objectives. For the evaluation of coverage of model, we will improve this experiment by asking students to imagine a pedagogical assistance system. They must show whether it can be defined by using blocs with one among the five modes of articulation or with a non-existing mode of articulation.

CONCLUSION AND FUTURE WORK

In this article, we presented the model of articulation between aLDEAS assistance rules which completes the aLDEAS language. This model explicitly expresses the notion of articulation between rules of an assistance system. It offers five modes of articulation corresponding to those we have identified in our study: independent, successive, simultaneous, progressive, and interactive. We implemented this model in the SEPIA system by adding the notion of bloc of rules articulated in a mode. This implementation have two main advantages: it makes explicit the definition of blocs of rules with graphical interface and it applies semi-automatically constraints on rules. With the introduction of this model in our approach, an assistance system is defined not only by a set of rules, but also by a set of blocs that explain the articulation between these rules. It allows teachers to view more explicitly as well as define more easily a complex assistance system. We evaluated our propositions by the experiment which showed some big potentials.

However, an assistance system can be described by many blocs of rules articulated in different modes. SEPIA just shows the graphical representation of a bloc but not the global graphical representation of all the blocs. The blocs are listed in a table that limits designer's view of a whole assistance system. Therefore, in the future, we will aim at a global graphical representation of assistance systems which will be more intuitive.

As part of thesis, we continue to evolve SEPIA which will facilitate the definition of pedagogical guidance. We will find out how existing applications or systems propose pedagogical activities suitable to learners. For example, the activities can be temporally planned or the proposition of activities can be constraint by states of previous activities (e.g. remediation activities). Then, with SEPIA, we try to define assistance systems which can also propose these activities in order to identify difficulties. Thus, SEPIA has not yet supported the concepts "pedagogical guidance" and "learning activity". It's difficult and complex for assistance designers who wish to define a pedagogical guidance. So, we aim to propose these concepts in SEPIA. Through our state of the art, we will enrich these concepts in SEPIA (for example, temporal attribute in pedagogical guidance, output states in pedagogical activity).

Figure 2 :

 2 Figure 2: aLDEAS rules pattern.

Figure 3 :

 3 Figure 3: Examples of modes of articulation between assistance elements.

Figure 5 :

 5 Figure 5: examples of model Figure 4: model of articulation between aLDEAS assistance rules

Figure 6 :

 6 Figure 6: Constraints on rules of successive mode of articulation in aLDEAS

Figure 7 :

 7 Figure 7: Detail of three rules articulated in successive mode in aLDEAS

Figure 9 :

 9 Figure 9: Execution of examples of assistance created by SEPIA with the implementation of our model

Figure 10 :

 10 Figure 10: number of success for the realization of tasks in our experiment


Manel Achichi

Rodolphe Bailly

Cécile Cecconi

Marie Destandau

Konstantin Todorov

Raphaël Troncy

DOREMUS: Doing Reusable Musical Data

This paper introduces DOREMUS-a semantic web project aiming to provide common knowledge models and shared multilingual vocabularies to cultural institutions, publishers, distributors and users in the musical domain. The project develops methods to describe, publish, connect and contextualize music catalogs on the web of data. Our focus is on the description of classical and traditional musical works as well as their interpretations (events).

Introduction

Music is everywhere-played, recorded, broadcasted and listened to. Files of recorded music are all over the web-stored, streamed, shared or sold. The knowledge about musical works is described in detail in the information systems of several cultural and media organizations around the world. The leading actors in France-the BnF (Bibliothèque Nationale de France), the Philharmonie de Paris and Radio France, as well as the Meaning Engines company join forces with three academic institutions within the DOREMUS project1 in order to make this knowledge available and re-usable on the web of data. The project covers three main objectives: [START_REF] Scharffe | Enabling linked data publication with the Datalift platform[END_REF] to publish and to link musical data on the web of data including the description of musical works as well as their interpretation (event); (2) to develop tools assisting the selection of pieces of music, for instance, to generate an original playlist for a specialized radio station or to illustrate a musical period or genre; and (3) to construct and validate educational tools that enable the deployment of standards, vocabularies and the underlying semantic web technologies in the cultural institutions in France and worldwide.

In this paper, we describe our approach for publishing and interlinking descriptions of musical data based on ontological models and vocabularies and we discuss several applications. The goal is to exploit as much as possible the rich data which is available and to open it to the public. The main contributions consist in (1) the provision of advanced ontological models for describing musical works and events and (2) the generation of automatic methods for opening musical data that are initially given in a specific input format, so far, disconnected from the semantic web world.

Musical works are complex objects. Expressing them comprehensively requires the description of their physical manifestations (recordings, scores) and all the events that define them (creation, publication, performance). The first aspect is relatively well-mastered, in library catalogues as well as in the music industry. Several models can be used to describe a musical work, some being specific to the music domain (MusicOntology), while others being more broadly designed for libraries in general (FRBR2 , UNIMARC3 ). The second aspect is rather new, although there is a growing need and interest in using even-based models. Several ontologies aim specifically to define events (Event4 , LODE5 ), but there are fewer examples of using them to describe the creation and publication processes. One of the difficulties with musical works is that although their expressions may differ significantly, they are still regarded as a single work. Modeling it requires to express the singleness of the work as well as the specificities of its expressions, and to show how events are connected to these expressions. Another issue is that an arrangement may be considered as an expression or as a new work, depending on the data producer. Our model shall also make it possible to specify the relations between works, for instance when one work is derived from another.

We are currently working on a model enabling the expression of all these aspects in a coherent way, while being flexible and powerful enough to interoperate with information systems dealing with any kind of cultural data. It occurred to us that the FRBRoo ontology had all the required qualities. It is based on the FRBR and CIDOC-CRM models, with a concern for the necessity to describe how events occur through the life of a thing and its resulting states. It is a formal ontology, based on the articulation of events and things, containing the notion of work, and offering precise elements to describe physical items. Its structure is adaptable and allows to model cultural metadata with fine granularity. Since the work, as defined in this model, is not specifically musical, we are extending the model by additional classes and properties that are recurrent and important in the description of musical works (Figure 1).

Data Lifting

As in many other domains, the DOREMUS project faces the problem of processing a very large amount of heterogeneous (in terms of formats and languages) and semi-structured data in order to transform them into RDF, describe them by (re-)using appropriate vocabularies, link these data to existing datasets and publish them on the web of data-a process also coined as data lifting [START_REF] Scharffe | Enabling linked data publication with the Datalift platform[END_REF]. We have followed and adapted the workflow of the Datalift platform (Figure 2). The methods for extracting RDF triples need to be adapted to the data source in the musical field, which consists of documents in the MARC (MAchine-Readable Cataloging) format, as well as XML documents.

Fig. 2. Data semantization and publishing workflow (adapted from [1]).

There exist several approaches for extracting RDF triples from XML documents [START_REF] Van Deursen | XML to RDF conversion: a generic approach[END_REF]. However, MARC (and its variants) documents has, so far, been rarely processed by semantic web applications. We have developed a two steps approach for transforming MARC data into RDF, while dealing with two variants of the MARC format: UNIMARC and INTERMARC. First, we propose an approach to extract raw RDF at the highest possible level of granularity by exploiting subareas coded in MARC with an alphanumeric character and a delimiter "$" (Figure 3). The principle is that each area and sub-area will be represented by an RDF property and the RDF object has the value of the area or one of its subfields. Second, we perform a manual mapping between the UNI/INTER-MARC areas and sub-areas with classes and properties defined in the models discussed in Figure 1. Thus, the description of musical works is performed at a lower level of granularity (due to merging of certain subfields taken as values of a property), while ensuring completeness of the information. Several documents within a single institution or across institutions describe the same entity (a particular musical work), but from different perspectives (scores, sound recordings, videos). We aim to develop generic data connectors [START_REF] Scharffe | Enabling linked data publication with the Datalift platform[END_REF] to automatically detect relationships (equivalence, transcription, interpretation) between musical entities. The multilingual nature of data-a characteristics that should be preserved in the data lifting process-will be handled by automating the alignment process of multilingual ontologies specific to the field of music.

Applications and Future Work

Once the musical data coming from the cultural institutions is on the web of data, we plan to develop a musical recommendation application. It will serve as an evaluation of our model, and enable us to control the quality of data, after being modeled, lifted and interlinked. It will also be an example of how to implement the DOREMUS model for other cultural institutions.

The application will offer features such as the creation of original playlists for specialized radio, or the selection of appropriate musical works and interpretations to illustrate the biography of a composer, a historical period, culture or genre. The targeted end-users range from a child building his/her musical culture, through music lovers or curious discoverers, to expert musicologists.

Fig. 1 .

 1 Fig. 1. Modeling a musical work: M-classes and U-properties are extensions of FRBRoo

Fig. 3 .

 3 Fig. 3. Transforming the UNIMARC version of the uniform musical title (TUM) of the "Moonlight Sonata" by L.V.Beethoven to RDF

http://www.doremus.org

http://www.ifla.org/publications/functional-requirements-for-bibliographic-records

http://www.ifla.org/publications/unimarc-formats-and-related-documentation

http://purl.org/NET/c4dm/event.owl

http://linkedevents.org/ontology/

Acknowledgments

This work has been partially supported by the French National Research Agency (ANR) within the DOREMUS Project, under grant number ANR-14-CE24-0020.


Claudio A Buzzi

Jeroen S W Lamb

Reversible Hamiltonian Liapunov center theorem

come

1. Introduction. It is well known that (purely) reversible and Hamiltonian dynamical systems have many striking features in common, see for instance [START_REF] Sevryuk | Reversible systems[END_REF][START_REF] Roberts | Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems[END_REF][START_REF] Lamb | Time-reversal symmetry in dynamical systems: a survey[END_REF]. Surprisingly, despite the frequent occurrence in applications, only relatively few theoretical studies have focussed on the category of dynamical systems that are both Hamiltonian and reversible at the same time.

In this paper, we discuss the dynamics in the neighbourhood of an equilibrium with some purely imaginary pairs of eigenvalues. We call such an equilibrium (partially) elliptic. Importantly, such types of equilibria occur generically (with codimension zero) in reversible, Hamiltonian, and reversible Hamiltonian dynamical systems. We discuss the existence of periodic solutions in the immediate vicinity of such equilibria. It is well known [START_REF] Abraham | Foundations of mechanics[END_REF] that generic (partially) elliptic equilibria in Hamiltonian vector fields are contained in two-dimensional manifolds containing one-parameter families of periodic solutions. Such families of periodic solutions are called Liapunov Center families. Devaney [START_REF] Devaney | Reversible diffeomorphisms and flows Trans[END_REF] was the first to prove a Liapunov Center theorem for (purely) reversible vector fields, see also Vanderbauwhede [START_REF] Vanderbauwhede | Local bifurcation and symmetry[END_REF] and Sevryuk [START_REF] Sevryuk | Reversible systems[END_REF].

In this paper we present a Liapunov Center theorem for purely reversible Hamiltonian vector fields, that are vector fields which are Hamiltonian and reversible at the same time, describing all periodic solutions in the neighbourhood of typical (partially) elliptic equilibria. Our only assumption is that the reversing symmetry respects the symplectic structure in the sense that it is symplectic or antisymplectic.

It turns out that the cases with symplectic and anti-symplectic reversing symmetry are essentially different. If the reversing symmetry acts anti-symplectically, generically there is a single isolated pair of purely imaginary eigenvalues, in which case one recovers the main conclusion of the Liapunov center theorem in the purely reversible and Hamiltonian contexts: the equilibrium is contained in a two-dimensional local smooth invariant manifold containing a one-parameter family of symmetric periodic solutions encircling the equilibrium point.

However, if the reversing symmetry acts symplectically there is typically a doubly degenerate pair of purely imaginary eigenvalues, and the structure of the set of periodic solutions is more complicated. The persistent double degeneracy of purely imaginary eigenvalues in this setting was (to our knowledge) first observed by Buzzi and Teixeira [START_REF] Buzzi | Time-reversible vector fields with symplectic symmetries[END_REF]. In fact, the double degeneracy is of the type of "semisimple 1 : -1 resonance", which in the present setting thus has codimension zero. In the absence of symmetry, the codimension-two Hamiltonian semisimple 1 : -1 resonance was studied in [START_REF] Chow | Bifurcation of periodic orbits for non-positive definite Hamiltonian systems[END_REF], and the codimension-one non-semisimple Hamiltonian 1 : -1 resonance in [START_REF] Van Der Meer | The Hamiltonian Hopf bifurcation[END_REF]. For related studies of codimension-one reversible equivariant and reversible equivariant Hamiltonian Hopf bifurcation, see [START_REF] Buzzi | Reversible equivariant Hopf bifurcation[END_REF][START_REF] Buzzi | Reversible equivariant Hamiltonian Hopf bifurcation[END_REF].

The overall aim is to extend our results to the reversible equivariant Hamiltonian category in the future, extending previous results in the reversible equivariant and reversible equivariant Hamiltonian categories obtained by Golubitsky, Krupa and Lim [START_REF] Golubitsky | Time-reversibility and particle sedimentation[END_REF] and Montaldi, Roberts and Stewart [START_REF] Montaldi | Periodic solutions near equilibria of symmetric Hamiltonian systems[END_REF][START_REF] Montaldi | Existence of nonlinear normal modes of symmetric Hamiltonian systems[END_REF], respectively. Although these works work with general types of equivariance, they assume certain conditions on the representation of the reversing symmetry. Also, ideally, the aim is to develop a full description of all robust families of periodic solutions near generic (partially) elliptic equilibria for reversible-equivariant and reversible-equivariant Hamiltonian vector fields. In [START_REF] Golubitsky | Time-reversibility and particle sedimentation[END_REF][START_REF] Montaldi | Periodic solutions near equilibria of symmetric Hamiltonian systems[END_REF][START_REF] Montaldi | Existence of nonlinear normal modes of symmetric Hamiltonian systems[END_REF], the existence of many families of periodic solutions is established, but the existence of additional periodic solutions is not ruled out.

Similar techniques may be used to study periodic solutions in the neighbourhood of degenerate (codimension one) elliptic equilibria, so-called reversible and/or Hamiltonian Hopf bifurcation [START_REF] Golubitsky | The constrained Liapunov-Schmidt procedure and periodic orbits, Normal forms and homoclinic chaos[END_REF]. Recently, in [START_REF] Buzzi | Reversible equivariant Hopf bifurcation[END_REF] we studied such Hopf bifurcation in reversible equivariant systems, where apart from splitting and passing eigenvalue scenarios, and alternative crossing scenario exists [START_REF] Lamb | Reversible equivariant linear systems[END_REF]. We intend to report on the corresponding bifurcations in Hamiltonian systems in the near future.

This paper is organized as follows. In Section 2, we introduce our setting of reversible Hamiltonian systems. In Section 3, we use results from linear normal form theory to study linear reversible Hamiltonian vector fields and establish the two different types of centers, depending on whether the reversing symmetry acts symplectically or anti-symplectically. In Section 4 we discuss the main tool by which we obtain our results: viewing the ODE as a functional differential operator on a loop space of periodic functions and then obtaining a finite-dimensional bifurcation problem by Liapunov-Schmidt reduction. We discuss the Liapunov Center theorems in the cases of anti-symplectic and symplectic reversing symmetries in Sections 5 and 6. Our main results are stated in Theorem 5.1 and Theorem 6.1.

2. Reversible Hamiltonian systems. In this section we define our setting of reversible Hamiltonian systems.

We consider a vector field f : R N → R N , generating a dynamical systems, i.e. the flow of

d dt x = f (x). (1)
Let R be a diffeomorphism of R N to itself. Then f is called R-reversible if R * f = -f , so that (1) has a solution x(t) if and only if Rx(-t) is too. Analogously, f is called S-equivariant if S * f = f in which case any solution x(t) implies the existence of the solution Sx(t). If the vector field is assumed not to have any nontrivial equivariance, it follows that R 2 = I and we say that f is purely reversible, to contrast the case in which the vector field is reversible-equivariant. In order to discuss Hamiltonian vector fields, we focus on the case that the phase space has even dimension N = 2n. Let ω be a symplectic form, i.e. a nondegenerate skew symmetric bilinear form on R 2n . Let < •, • > denote the standard inner product on R 2n . Then there is a (linear) structure map J satisfying J * = -J (with * denoting transpose) and J 2 = -I such that ω(x, y) =< x, Jy > for all x, y ∈ R 2n . f is a Hamiltonian vector field if its flow is symplectic, i.e. preserves the symplectic form ω. This coincides with the condition that f = J∇H, where H : R 2n → R is the so-called Hamiltonian.

By the Darboux Theorem [START_REF] Abraham | Foundations of mechanics[END_REF], we may assume without loss of generality that J assumes the standard form

J = 0 -I n I n 0 . (2)
On the other hand, by Bochner's Theorem [START_REF] Bochner | Compact groups of differentiable transformation[END_REF] in the case of reversible vector fields one may assume without loss of generality that R acts linearly. It remains the question whether R and J can be simultaneously brought into these forms. In general the answer to this question is negative. However, when we assume that the reversing symmetry R acts (anti-)symplectically, it follows that one can in fact linearize R using a symplectic coordinate transformation. We recall that R is symplectic if ω(Rx, Ry) = ω(x, y) for all x, y ∈ R 2n and anti-symplectic if ω(Rx, Ry) = -ω(x, y) for all x, y ∈ R 2n . In other words, R is (anti-)symplectic if RJ = ±JR * . Hence, whenever R is (anti-)symplectic it preserves the absolute value of the symplectic form, which can be viewed as some kind of symplectic volume.

Although examples of Hamiltonian vector fields with non-(anti-)symplectic symmetries and reversing symmetries can be constructed, from a theoretical point of view -in the context of dynamical systems with symplectic structure -it appears natural to focus on (reversing) symmetries that act (anti-)symplectically. In fact, in the literature -in particular in mechanics -it is often assumed that reversing symmetries act anti-symplectically, which coincides with the property that the Hamilton (energy) function H is invariant under all symmetry transformations. However, in general, it may happen that some symmetries act anti-symplectically and some reversing symmetries symplectically, in which case such (reversing) symmetries only fix the absolute value of H, but change its sign (see for instance [START_REF] Heinz | Dynamik und Stabilität eines Hamiltonschen Systems[END_REF] for an example where symplectic time-reversal symmetries arise).

Thus by assumption of the (anti-)symplectic action of R, R and J can be chosen to be simultaneously linear. Moreover, by [START_REF] Hoveijn | Normal forms and unfoldings of linear systems in the eigenspaces of (anti-)automorphisms of order two[END_REF][appendix B], R and J can be chosen to be simultaneously orthogonal (with respect to the natural inner product).

3. Linear reversible Hamiltonian vector fields with purely imaginary eigenvalues. We now consider in more detail the persistent occurrence of purely imaginary eigenvalues in reversible Hamiltonian vector fields.

Let L ∈ sp J (2n, R) represent a linear Hamiltonian vector field so that LJ = -JL * .

Using the Darboux Theorem one may assume without loss of generality that J is in the standard form [START_REF] Bochner | Compact groups of differentiable transformation[END_REF]. By the discussion in the previous section, the reversing symmetry R can at the same time be chosen to be linear and orthogonal. Rreversibility of L then implies that LR = -RL.

We now use the important assumption that R is (anti-)symplectic

RJ = ±JR.
The linear normal form theory for reversible Hamiltonian systems can be studied along the lines set out by Hoveijn, Lamb and Roberts [START_REF] Hoveijn | Normal forms and unfoldings of linear systems in the eigenspaces of (anti-)automorphisms of order two[END_REF], as the structure is characterized by two commuting Lie-algebra (anti-) automorphisms

ψ J (L) = J -1 L * J and φ R (L) = R -1 LR. In fact, L ∈ gl(2n, R) is precisely reversible Hamiltonian if it is in the -1 eigenspaces of both ψ J and φ R . The action of Z 2 (ψ J )×Z 2 (φ R ) on gl(2n, R) induces the following isotypic decomposition of gl(2n, R): gl(2n, R) = gl J+,R+ (2n, R) ⊕ gl J+,R-(2n, R) ⊕ gl J-,R+ (2n, R) ⊕ gl J-,R-(2n, R),
where the righthand components are exactly the combined ±1 eigenspaces of ψ J and φ R . The component corresponding to linear reversible Hamiltonian vector fields is thus gl J-,R-(2n, R).

Hoveijn et al. [START_REF] Hoveijn | Normal forms and unfoldings of linear systems in the eigenspaces of (anti-)automorphisms of order two[END_REF] treat the linear normal form theory for such components. Normal forms are based on the construction of minimal < J, R >-invariant subspaces. By [START_REF] Hoveijn | Normal forms and unfoldings of linear systems in the eigenspaces of (anti-)automorphisms of order two[END_REF], as we are only interested in codimension zero cases, we can restrict our attention to such minimal invariant subspaces on which L is semi-simple (i.e. is diagonalizable over C).

As we are interested in (partially) elliptic equilibria, we furthermore assume that L has at least one pair of purely imaginary eigenvalues λ, -λ with corresponding (real) invariant subspace spanned by e 1 , ē1 . Note that if e 1 has eigenvalue λ then ē1 has eigenvalue -λ = λ. Now we consider the situation that L is not only Hamiltonian, but also (purely) R-reversible. This implies that if e is an eigenvector for eigenvalue λ, then Re is an eigenvector for eigenvalue -λ. The type of minimal invariant subspace now depends on whether R acts symplectically or anti-symplectically. Lemma 3.1. Consider a linear (purely) R-reversible Hamiltonian vector field L, with R acting (anti-)symplectically. Let V be a minimal (L, J, R)-invariant subspace on which L has purely imaginary eigenvalues. Then, L| V , J| V and R| V can be brought in the following normal forms:

• R acting anti-symplectically: dim V = 2 and

R| V = 1 0 0 -1 , J| V = 0 -1 1 0 , L| V = 0 -1 1 0 .
• R acting symplectically: dim V = 4 and

R| V =     0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0     , J| V =     0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 1 0     , L| V =     0 -1 0 0 1 0 0 0 0 0 0 1 0 0 -1 0     .
The above situations arise generically in (purely) reversible Hamiltonian vector fields when R acts anti-symplectically or symplectically, respectively.

Proof. Let W be a 2-dimensional symplectic subspace so that L has purely imaginary eigenvalues. Then, by standard Hamiltonian theory, L and J can be normalized to take the same form on W (taking into account multiplication of time by a scalar). If the reversing symmetry R acts anti-symplectically we have RL = -LR and RJ = -JR, and R maps eigenvectors of L and J with eigenvalue λ into eigenvectors with eigenvalue -λ. Because L and J commute on W , they share their eigenvectors. Let e 1 and e 1 be the (complexified) eigenvectors of L, then -taking into account the anti-commutation between R and L, one obtains a minimal invariant subspace by choosing Re 1 = e 1 . As J = L on W , the anti-commutation relation with J is automatically satisfied.

If R acts symplectically the dimension of the minimal invariant subspace can no longer be taken to be equal to two. Namely, starting with a 2-dimensional W as above, it follows that if R leaves W invariant and anti-commutes with L it also must anti-commute with J. Hence, R(W ) = W = W and a minimal invariant subspace is given by V = W ⊕ W , so that dim V = 4. Because of the commutation relations, when we now choose J| W = L| W and have J acting identically on W and W , then

J W = -L W .
The generic (codimension zero) occurrence of invariant subspaces with purely imaginary eigenvalues of the above kind, can be easily verified by calculation of the centralizer unfolding, the versality of which has been proven in [START_REF] Hoveijn | Normal forms and unfoldings of linear systems in the eigenspaces of (anti-)automorphisms of order two[END_REF]. Such unfolding of L moves the eigenvalues along the imaginary axis.

In this paper we focus on the situation in which L has purely imaginary eigenvalues ±i, either a single or double pair depending on whether R acts anti-symplectically or symplectically. Furthermore, we assume that these imaginary eigenvalues of L are nonresonant in the sense that L has no other eigenvalues of the form ±ki with k ∈ Z. This condition is clearly generic (codimension zero). The main aim is to describe the (families of) periodic solutions in the neighbourhood of the equilibrium point. The main technique we employ is a Liapunov-Schmidt reduction, which we discuss in the next section.

4. Liapunov-Schmidt reduction. In this section we discuss the Liapunov-Schmidt reduction procedure for periodic solution near equilibria for reversible equivariant Hamiltonian systems. We assume our vector field f : R N → R N has a reversing symmetry group G, implying the existence of representations ρ : G → O(N ) and σ : G → {±1} (reversible sign) such that

f ρ(g) = σ(g)ρ(g)f, ∀ g ∈ G.
We closely follow the exposition of Liapunov-Schmidt reduction in [START_REF] Golubitsky | Singularities and groups in bifurcation theory[END_REF][START_REF] Golubitsky | The constrained Liapunov-Schmidt procedure and periodic orbits, Normal forms and homoclinic chaos[END_REF]. Define Φ :

C 1 2π × R → C 2π by Φ(u, τ ) = (1 + τ ) du dt -f (u) (3)
where C 2π is the Banach Space of R N -valued, 2π-periodic mappings and C 1 2π consists of those u(t) ∈ C 2π that are continuously differentiable. Observe that solutions to Φ(u, τ ) = 0 corresponds to 2π 1+τ -periodic solutions of (1). We can define an action

T : G × C 2π → C 2π , or in C 1 2π , by (T g u)(t) = ρ(γ)(u(σ(γ)t + θ)) (4)
where

g = γθ is an element of G = G S 1 .
From the G-reversible equivariance of f , it is readily verified that Φ is Greversible equivariant:

Φ(T g u, τ ) = σ(γ)T g Φ(u, τ ) ∀g = γθ ∈ G = G S 1 (5)
In the first step of the Liapunov-Schmidt reduction we consider the linear part of Φ L = (dΦ) 0,0 ,

where

Lu = du dt -Lu,
and L = (df ) 0 . The elements of Ker L correspond to solutions of the linear system du dt = Lu satisfying u(t) = u(t + 2π). Hence, Ker L is the union of the (generalized) eigenspaces of L with eigenvalues of the form ki, with k ∈ Z. Naturally, L is also G-reversible equivariant

LT g = σ(γ)T g L,
and it follows that T g preserves Ker L and Range L.

The second step in the Liapunov-Schmidt reduction is to consider the splittings

C 1 2π = Ker L ⊕ (Ker L) ⊥ and C 2π = (Range L) ⊥ ⊕ Range L. (6)
Here, we choose the orthogonal complement to be taken with respect to the natural inner product in

C 2π and C 1 2π [u, v] = G < T g u, T g v > dµ (7)
where < u, v >= Using the fact that (Range L) and (Range L) ⊥ are T g -invariant, it follows that the projections

E : C 2π → Range L (I -E) : C 2π → (Range L) ⊥ (8) commute with T g . Namely, if u ∈ C 2π , u = v + w, v ∈ (Range L) ⊥ and w ∈ Range L. In this case, T g (E(u)) = T g w = E(T g w) = E(T g v + T g w) = E(T g u) and T g [(I -E)(u)] = T g v = (I -E)(T g v) = (I -E)(T g v + T g w) = (I -E)(T g u).
The third step in the Liapunov-Schmidt reduction procedure is to solve the equation EΦ(v + w, τ ) = 0 (9) by using the implicit function theorem to find a function W : Ker L×R → (Ker L) ⊥ such that EΦ(v + W (v, τ ), τ ) = 0 for all v ∈ Ker L.

It is easy to verify that W commutes with T g . Namely, define

W g : Ker L × R → (Ker L) ⊥ by W g (v, τ ) = T g -1 W (T g v, τ ). Observe that EΦ(v + W g (v, τ ), τ ) = EΦ(T g -1 (T g v + W (T g v, τ ), τ )) = σ(γ)T g -1 EΦ(T g v + W (T g v, τ ), τ ) = 0 since T g v ∈ Ker L.
Then W g also solves the equation ( 9) and W g (0, 0) = W (0, 0) = 0, so we have by uniqueness W g (v, τ ) = W (v, τ ), which implies that W commutes with T g .

The above Liapunov-Schmidt reduction thus reduces our problem to the problem of finding zeros of the map ϕ : Ker L × R → (Range L) ⊥ that is defined by

ϕ(u, τ ) = (I -E)Φ(v + W (v, τ ), τ ). ( 10
)
We refer to ϕ as the bifurcation map. Importantly, the bifurcation map is Greversible equivariant:

Proposition 4.1. If f is G reversible equivariant then the bifurcation map ϕ is also G reversible equivariant: ϕ(T g u, τ ) = σ(γ)T g ϕ(u, τ ) ∀g ∈ G S 1 (11)
Proof. This is a direct consequence of the G-equivariance of W and the projection (I -E).

Having seen above how symmetry enters the bifurcation equation ϕ, we now consider the consequences of the Hamiltonian structure, following the treatment of constrained Liapunov-Schmidt reduction in [START_REF] Golubitsky | The constrained Liapunov-Schmidt procedure and periodic orbits, Normal forms and homoclinic chaos[END_REF].

Since the vector field f is Hamiltonian, i.e. f = X H , the identity

ω(X H (u), v) =< dH(u), v >
holds for all v ∈ R 2n . One may define the map

F : C 1 2π × R × C 0 2π → [C 1 2π ] * by F (u, τ, v).U = 2π 0 ω v -(1 + τ ) du ds , U + dH(u).U ds.
Then, the implicit constraint

F (u, τ, Φ(u, τ )) = 0
is a restatement of the condition that the vector field f is Hamiltonian with Hamiltonian function H. Alternatively, Φ may be regarded as a (parameter-dependent) vector field on C 1 2π that is Hamiltonian with respect to the weak symplectic form

Ω(u, v) = 1 2π 2π 0 ω(u(s), v(s))ds,
and with the Hamiltonian function

H(u, τ ) = 1 2π 2π 0 1 2 ω (1 + τ ) du ds , u -H(u) ds. (12)
In the Hamiltonian context, we now assume that the actions of G are (anti-)symplectic. Hence we introduce a symplectic sign χ :→ {±1}, so that ω(gx, gy) = χ(g)ω(x, y)

Consequently, Ω is G = G S 1 (anti-)invariant, Ω(γu, γv) = χ(g)Ω(u, v), γ = gθ, θ ∈ S 1 ,
and the Hamiltonian H satisfies

H(γu, γv) = σ(g)χ(g)H(u, v). (13)
By [9, Theorem 6.2], it follows that as long as the kernel of L is symplectic and invariant under the complex structure J, i.e.

Ker L = Ker L * , (14)
then the bifurcation equation is a Hamiltonian vector field. To apply the above theory, one thus needs to work out the linearized equations and calculate the induced symplectic structure on the kernel of the linearization. Thereafter the induced Hamiltonian can be derived and the resulting bifurcation equation studied.

In the situations we consider in this paper, Ker L is finite dimensional and ( 14) is satisfied. In that case, the bifurcation equation is thus a Hamiltonian vector field. The corresponding symplectic form is the restriction of Ω to Ker L and the corresponding Hamiltonian satisfies the same (anti-)invariance relations [START_REF] Lamb | Time-reversal symmetry in dynamical systems: a survey[END_REF], of course with respect to the actions of G restricted to Ker L. Also, in this paper we always have G = Z 2 and G = Z 2 S 1 ∼ = O(2).

5.

Anti-symplectic time-reversal symmetry. We first discuss the case that the reversing symmetry R is anti-symplectic. Anti-symplectic reversing symmetries are well known to arise in classical mechanics, corresponding to the "natural" timereversal transformation (x, p, t) → (x, -p, -t).

From the linear analysis in Section 3 above, we know that purely imaginary eigenvalue pairs typically arise isolated. We are interested in the existence of periodic solutions close to an equilibrium point with such an isolated pair of purely imaginary eigenvalues. We are also interested in the symmetry properties of such periodic solutions. Recall that an orbit is called "symmetric" if it is (setwise) R-invariant. Theorem 5.1. Consider a symmetric equilibrium 0 of a purely reversible Hamiltonian vector field f , with the reversing symmetry acting anti-symplectically. Suppose that Df (0) has one purely imaginary pair of eigenvalues ±i, and that the spectrum contains no other eigenvalues of the form ±ki, k ∈ Z. Then, typically, the equilibrium is contained in a smooth two-dimensional flow-invariant manifold consisting of a one-parameter family of symmetric periodic solutions whose period tends to 2π as they approach the equilibrium. Furthermore, there are no other periodic solutions with period close to 2π in the neighbourhood of 0.

It is not surprising that the Liapunov Center Theorem takes the above form. In this case there are many ways in which the existence of a Liapunov Center family of periodic solutions can be proven, using the fact that the flow has an integral [START_REF] Golubitsky | The constrained Liapunov-Schmidt procedure and periodic orbits, Normal forms and homoclinic chaos[END_REF], using the fact that the flow is reversible [START_REF] Vanderbauwhede | Local bifurcation and symmetry[END_REF][START_REF] Sevryuk | Reversible systems[END_REF][START_REF] Golubitsky | The constrained Liapunov-Schmidt procedure and periodic orbits, Normal forms and homoclinic chaos[END_REF], and using the fact that the flow is Hamiltonian [START_REF] Abraham | Foundations of mechanics[END_REF][START_REF] Golubitsky | The constrained Liapunov-Schmidt procedure and periodic orbits, Normal forms and homoclinic chaos[END_REF]. Moreover, the linear analysis shows that in case R acts antisymplectically, one would not expect more periodic solutions than a one-parameter family, in which case by uniqueness it also follows that all orbits in this family then should be symmetric.

For completeness, we give a proof using the Liapunov-Schmidt strategy set out above, following closely the treatment in [START_REF] Golubitsky | The constrained Liapunov-Schmidt procedure and periodic orbits, Normal forms and homoclinic chaos[END_REF].

Proof (of Theorem 5.1) The linearization of Φ at the equilibrium 0 is

L : C 1 2π → C 2π , Lv = dv ds -Lv,
where L = Df (0). Since L is Fredholm of index zero, we may use the orthogonal splittings

C 1 2π = Ker L ⊕ Range L * , C 2π = Range L ⊕ Ker L * .
As L has a single pair of eigenvalues ±i, dim Ker L = 2, with Ker L ⊂ C 2π (V ) being generated by

v 1 = sin(s) -cos(s) , v 2 = cos(s) sin(s) ,
where V ∼ = R 2 is the eigenspace for ±i of L, assuming without loss of generality that L| V takes the (normal) form of Lemma 3.1. Consequently, the symplectic form Ω for the reduced bifurcation equation satisfies

Ω(v n , v m ) = 2π 0 < v n , Jv m > ds = J n,m
where J is the symplectic structure map of Lemma 3.1, and J n,m with n, m ∈ {1, 2} denote its components as usual. Hence with basis {v 1 , v 2 } for Ker L ∼ = R 2 we obtain Ω(x, y) =< x, Jy >, for all x, y ∈ Ker L, so that our reduced bifurcation equation has the same symplectic form as the original Hamiltonian vector field.

By similar calculation, the reduced Hamiltonian can be computed from H to take the form

h(x, y, τ ) = τ 2 (x 2 + y 2 ) + (x 2 + y 2 ) h(x 2 + y 2 , τ ),
with h smooth and h(0, 0) = 0. We note that the eigenvalue conditions and Liapunov-Schmidt reduction coordinate τ arise prominently in the lowest order (quadratic) part of the reduced Hamiltonian. The higher order terms of h can be characterized by its G (anti-)invariance properties that are induced from H. Here,

G = Z 2 (R) S 1 ∼ = O(2),
acts on Ker L as the matrix in Lemma 3.1 and θ ∈ S 1 acts as the matrix cos θ sin θ -sin θ cos θ .

Because of the S 1 -invariance of h we have h(x, y, τ ) = ĥ(x 2 + y 2 , τ ). But then h is automatically also R-invariant, with R(x, y) = (x, -y). (Hence the presence of reversibility has few consequences.) Writing x 2 + y 2 = r 2 , we may write

h(r 2 , τ ) = r 2 ( τ 2 + h(r 2 , τ )).
The equilibria for the corresponding vector field are given by: r = 0 (trivial equilibrium) and

τ 2 + h(r 2 , τ ) + r 2 h (r 2 , τ ) = 0.
By application of the implicit function theorem, there is a function τ (r 2 ) so that for all r 2 sufficiently small

τ (r 2 ) 2 + h(r 2 , τ (r 2 )) + r 2 h (r 2 , τ (r 2 )) = 0.
Thus we obtain S 1 -orbits of solutions, i.e. circles if r 2 > 0, each circle representing a periodic orbit. They are symmetric because they are setwise R-invariant (and hence intersect FixR twice). Alternatively, it is readily verified that every point in Ker L fixed by θR for some θ ∈ S 1 , so that all solutions must be symmetric. Finally, we note that τ → 0 as r 2 → 0. By the definition of τ in Section 4, the corresponding family of periodic orbits thus have their period tending to 2π as r → 0.

6. Symplectic time-reversal symmetry. We now consider the case that the time-reversal symmetry acts symplectically. In this case, (partially) elliptic equilibria typically have all their purely imaginary eigenvalues doubly degenerate. The structure of nearby periodic solutions is also more complicated. Theorem 6.1. Consider a symmetric equilibrium 0 of a purely reversible Hamiltonian vector field f , with the reversing symmetry acting symplectically. Suppose that Df (0) has two purely imaginary pairs of eigenvalues ±i, and that the spectrum contains no other eigenvalues of the form ±ki, k ∈ Z. Then, typically, the equilibrium is contained in a three-dimensional flow-invariant surface consisting of a two-parameter family of symmetric periodic solutions whose period tends to 2π as they approach the equilibrium. Moreover, typically, the equilibrium is contained in two smooth two-dimensional flow-invariant manifolds each containing a oneparameter family of non-symmetric periodic solutions whose period tends to 2π as they approach the equilibrium. Furthermore, there are no other periodic solutions with period close to 2π in the neighbourhood of 0.

The remainder of this section is devoted to the proof of this theorem, which consists of Lemma 6.2 and Lemma 6.4. Moreover, more detail on the structure of two-parameter family of symmetric periodic solutions is described in Lemma 6.3.

We assume without loss of generality that the actions of R, J and L on the four-dimensional eigenspace V of ±i for L are as in Lemma 3.1.

We proceed to apply the Liapunov-Schmidt reduction of Section 4. First of all, we find that dim Ker L = 4, and we can find a basis for Ker L such the symplectic structure map J and reversing symmetry R act in the same way as they act on V in Lemma 3.1. Clearly, again Ker L satisfies the conditions of [9, Theorem 6.2], and consequently the reduced bifurcation equation ϕ is a Hamiltonian vector field.

To facilitate the notation, we furthermore identify Ker L ∼ = R 4 with C 2 in a natural way

R 4 (x 1 , y 1 , x 2 , y 2 ) (x 1 + iy 1 , x 2 + iy 2 ) = (z 1 , z 2 ) ∈ C 2 ,
so that our bifurcation map takes the form

ϕ : C 2 × R → C 2 , ϕ is Hamiltonian ϕ = 2J∇ z h (15)
with Hamilton function h :

C 2 × R → R. Moreover, h is S 1 -invariant, h • θ = h with θ ∈ S 1 acting on C 2 as θ(z 1 , z 2 ) = (e iθ z 1 , e -iθ z 1 ),
and

h is R-anti-invariant, h • R = -h, with R(z 1 , z 2 ) = (z 2 , z 1 ).
Finally, J acts on C 2 simply as multiplication by i.

By S 1 -invariance we have

h = h(|z 1 | 2 , |z 2 | 2 , z 1 z 2 , z 1 z 2 , τ ).
Using moreover the R-anti-invariance and reality of h, we have

h = (|z 1 | 2 -|z 2 | 2 )g(|z 1 | 2 + |z 2 | 2 , |z 1 z 2 | 2 , z 1 z 2 , z 1 z 2 , τ ), = (|z 1 | 2 -|z 2 | 2 ) g 1 (|z 1 | 2 + |z 2 | 2 , |z 1 z 2 | 2 , τ ) +2Re[z 1 z 2 g 2 (|z 1 | 2 + |z 2 | 2 , |z 1 z 2 | 2 , z 1 z 2 , τ )] , (16)
where g is decomposed as g = g 1 + 2Re(z 1 z 2 g 2 ) so as to remove the ambiguity of coefficients for the Taylor expansion of g. Finally, from [START_REF] Hoveijn | Normal forms and unfoldings of linear systems in the eigenspaces of (anti-)automorphisms of order two[END_REF] the τ -dependence of the lowest (quadratic) order contribution to h can be calculated to yield

h = (|z 1 | 2 -|z 2 | 2 ) τ 2 + hot,
or equivalently that g 1 (0, 0, τ ) = τ 2 . 6.1. Symmetric periodic solutions. As h•R = -h it follows that all symmetric (R-invariant) periodic solutions must be presented by (S 1 -orbits of) zeroes of the bifurcation map ϕ that lie in the level set h = 0. In fact, it turns out that all points in This level set are zeroes of the bifurcation map, corresponding to symmetric periodic solutions: Lemma 6.2. In the neighbourhood of the equilibrium (z 1 , z 2 , τ ) = (0, 0, 0), all points in the level set h = 0 are solutions of the bifurcation map ϕ.

Proof. The bifurcation map [START_REF] Montaldi | Periodic solutions near equilibria of symmetric Hamiltonian systems[END_REF], evaluated in Fix(R) = {(z, z) ∈ C 2 }, yields z = 0, or g(z, τ ) = 0.

The orbit of (z, z) is an equilibrium if and only if z = 0. By the form of g we know that ∂g/∂τ (0, 0) = 1 2 = 0. Subsequent application of the Implicit Function Theorem yields that for each nonzero small z there exists a τ such that (z, z) lies on a periodic solution with period 2π/(τ + 1). For a more geometrical approach to the proof of Lemma 6.2, we consider the simplest situation, that is a two-degree of freedom Hamiltonian system with symplectic reversing symmetry.

In that case, in complex coordinates, the Hamiltonian has the form

H = (|z 1 | 2 -|z 2 | 2 )g(|z 1 | 2 + |z 2 | 2 , |z 1 z 2 | 2 , z 1 z 2 , z 1 z 2 ),
with g(0, 0, 0, 0) = 1. Clearly, the origin lies in FixR = {z 1 = z 2 } and in the level set H = 0. As g(0, 0, 0, 0) = 0, locally near the origin, this level set is given by |z 1 | 2 -|z 2 | 2 = 0, or in polar coordinates r 1 = r 2 . Considering a small sphere around the origin, the level set H = 0 is hence a three-dimensional surface.

By restricting this surface to some nonzero r, one obtains a two-torus, parametrized by the two angles φ 1 and φ 2 . The surface thus looks like a one-parameter (r) family of two-tori, that is pinched at the origin, yielding a singular point at the origin with conical structure. Now we consider FixR = {z 1 = z 2 }. FixR intersects each of the above two-tori in exactly one line φ 1 = φ 2 . The flow induced on the two-tori by the linear part of the vector field φ1 = 1, φ2 = -1, traces orbits on the torus, intersecting FixR exactly twice, see the illustration in Fig. 1.

Taking r again into account, we observe that all orbits of the linear vector field in the level set H = 0 intersect FixR transversally. We note that FixR is a twodimensional submanifold of the three-dimensional level set H = 0, and thus has codimension one. Hence, it is not at all surprising that when taking higher order terms of the nonlinear vector field into account, all orbits continue to intersect FixR transversally in two points, and hence are symmetric periodic orbits. The latter follows from the well-known fact that an orbit of a reversible flow is a symmetric periodic orbit if and only if it intersects FixR (exactly) twice, see [START_REF] Lamb | Time-reversal symmetry in dynamical systems: a survey[END_REF].

There is one remaining feature that we would like to highlight. Namely, examination of the period distribution within the set of periodic solutions reveals an Elliptic Hyperbolic interesting feature. Let z = x + iy, then the level sets of τ (x, y) can be typically shown to be one of two types: in suitable coordinates (x, ỹ), near (0, 0) τ is equivalent to τ = ε 1 x2 + ε 2 ỹ2 with ε j = ±1, where the signs depend on details of h (and H). If the singularity is elliptic (ε 1 ε 2 = 1) the level sets form circles and τ typically increases or decreases monotonically with increasing radius. If it is hyperbolic (ε 1 ε 2 = -1), the level sets form two families of hyperbolae, one family with positive increasing τ and one with negative decreasing τ . The elliptic and hyperbolic situations are sketched in Fig. 2. Recall that the periodic solutions are in a one-to-two relationship with the zeroes of τ , as the points (x, y) and (-x, -y) refer to two points on the same orbit, half-a-period apart.

It remains to give an interpretation of this feature. In the elliptic case, the equilibrium is surrounded by periodic solutions with period close but not equal to 2π, and the periods converge to 2π, from one side, as they approach the equilibrium. In the hyperbolic case, however, there are two smooth local two-dimensional invariant manifolds, each containing a one-parameter family of symmetric periodic solutions with period exactly equal to 2π. In this case, the remainder of the symmetric periodic solutions have period slightly higher or lower: on one side the period is higher (τ < 0) and on the other lower (τ > 0), cf. Fig. 2. Recall in this respect that the points (x, y) and (-x, -y) correspond to the same periodic solution.

We summarize this conclusion in the following lemma: Lemma 6.3. Depending on quartic terms of the Hamiltonian (and quadratic terms of the function g), among the three-dimensional surface of symmetric periodic solutions near the equilibrium point, typically either there exist two local two-dimensional invariant manifolds containing periodic solutions whose period is exactly equal to 2π (hyperbolic case), or no such periodic solutions exist (elliptic case).

Proof. We already discussed the main arguments of the proof, above. We here give some details on the computation of τ (x, y), and explicit conditions on the function g for the singularity to be elliptic or hyperbolic.

From the proof of Lemma 6.2, we see that the profile of τ (z) is determined by the equation g(z, τ ) = 0. The lowest (quadratic) order terms of this equation read:

τ 2 = 2|z| 2 g 1 1 (0, 0, 0) + 2Re[z 2 g 2 (0, 0, 0, 0)], = 2g 1 1 (0, 0, 0)(x 2 + y 2 ) + 2Re[g 2 (0, 0, 0, 0)](x 2 -y 2 ) -4Im[g 2 (0, 0, 0, 0)]xy, [START_REF] Roberts | Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems[END_REF] where z = x + iy. The shape of τ (x, y) near 0 is typically determined by the quadratic terms by Morse's Lemma. The nature of the singularity is determined by the determinant D = g 1 1 (0, 0, 0) 2 -(Re[g 2 (0, 0, 0, 0)]) 2 -4(Im[g 2 (0, 0, 0, 0)]) 2 . It is elliptic if D > 0 and hyperbolic if D < 0.

When we write τ = ε 1 x 2 + ε 2 y 2 , we have sign(D) = ε 1 ε 2 and, in the elliptic case (where sign(ε 1 ) has a meaining), ε 1 = -sign(g 1 1 (0, 0, 0)). 6.2. Non-symmetric periodic solutions. It now remains to be determined if there are also non-symmetric periodic solutions close to the equilibrium point. Lemma 6.4. In addition to the symmetric Liapunov Center families described in Lemma 6.2, there are typically two nonsymmetric Liapunov Center families of periodic solutions, each filling out a local two-dimensional smooth manifold with the period of the periodic solutions converging to 2π as the solutions tend to the equilibrium point.

Proof. We divide the proof into the discussion of three cases: z 1 = 0 and z 2 = 0, z 1 = 0 and z 2 = 0, and z 1 = 0 and z 2 = 0. z 1 = 0 and z 2 = 0 As before, we write h = I 1 g, with I 1 = (|z 1 | 2 -|z 2 | 2 ). As J is invertible, we have ϕ = 0 ⇔ ∇ z h = 0.

The latter equality can be rewritten as z 1 -z 2 g + I 1 ∇ z g = 0.

After writing out these equations, using [START_REF] Montaldi | Existence of nonlinear normal modes of symmetric Hamiltonian systems[END_REF], and multiplying the first one by z 1 and the second one by z 2 (looking for solutions with z 1 = 0 and z 2 = 0), we obtain

0 = |z 1 | 2 g + I 1 (g 1 1 + |z 2 | 2 g 1 2 + 2Re[z 1 z 2 (g 2 1 + |z 2 | 2 g 2 2 )] + z 1 z 2 (g 2 + z 1 z 2 g 2 2 ), 0 = |z 2 | 2 g + I 1 (g 1 1 + |z 1 | 2 g 1 2 + 2Re[z 1 z 2 (g 2 1 + |z 1 | 2 g 2 2 )
] + z 1 z 2 (g 2 + z 1 z 2 g 2 3 ), [START_REF] Sevryuk | Reversible systems[END_REF] where g j i the partial derivative of g j with respect to X i , where

X 1 = |z 1 | 2 + |z 2 | 2 , X 2 = |z 1 z 2 | 2 , and X 3 = z 1 z 2 .
We now note that all but the last identical term of (18) are real. Hence the imaginary part of this last term must be equal to zero:

Im[z 1 z 2 (g 2 + z 1 z 2 g 2 3 )] = 0. ( 19
)
We now write z j = r j e iφj . We now note that on each S 1 -orbit there lies a point (z 1 , z 2 ) with φ 1 = φ 2 = φ for some φ. Since periodic orbits are represented by S 1 -orbits of solutions of the bifurcation map, without loss of generality we may confine our search for solutions to points (z 1 , z 2 ) = (r 1 e iφ , r 2 e iφ ).

2π0

  [u(t)] t v(t)dt and µ is a normalized Haar measure for G. It is readily verified that [T g u, T g v] = [u, v] for all g ∈ G so that the complements (Ker L) ⊥ and (Range L) ⊥ , and hence the splittings (6), are T g -invariant.

Figure 1 .

 1 Figure 1. Sketch of typical orbit of the linear part of the vector field on the two-tori characterized by r 1 = r 2 = r, and parameterized by φ 1 , φ 2 . Each orbit intersects FixR = {φ 1 = φ 2 } exactly twice.

Figure 2 .

 2 Figure 2. Structure of period-distribution for symmetric periodic solutions: sketch of the level sets of τ = ε 1 x 2 + ε 2 y 2 .

Acknowledgements. CAB gratefully acknowledges support by Procad/Capes-Brazil (grant 0092/01-0), Fapesp-Brazil (grant 02/06531-3). JSWL gratefully acknowledges the support of the UK-EPSRC through its Advanced Fellowship scheme. CAB and JSWL have also been supported by UK-EPSRC grant GR/R93025/01. Finally, the authors would like to thank IMPA (Rio de Janeiro) for their hospitality during visits in which part of this work was done.

In these coordinates, [START_REF] Vanderbauwhede | Local bifurcation and symmetry[END_REF] reads Im[r 1 r 2 e 2iφ (g 2 + r 1 r 2 e 2iφ g 2

3 )] = 0. We now perform a "blow-up" by dividing this equation by r 1 r 2 , and setting r 1 = r 2 = τ = 0, yielding Im[e 2iφ g 2 (0, 0, 0, 0)] = 0. Writing g 2 (0, 0, 0) = a + ib, with a, b ∈ R, we find that

Assuming that |g 2 (0, 0, 0, 0)| = 0, this yields four solutions for φ, each π 2 apart from each other. In order to extend this result to nonzero r 1 and r 2 , we would like to invoke the Implicit Function Theorem with respect to φ. We have

which, evaluated at the value of φ satisfying [START_REF] Van Der Meer | The Hamiltonian Hopf bifurcation[END_REF], amounts to

This expression is nonzero whenever |g 2 (0, 0, 0, 0)| = 0. Clearly, this condition holds generically. And when this derivative is nonzero, by application of the Implicit Function Theorem, we have four solutions for φ for all sufficiently small r 1 , r 2 and τ .

We thus assume now that φ has been fixed by the above, and we are left with the variables r 1 , r 2 and τ to solve the pair of real equations

], (21) Subtracting the second equation of (21) from the first we obtain

) = 0. As we are looking for nonsymmetric solutions, we have I 1 = 0. However, the remaining expression examined at r 1 = r 2 = τ = 0 yields g 1 2 (0, 0, 0) which is typically nonzero. Hence, we find no small solutions to the bifurcation map with z 1 = 0 and z 2 = 0. z 1 = 0 and z 2 = 0 The same procedure is followed as in the previously discussed case. However, due to the fact that z 1 = 0, the equations ( 18) simplify significantly to the single equation

)) = 0. Now we use the fact that g 1 (0, τ ) = τ 2 . Hence, since g 1 (0, 0) = 0, we can use the Implicit Function Theorem with respect to τ , to find a smooth function τ (|z 2 |) with τ (0) = 0 so that for sufficiently small |z 2 | we have g

)) = 0. Correspondingly, we have a one-parameter family of nonsymmetric periodic solutions filling out a local smooth two-dimensional invariant manifold. z 1 = 0 and z 2 = 0 The analysis in this case is identical to the previous one, yielding another oneparameter family which is in fact the R-image of the family with z 1 = 0.