## Final Report Summary - MUDIBI (Multiple-Discontinuity Induced Bifurcations in Theory and Applications)

The objectives of the MuDiBi project have been successfully achieved. Below it is reported how the tasks of specific Workpackages (WPs) of the MuDiBi Project have been proceeded and solved.

The piecewise linear map with two kink points proposed for the WP 1.1 is investigated, the relevant bifurcations structures are identified. The calculation technique (map replacement technique, revisited in [T2]) proposed for the WP 1.2 is successfully adjusted so that its applicability to maps of the considered class is confirmed. Bifurcation structures of the considered map are investigated in [P2] in the case in which the outermost slopes are positive and less than one. Three generic bifurcation structures are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with periodic points on two adjacent branches of the map, the period adding structure is related to periodic points on the outermost branches, and the fin structure is contiguous with the period adding structure and is associated with attracting cycles with at most one point on the middle branch. Analytical expressions for the periodicity region boundaries are obtained using the map replacement technique, as proposed in the project plan.

The work started in [P2] has been continued in [P14] focusing on chaotic dynamics. In particular, the bifurcation structure formed by parameter regions corresponding to chaotic attractors with different number of bands are described, which are adjacent to the regions belonging to the period adding and fin structures. The general mechanism leading the number of bands of a chaotic attractor to change is related to homoclinic bifurcation of a repelling cycle belonging to the immediate basin boundary of this attractor. A classification of possible effects of these bifurcations is presented in [P4]. In particular, one can observe gradual increase in size of the pieces of a chaotic attractor, which then merge pairwise. In general, such a transformation, which is called a merging bifurcation, is associated with a homoclinic bifurcation of a cycle with negative multiplier. The number of pieces of an attractor can also change due to an expansion bifurcation characterized by an abrupt increase in size of the attractor with uneven density of points after the bifurcation. This bifurcation is associated with a homoclinic bifurcation of a cycle with positive multiplier. Taking into account that for the considered bimodal map the conditions of homoclinic bifurcations are related to images of two critical points, it was established a basic technique for an analytical description of the bifurcation structures formed by merging and expansion bifurcations in this map. Additionally, it turned out to be possible to simplify the description by taking advantage of the results known for the skew tent map and by application of the map replacement technique, already applied in [P2].

As the region in the parameter space relevant for the electronic circuit considered in WP 1.3 differs from the one considered in the WPs 1.1 and 1.2 the results of these WPs could not be adopted directly. However, the calculation technique considered in the WP 1.2 was successfully applied also within WP 1.3 as expected. In collaboration with Prof. D. Fournier-Prunaret at INSA (Instutute National des Science Appliquées, Toulouse, France) established within this project a continuous piecewise-linear bimodal map was investigated, which results from the modeling of an electronic circuit previously proposed by Prof. D. Fournier-Prunaret and her collaborators for generation of chaotic signals. Such signals are used for purposes of secure communication and the level of security depends on the properties of the signals. It has been shown in our work that the bifurcation structure of the considered map is not generic, due to some particular features of the map caused by the modelled circuit. Still, the components of this bifurcation structure are generic and could be identified and calculated analytically in an efficient way using the technique proposed in the project plan. The collaborations started within this Marie Curie Fellowship continue beyond the time of the project, as immediately after finishing this project a next joint research activity on a topic similar to the one already proceeded have been started.

For the WP 1.4 a more challenging task than initially planned have been identified. Due to a collaboration with the group of Prof. Mosekilde (Copenhagen) and Prof. Zhusubaliev (Kursk) established within the scope of this Marie Curie Fellowship a new application area for maps with multiple discontinuities was identified. These maps, modelling the dynamic behavior of power electronic DC/AC converters, have by construction a very high number of break points. Due to this, and because of the increasing importance of the considered class of power electronic DC/AC converters (based, among other reasons, on the fact that such converters are unavoidable in solar panels), this work was done within the scope of the WP 1.4. Supported by physical experiments, [P13] considers an unusual transition from the domain of stable fixed points (corresponding to the desired mode of operation of the converter) to chaotic dynamic. The behaviour of the converter is studied by using a 1D stroboscopic map which by construction has a high number of border points. It is shown that due to this high number of border points the transition occurs stepwise, via irregular cascades of different border-collision bifurcations. Although particular bifurcations occurring in the system belong to well-known classes and can be explained using existing theory, such cascades have never been reported before. Remarkably, the obtained results are valid not only for the particular model considered in [P13] but for a broad class of models whose behaviour is essentially influenced by a high number of border points resulting from the mode of operation of DC/AC converters. The importance of this topic has been confirmed by the great interest of the audience for the talks [T16], [T17], [T19] in which the obtained results have been presented. The discussions followed the talks lead to some collaborations which are planned for the time after this project.

The obtained results have been extended in [P7]. In this work it is reported that not only the boundary between regular and chaotic domains in the considered model is predominantly influenced by multiple borders, but also interior structure of the regular domain. In this domain, close to its boundary, the behavior of the system is influenced by cascades of border collisions of so-called persistence type. The mechanism causing such cascades to occur is explained in [P7]. Note that as stable fixed points exist before, at the moment, and after border collisions forming such cascades, they can hardly be detected using standard numerical tools of bifurcation analysis. Still, their detection is an important task from the practical point of view, since the presence of such border collisions significantly decreases the quality of the output signal of the considered DC/AC converters. Although first steps towards solving this tasks are already done in [P7], further improvements of the quality of numerical results are necessary.

The task of the WP 1.5 is solved; the map to be considered in this WP is investigated. This map represents a piecewise linear model which describes dynamics of a business cycle model introduced by Day and Shafer. As initially assumed, the results obtained within the WPs 1.2 and 1.3 can be successfully transferred to this map. The results of this work are presented in [P9]. In particular, in this publication analytical expressions of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and related fin structure) are reported. Note that outside of these region the map has chaotic behaviour (robust chaos) so that the results presented in [P14] apply.

The work on organizing centres performed within the scope of the WP 1.6 represents one of the most significant results of the project. It was rigorously demonstrated that the appearance of organizing centres in 1D maps with an arbitrary number of discontinuities can be predicted in the same way as it is possible in maps with one discontinuity only. The sufficient conditions for different types of organizing centres are derived. The obtained results are presented in [P3] and in the talks [T1], [T4], [T7]. In this work a two-parametric family of 1D piecewise smooth maps with one discontinuity point is considered. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. The obtained results are applicable, however, to 1D piecewise smooth maps with any number of discontinuity points, under the assumption that the border-collisions occur at the same discontinuity. We describe the case of the collision of two stable cycles of any period and any symbolic sequences. For this case, it is proved that the local monotonicity of the functions constituting the first return map defined in a neighborhood of the border point at the parameter values related to the codimension-2 bifurcation point determines, under suitable conditions, the kind of bifurcation structure originating from this point; this can be either a period adding structure, or a period incrementing structure, or simply associated with the coupling of colliding cycles.

One more significant result obtained within this WP regards the possibility to adopt the ideas known for smooth bimodal map for explanation of the appearance of organizing centers in piecewise-smooth maps defined on many partitions. The hypothesis formulated in the project proposal was that these results can be adopted, and the possibility that they can not was mentioned in the risk assessment. However, it follows directly from the results presented in [P2] that in general this hypothesis can be seen as disproved. This result influenced the work done within the project up to some extent, since it has been considered as more efficient to reduce the effort related to the concept of organizing centers and to focus the work on more promising concepts.

The task of the WP 2.1 is solved up to a large extent. The numerical investigations are done for several systems, and the applicability of the map replacement technique for 2D maps is confirmed. The obtained results served as basis for several works, in particular related to the WP 2.2.

The tasks of the WP 2.2 are solved in a much larger extent than initially planned. In particular, the work presented in [P6] is devoted to an adaptive segregation model of Shelling’s type introduced by Bischi and Merlone (2011). The importance of this work is, among other reasons, based on the actuality of segregation problems for European Community. Indeed, the open borders policies of the European community facilitate migration from country to country of people of different nationalities, languages, skills and cultures. This process rises the necessity of integration between the indigenous dwellers and the newcomers. The main force that prevents integration between members of different groups is the limited tolerance of members of one group towards members of other groups. Aware of this, policy makers tend to avoid segregation by combining some integration policies (for example, promoting multiculturalism) with more drastic measures such as the imposition of entry constraints for the members of the different groups. The effect of such constrains was investigates within the scope of WP 2.2. The considered model describes dynamics of entry and exit of individuals belonging to two populations into a system, based on a limited tolerance of the members of each group to the presence of the members from the other group. The constraints are given by the upper limits for the number of individuals of a population that are allowed to enter the system. They represent possible exogenous controls imposed by an authority in order to regulate the system. Using analytical, geometric and numerical methods, we investigated the border collision bifurcations generated by these constraints. In [P6] the case is considered in which the two groups have similar characteristics and have the same level of tolerance toward the members of the other group. The results of this paper are extended in [P8]. In this work the effects of the entry constraints on the dynamics is investigated for the case in which two populations involved differ by the number of their members, for the level of tolerance toward members of the other population and for the speed of reaction. The investigation of the dynamics of the segregation model for an asymmetric setting of the parameters reveals that other types of bifurcations can occur than in the symmetric case studied before. The investigation reveals that asymmetries in the level of tolerance of the two populations involved may led to effects of overreaction in the adjustment process. In order to avoid the risk of segregation, suitable entry limitations must be imposed at least on the more tolerant population. This is an interesting and surprising result contradicting the common opinion that to avoid segregation an entry constraint should be imposed on the less tolerant population.

The results of WP 2.2 related to the Braess paradox and the ternary choice problem are summarized in [P10]. The Braess paradox is a well-known example in which individual rationality of the participants may lead to a collective irrationality. By contrast to the models considered in the literature, a system consisting of non-impulsive commuters changing their decisions proportionally to the cost difference is considered. For a simplified version of the problem associated with a binary choice a rigorous proof of the existence of a unique fixed point is provided, showing that it is globally attracting even if it may be locally unstable. This is possible because modelling of this system results in a 1D continuous map. In the case of a ternary choice, the system is necessarily modelled by a discontinuous 2D map. In this case it can also be proved that a unique fixed point exists, but its global attractivity is numerically evidenced, also when the fixed point is locally unstable. The work on Braess paradox models has been continued within the scope of WP 3 as described below.

The tasks of the WPs 3.1 and 3.2 are solved. The obtained results have been embedded as technical background in the work done within the WP 3.5. Regarding the WPs 3.3 and 3.4 the hypothesis regarding the transfer of the results known for smooth bimodal maps and double-superstable orbits has been disproved, as already mentioned. Hence, the task of these WPs is solved, and additionally it became possible to devote the time initially proposed for these WPs (as mentioned in the proposal, the WP 3.3 was designed as a preparation for the WP 3.4) to more challenging tasks. As such a task the dynamics of 2D discontinuous maps defined on three partitions has been selected which has been left out in the initial structuring of the project due to lack of time. Accordingly, adjusting the task on the prediction of the occurrence of organizing centres on the results obtained so far, the work on Braess paradox models has been continued. The considered model of a network with two roads is given by a piecewise linear discontinuous 2D map defined on two partition. Introducing the third road corresponds in the map to the appearance of a third partition. Possible dynamics of the model on two partition and their organization in bifurcation structures have been completely understood in the previous works. Within this WP the effect of introducing the third partition on these bifurcation structures is investigated. It is shown that when the third partition is introduced, the bifurcation structure of the parameter plane of interest changes completely. A high number of periodicity regions appears, associated with cycles with points located in all three partitions. As any cycle of the investigated map is necessarily stable, the boundaries of these regions are given by border collision bifurcation curves. Due to the linearity of the map in all three partitions it is possible to calculate these bifurcation curves analytically for cycles of any period, provided that the symbolic sequence associated with the cycle is known. However, it turns out to be a circumstantial task to identify the regularities in the appearance of cycles in the parameter plane of interest. To solve this problem, a two-step approach is applied. First, a different parameter plane is investigated, at the particular parameter values for which the investigated map has a constant value on one on the partitions. This makes it possible – at least, in principle -- to identify all families of cycles existing in this parameter plane, as they are determined by a few organizing centers. Then, as a next step, the results obtained in this plane can easily be transferred back to the original parameter plane of interest. Although set of cycles which occur in the case that map has a constant value on one on the partitions is only a subset of all possible cycles, the obtained results show how several periodicity regions appear, become split in parts and disappear. It has also been shown that the period adding structure occurring between the discussed periodicity regions may be enriched by a regular appearance of coexisting cycles. The obtained results are presented [P16]. The work has been done in collaboration with the authors of the considered model, as proposed by the project plan. Also these collaborations will continue beyond the time of this Marie Curie Fellowship.

The tasks proposed for the WP 3.5 are solves. As an example of a practical application considered within the scope of this WP, the dynamics of a growth model formulated in the tradition of Kaldor and Pasinetti is investigated, where the accumulation of the ratio capital/workers is regulated by a 2D discontinuous map with triangular structure. The obtained results are reported in [P15] and [P12]. In these works the border collision bifurcation boundaries of periodicity regions related to attracting cycles are determined analytically, showing that in a two-dimensional parameter plane these regions are organized in the period adding structure. It is also shown that the cascade of flip bifurcations in the base one-dimensional map corresponds for the two-dimensional map to a sequence of pitchfork and flip bifurcations for cycles of even and odd periods, respectively.

One more application investigated within the scope of this WP is a one-dimensional piecewise-linear map with two discontinuity points describing endogenous bull and bear market dynamics arising from a simple asset-pricing model. As proposed by the Project plan, the work has been performed in collaboration with Dr. I. Sushko. An important feature of the considered model is that some speculators only enter the market if the price is sufficiently distant to its fundamental value. As a first step, the particular case has been investigated in which the map is symmetric with respect to the origin. This case associated with equal market entry thresholds in the bull and bear market. Starting from these results, the analysis has been generalized exploring the bifurcation structures appearing when the symmetry of the map is broken. It is shown that the standard bifurcation structures associated with one discontinuity (period adding and period incrementing structures) persist under symmetry breaking, and are modified only quantitatively. By contrast, bifurcation structures caused by the presence of two discontinuity points give rise to new substructures. The obtained results are presented in the conference talk [T19] and in the paper [P17] which provides also the interpretation of the obtained results from the economical point of view.

The piecewise linear map with two kink points proposed for the WP 1.1 is investigated, the relevant bifurcations structures are identified. The calculation technique (map replacement technique, revisited in [T2]) proposed for the WP 1.2 is successfully adjusted so that its applicability to maps of the considered class is confirmed. Bifurcation structures of the considered map are investigated in [P2] in the case in which the outermost slopes are positive and less than one. Three generic bifurcation structures are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with periodic points on two adjacent branches of the map, the period adding structure is related to periodic points on the outermost branches, and the fin structure is contiguous with the period adding structure and is associated with attracting cycles with at most one point on the middle branch. Analytical expressions for the periodicity region boundaries are obtained using the map replacement technique, as proposed in the project plan.

The work started in [P2] has been continued in [P14] focusing on chaotic dynamics. In particular, the bifurcation structure formed by parameter regions corresponding to chaotic attractors with different number of bands are described, which are adjacent to the regions belonging to the period adding and fin structures. The general mechanism leading the number of bands of a chaotic attractor to change is related to homoclinic bifurcation of a repelling cycle belonging to the immediate basin boundary of this attractor. A classification of possible effects of these bifurcations is presented in [P4]. In particular, one can observe gradual increase in size of the pieces of a chaotic attractor, which then merge pairwise. In general, such a transformation, which is called a merging bifurcation, is associated with a homoclinic bifurcation of a cycle with negative multiplier. The number of pieces of an attractor can also change due to an expansion bifurcation characterized by an abrupt increase in size of the attractor with uneven density of points after the bifurcation. This bifurcation is associated with a homoclinic bifurcation of a cycle with positive multiplier. Taking into account that for the considered bimodal map the conditions of homoclinic bifurcations are related to images of two critical points, it was established a basic technique for an analytical description of the bifurcation structures formed by merging and expansion bifurcations in this map. Additionally, it turned out to be possible to simplify the description by taking advantage of the results known for the skew tent map and by application of the map replacement technique, already applied in [P2].

As the region in the parameter space relevant for the electronic circuit considered in WP 1.3 differs from the one considered in the WPs 1.1 and 1.2 the results of these WPs could not be adopted directly. However, the calculation technique considered in the WP 1.2 was successfully applied also within WP 1.3 as expected. In collaboration with Prof. D. Fournier-Prunaret at INSA (Instutute National des Science Appliquées, Toulouse, France) established within this project a continuous piecewise-linear bimodal map was investigated, which results from the modeling of an electronic circuit previously proposed by Prof. D. Fournier-Prunaret and her collaborators for generation of chaotic signals. Such signals are used for purposes of secure communication and the level of security depends on the properties of the signals. It has been shown in our work that the bifurcation structure of the considered map is not generic, due to some particular features of the map caused by the modelled circuit. Still, the components of this bifurcation structure are generic and could be identified and calculated analytically in an efficient way using the technique proposed in the project plan. The collaborations started within this Marie Curie Fellowship continue beyond the time of the project, as immediately after finishing this project a next joint research activity on a topic similar to the one already proceeded have been started.

For the WP 1.4 a more challenging task than initially planned have been identified. Due to a collaboration with the group of Prof. Mosekilde (Copenhagen) and Prof. Zhusubaliev (Kursk) established within the scope of this Marie Curie Fellowship a new application area for maps with multiple discontinuities was identified. These maps, modelling the dynamic behavior of power electronic DC/AC converters, have by construction a very high number of break points. Due to this, and because of the increasing importance of the considered class of power electronic DC/AC converters (based, among other reasons, on the fact that such converters are unavoidable in solar panels), this work was done within the scope of the WP 1.4. Supported by physical experiments, [P13] considers an unusual transition from the domain of stable fixed points (corresponding to the desired mode of operation of the converter) to chaotic dynamic. The behaviour of the converter is studied by using a 1D stroboscopic map which by construction has a high number of border points. It is shown that due to this high number of border points the transition occurs stepwise, via irregular cascades of different border-collision bifurcations. Although particular bifurcations occurring in the system belong to well-known classes and can be explained using existing theory, such cascades have never been reported before. Remarkably, the obtained results are valid not only for the particular model considered in [P13] but for a broad class of models whose behaviour is essentially influenced by a high number of border points resulting from the mode of operation of DC/AC converters. The importance of this topic has been confirmed by the great interest of the audience for the talks [T16], [T17], [T19] in which the obtained results have been presented. The discussions followed the talks lead to some collaborations which are planned for the time after this project.

The obtained results have been extended in [P7]. In this work it is reported that not only the boundary between regular and chaotic domains in the considered model is predominantly influenced by multiple borders, but also interior structure of the regular domain. In this domain, close to its boundary, the behavior of the system is influenced by cascades of border collisions of so-called persistence type. The mechanism causing such cascades to occur is explained in [P7]. Note that as stable fixed points exist before, at the moment, and after border collisions forming such cascades, they can hardly be detected using standard numerical tools of bifurcation analysis. Still, their detection is an important task from the practical point of view, since the presence of such border collisions significantly decreases the quality of the output signal of the considered DC/AC converters. Although first steps towards solving this tasks are already done in [P7], further improvements of the quality of numerical results are necessary.

The task of the WP 1.5 is solved; the map to be considered in this WP is investigated. This map represents a piecewise linear model which describes dynamics of a business cycle model introduced by Day and Shafer. As initially assumed, the results obtained within the WPs 1.2 and 1.3 can be successfully transferred to this map. The results of this work are presented in [P9]. In particular, in this publication analytical expressions of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and related fin structure) are reported. Note that outside of these region the map has chaotic behaviour (robust chaos) so that the results presented in [P14] apply.

The work on organizing centres performed within the scope of the WP 1.6 represents one of the most significant results of the project. It was rigorously demonstrated that the appearance of organizing centres in 1D maps with an arbitrary number of discontinuities can be predicted in the same way as it is possible in maps with one discontinuity only. The sufficient conditions for different types of organizing centres are derived. The obtained results are presented in [P3] and in the talks [T1], [T4], [T7]. In this work a two-parametric family of 1D piecewise smooth maps with one discontinuity point is considered. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. The obtained results are applicable, however, to 1D piecewise smooth maps with any number of discontinuity points, under the assumption that the border-collisions occur at the same discontinuity. We describe the case of the collision of two stable cycles of any period and any symbolic sequences. For this case, it is proved that the local monotonicity of the functions constituting the first return map defined in a neighborhood of the border point at the parameter values related to the codimension-2 bifurcation point determines, under suitable conditions, the kind of bifurcation structure originating from this point; this can be either a period adding structure, or a period incrementing structure, or simply associated with the coupling of colliding cycles.

One more significant result obtained within this WP regards the possibility to adopt the ideas known for smooth bimodal map for explanation of the appearance of organizing centers in piecewise-smooth maps defined on many partitions. The hypothesis formulated in the project proposal was that these results can be adopted, and the possibility that they can not was mentioned in the risk assessment. However, it follows directly from the results presented in [P2] that in general this hypothesis can be seen as disproved. This result influenced the work done within the project up to some extent, since it has been considered as more efficient to reduce the effort related to the concept of organizing centers and to focus the work on more promising concepts.

The task of the WP 2.1 is solved up to a large extent. The numerical investigations are done for several systems, and the applicability of the map replacement technique for 2D maps is confirmed. The obtained results served as basis for several works, in particular related to the WP 2.2.

The tasks of the WP 2.2 are solved in a much larger extent than initially planned. In particular, the work presented in [P6] is devoted to an adaptive segregation model of Shelling’s type introduced by Bischi and Merlone (2011). The importance of this work is, among other reasons, based on the actuality of segregation problems for European Community. Indeed, the open borders policies of the European community facilitate migration from country to country of people of different nationalities, languages, skills and cultures. This process rises the necessity of integration between the indigenous dwellers and the newcomers. The main force that prevents integration between members of different groups is the limited tolerance of members of one group towards members of other groups. Aware of this, policy makers tend to avoid segregation by combining some integration policies (for example, promoting multiculturalism) with more drastic measures such as the imposition of entry constraints for the members of the different groups. The effect of such constrains was investigates within the scope of WP 2.2. The considered model describes dynamics of entry and exit of individuals belonging to two populations into a system, based on a limited tolerance of the members of each group to the presence of the members from the other group. The constraints are given by the upper limits for the number of individuals of a population that are allowed to enter the system. They represent possible exogenous controls imposed by an authority in order to regulate the system. Using analytical, geometric and numerical methods, we investigated the border collision bifurcations generated by these constraints. In [P6] the case is considered in which the two groups have similar characteristics and have the same level of tolerance toward the members of the other group. The results of this paper are extended in [P8]. In this work the effects of the entry constraints on the dynamics is investigated for the case in which two populations involved differ by the number of their members, for the level of tolerance toward members of the other population and for the speed of reaction. The investigation of the dynamics of the segregation model for an asymmetric setting of the parameters reveals that other types of bifurcations can occur than in the symmetric case studied before. The investigation reveals that asymmetries in the level of tolerance of the two populations involved may led to effects of overreaction in the adjustment process. In order to avoid the risk of segregation, suitable entry limitations must be imposed at least on the more tolerant population. This is an interesting and surprising result contradicting the common opinion that to avoid segregation an entry constraint should be imposed on the less tolerant population.

The results of WP 2.2 related to the Braess paradox and the ternary choice problem are summarized in [P10]. The Braess paradox is a well-known example in which individual rationality of the participants may lead to a collective irrationality. By contrast to the models considered in the literature, a system consisting of non-impulsive commuters changing their decisions proportionally to the cost difference is considered. For a simplified version of the problem associated with a binary choice a rigorous proof of the existence of a unique fixed point is provided, showing that it is globally attracting even if it may be locally unstable. This is possible because modelling of this system results in a 1D continuous map. In the case of a ternary choice, the system is necessarily modelled by a discontinuous 2D map. In this case it can also be proved that a unique fixed point exists, but its global attractivity is numerically evidenced, also when the fixed point is locally unstable. The work on Braess paradox models has been continued within the scope of WP 3 as described below.

The tasks of the WPs 3.1 and 3.2 are solved. The obtained results have been embedded as technical background in the work done within the WP 3.5. Regarding the WPs 3.3 and 3.4 the hypothesis regarding the transfer of the results known for smooth bimodal maps and double-superstable orbits has been disproved, as already mentioned. Hence, the task of these WPs is solved, and additionally it became possible to devote the time initially proposed for these WPs (as mentioned in the proposal, the WP 3.3 was designed as a preparation for the WP 3.4) to more challenging tasks. As such a task the dynamics of 2D discontinuous maps defined on three partitions has been selected which has been left out in the initial structuring of the project due to lack of time. Accordingly, adjusting the task on the prediction of the occurrence of organizing centres on the results obtained so far, the work on Braess paradox models has been continued. The considered model of a network with two roads is given by a piecewise linear discontinuous 2D map defined on two partition. Introducing the third road corresponds in the map to the appearance of a third partition. Possible dynamics of the model on two partition and their organization in bifurcation structures have been completely understood in the previous works. Within this WP the effect of introducing the third partition on these bifurcation structures is investigated. It is shown that when the third partition is introduced, the bifurcation structure of the parameter plane of interest changes completely. A high number of periodicity regions appears, associated with cycles with points located in all three partitions. As any cycle of the investigated map is necessarily stable, the boundaries of these regions are given by border collision bifurcation curves. Due to the linearity of the map in all three partitions it is possible to calculate these bifurcation curves analytically for cycles of any period, provided that the symbolic sequence associated with the cycle is known. However, it turns out to be a circumstantial task to identify the regularities in the appearance of cycles in the parameter plane of interest. To solve this problem, a two-step approach is applied. First, a different parameter plane is investigated, at the particular parameter values for which the investigated map has a constant value on one on the partitions. This makes it possible – at least, in principle -- to identify all families of cycles existing in this parameter plane, as they are determined by a few organizing centers. Then, as a next step, the results obtained in this plane can easily be transferred back to the original parameter plane of interest. Although set of cycles which occur in the case that map has a constant value on one on the partitions is only a subset of all possible cycles, the obtained results show how several periodicity regions appear, become split in parts and disappear. It has also been shown that the period adding structure occurring between the discussed periodicity regions may be enriched by a regular appearance of coexisting cycles. The obtained results are presented [P16]. The work has been done in collaboration with the authors of the considered model, as proposed by the project plan. Also these collaborations will continue beyond the time of this Marie Curie Fellowship.

The tasks proposed for the WP 3.5 are solves. As an example of a practical application considered within the scope of this WP, the dynamics of a growth model formulated in the tradition of Kaldor and Pasinetti is investigated, where the accumulation of the ratio capital/workers is regulated by a 2D discontinuous map with triangular structure. The obtained results are reported in [P15] and [P12]. In these works the border collision bifurcation boundaries of periodicity regions related to attracting cycles are determined analytically, showing that in a two-dimensional parameter plane these regions are organized in the period adding structure. It is also shown that the cascade of flip bifurcations in the base one-dimensional map corresponds for the two-dimensional map to a sequence of pitchfork and flip bifurcations for cycles of even and odd periods, respectively.

One more application investigated within the scope of this WP is a one-dimensional piecewise-linear map with two discontinuity points describing endogenous bull and bear market dynamics arising from a simple asset-pricing model. As proposed by the Project plan, the work has been performed in collaboration with Dr. I. Sushko. An important feature of the considered model is that some speculators only enter the market if the price is sufficiently distant to its fundamental value. As a first step, the particular case has been investigated in which the map is symmetric with respect to the origin. This case associated with equal market entry thresholds in the bull and bear market. Starting from these results, the analysis has been generalized exploring the bifurcation structures appearing when the symmetry of the map is broken. It is shown that the standard bifurcation structures associated with one discontinuity (period adding and period incrementing structures) persist under symmetry breaking, and are modified only quantitatively. By contrast, bifurcation structures caused by the presence of two discontinuity points give rise to new substructures. The obtained results are presented in the conference talk [T19] and in the paper [P17] which provides also the interpretation of the obtained results from the economical point of view.