Periodic Reporting for period 2 - Dynamics (Contributions to codimension k bifurcations in dynamical systems theory)
Période du rapport: 2020-04-01 au 2024-03-31
Dynamical systems theory (DST) and its applications is a vast area of research where new knowledge is expected to be discovered. The problem addressed in this project is to study different classes of continuous and discrete dynamical systems (smooth or non-smooth). The problem is important for society because new results in this area lead to a better understanding of real-life phenomena, met in biology, medicine, engineering and ecology. The overall project objectives are to produce new knowledge in DST and provide training to early-stage researchers. As a general conclusion, the project’s success can be measured by the new results reported in the more 60 articles published mostly in high-ranking international journals. About 45 researchers took part in project activities and benefited from valuable networking opportunities during the 225 months of secondments on the project.
The project has six precise tasks, T1.1-T1.4 and T2.1-T2.2 which have been approached during the 72 months of the project.
T1.1 Study degenerate Bautin bifurcations. Results have been obtained on this topic in the first part of the project, when a normal form which involves the third Lyapunov coefficient and a method for determining the coefficient have been obtained.
T1.2 Study degenerate Hopf-Hopf bifurcation. We approached this bifurcation, known also as double-Hopf, in a degenerate context given by the elimination of one generic condition at once used for determining a normal form in the non-degenerate case. Six such degenerate cases have been studied and the results are published in three papers [1]-[3].
T1.3 Study other codimension k bifurcations in continuous (smooth) systems. We have studied: a model for the tuberculosis and the Dengue fever [4], a system related to the two-body problem [5], a finance model [6], a dynamical system with an equilibrium having two zero eigenvalues [7], the birth and death of families of algebraic limit cycles of quadratic polynomial differential systems (pde) [8], the criticality of some planar systems of pde having a center for various low degrees n [9], the phase portraits of a 3-dimensional Lotka–Volterra system for all the values of two parameters [10], the dynamics of a 3-dimensional Muthuswamy-Chua system in the infinity of the Poincare ball for all values of the parameter [11].
T1.4 Study other codim k bifurcations in discrete systems. Related to this task we have studied: degenerate Chenciner bifurcation [27]-[29], the fixed points of a return map [30], continuous self-maps on topological graphs [31], the set of periods of the Morse–Smale diffeomorphisms in several cases [32].
T2.1 Study codim k bifurcations in non-smooth systems. We have studied: discontinuous planar piecewise differential systems (pds) formed by linear centers and separated by two concentric circles [33], the number of limit cycles of planar pds separated by a branch of an algebraic curves [34], discontinuous pds formed by two linear centers separated by a nonregular straight line [35], the conjecture of Markus and Yamabe to continuous and discontinuous pds [36].
T2.2 Study bifurcations in non-smooth systems with impacts and applications of mathematics and other disciplines in the medical field.
We have studied: a mathematical model for the interactions between human immune system and a pathogenic virus, such as Covid-19 [41], several aspects in computer-assisted surgery [42], a SIR system [43], a SEIR model [44], a predator–prey system [45], two NZP models [46], a model for cancerous tumors [47] and [48].
As an overview, the project main results are in dynamical systems theory and its applications, where they can be exploited. The results are disseminated as green or gold open access articles.
[1] G. Tigan, 2021a.
[2]-[3] G. Moza, 2022a, 2022b.
[4]-[5] J. Llibre, 2019, 2018.
[6] M.R. Candido, 2018.
[7] C. Rocsoreanu, 2019.
[8] J. Llibre, 2021a.
[9] I. Sánchez-Sánchez, 2021a.
[10] J. Llibre, 2020a.
[11] Y. P. Martínez, 2024.
[12]-[13] J. Llibre, 2022, 2021b.
[14] R. Oliveira, 2022.
[15] I. Sánchez-Sánchez, 2021b.
[16] C. A. Buzzi, 2021.
[17] G. Moza, 2022c.
[18] J. Itikawa, 2022.
[19] I. Sánchez-Sánchez, 2022.
[20] M. Boureanu, 2022.
[21] E. Musafirov, 2022.
[22] C-P. Danet, 2022.
[23] C. Ionescu, 2022.
[24]-[25] G. Moza, 2023a, 2024.
[26] J. Llibre, 2023.
[27]-[28] G. Tigan, 2020, 2021b.
[29] S. Lugojan, 2022.
[30]-[31] J. Llibre, 2020b, 2021c.
[32] C. Cufi-Cabre, 2022.
[33] M.E. Anacleto, 2021.
[34] A. Gasull, 2020.
[35] M. Esteban, 2021.
[36] J. Llibre, 2021d.
[37] E. Freire, 2021.
[38] X. Chen, 2022.
[39] L. P. C. da Cruz, 2022.
[40] M. E. Anacleto, 2023.
[41] G. Moza, 2023b.
[42] J. Ehrlich, 2022.
[43] D. Constantinescu, 2022.
[44] L. Bucur, 2022.
[45] F. Munteanu, 2022.
[46] M. Sterpu, 2023.
[47] D. A. Drexler, 2024.
[48] L. Kovács, 2024.
T1.1 Study degenerate Bautin bifurcations. Results have been obtained on this topic in the first part of the project, when a normal form which involves the third Lyapunov coefficient and a method for determining the coefficient have been obtained.
T1.2 Study degenerate Hopf-Hopf bifurcation. We approached this bifurcation, known also as double-Hopf, in a degenerate context given by the elimination of one generic condition at once used for determining a normal form in the non-degenerate case. Six such degenerate cases have been studied and the results are published in three papers [1]-[3].
T1.3 Study other codimension k bifurcations in continuous (smooth) systems. We have studied: a model for the tuberculosis and the Dengue fever [4], a system related to the two-body problem [5], a finance model [6], a dynamical system with an equilibrium having two zero eigenvalues [7], the birth and death of families of algebraic limit cycles of quadratic polynomial differential systems (pde) [8], the criticality of some planar systems of pde having a center for various low degrees n [9], the phase portraits of a 3-dimensional Lotka–Volterra system for all the values of two parameters [10], the dynamics of a 3-dimensional Muthuswamy-Chua system in the infinity of the Poincare ball for all values of the parameter [11].
T1.4 Study other codim k bifurcations in discrete systems. Related to this task we have studied: degenerate Chenciner bifurcation [27]-[29], the fixed points of a return map [30], continuous self-maps on topological graphs [31], the set of periods of the Morse–Smale diffeomorphisms in several cases [32].
T2.1 Study codim k bifurcations in non-smooth systems. We have studied: discontinuous planar piecewise differential systems (pds) formed by linear centers and separated by two concentric circles [33], the number of limit cycles of planar pds separated by a branch of an algebraic curves [34], discontinuous pds formed by two linear centers separated by a nonregular straight line [35], the conjecture of Markus and Yamabe to continuous and discontinuous pds [36].
T2.2 Study bifurcations in non-smooth systems with impacts and applications of mathematics and other disciplines in the medical field.
We have studied: a mathematical model for the interactions between human immune system and a pathogenic virus, such as Covid-19 [41], several aspects in computer-assisted surgery [42], a SIR system [43], a SEIR model [44], a predator–prey system [45], two NZP models [46], a model for cancerous tumors [47] and [48].
As an overview, the project main results are in dynamical systems theory and its applications, where they can be exploited. The results are disseminated as green or gold open access articles.
[1] G. Tigan, 2021a.
[2]-[3] G. Moza, 2022a, 2022b.
[4]-[5] J. Llibre, 2019, 2018.
[6] M.R. Candido, 2018.
[7] C. Rocsoreanu, 2019.
[8] J. Llibre, 2021a.
[9] I. Sánchez-Sánchez, 2021a.
[10] J. Llibre, 2020a.
[11] Y. P. Martínez, 2024.
[12]-[13] J. Llibre, 2022, 2021b.
[14] R. Oliveira, 2022.
[15] I. Sánchez-Sánchez, 2021b.
[16] C. A. Buzzi, 2021.
[17] G. Moza, 2022c.
[18] J. Itikawa, 2022.
[19] I. Sánchez-Sánchez, 2022.
[20] M. Boureanu, 2022.
[21] E. Musafirov, 2022.
[22] C-P. Danet, 2022.
[23] C. Ionescu, 2022.
[24]-[25] G. Moza, 2023a, 2024.
[26] J. Llibre, 2023.
[27]-[28] G. Tigan, 2020, 2021b.
[29] S. Lugojan, 2022.
[30]-[31] J. Llibre, 2020b, 2021c.
[32] C. Cufi-Cabre, 2022.
[33] M.E. Anacleto, 2021.
[34] A. Gasull, 2020.
[35] M. Esteban, 2021.
[36] J. Llibre, 2021d.
[37] E. Freire, 2021.
[38] X. Chen, 2022.
[39] L. P. C. da Cruz, 2022.
[40] M. E. Anacleto, 2023.
[41] G. Moza, 2023b.
[42] J. Ehrlich, 2022.
[43] D. Constantinescu, 2022.
[44] L. Bucur, 2022.
[45] F. Munteanu, 2022.
[46] M. Sterpu, 2023.
[47] D. A. Drexler, 2024.
[48] L. Kovács, 2024.
T1.1. A method to find the third Lyapunov coefficient has been obtained. Since other similar methods exist, the results diversify the existing methods but do not go beyond the state of the art.
T1.2. The results we have obtained in the six degenerate cases of the double-Hopf bifurcation go beyond the state of the art. New bifurcation diagrams and properties have emerged due to the degeneracies.
T1.3. New results are reported in the studied systems: four new theorems [5], a new method related to limit cycles [6], new results on a 3D Lotka–Volterra system [7] and Muthuswamy-Chua system [8], all the phase portraits of a class of systems [9], the existence of at least one limit cycle in a Belousov–Zhabotinsky model [10], a method to study the simultaneity [11], new results on the maximum number of critical periods [12], the center-focus and cyclicity problems [13], a new system for modeling inflation rate [14], a compact expression for the first-order Taylor series of the Melnikov function [15], a quintic system with 92 limit cycles [16], new results on boundary value [17], the existence of an infinite set of limit cycles in a Langford system [18], new results for a semi linear sixth-order ODE [19], a method for finding traveling wave solutions [20], two new models in economy [21,22], and new results on the existence of first integrals in a system [23]. The results go beyond the state of the art.
T1.4. The results for this task go beyond the state of the art, such as: new results on the degenerate Chenciner bifurcation [24,27,28], new insights on the limit cycles using return map [25], new results for self-maps on topological graphs [26], and the Morse–Smale diffeomorphisms [29].
T2.1. New results which go beyond the state of the art are reported for this topic, such as: contributions to the 16th Hilbert problem for piecewise differential systems [30], [32], [37], new insights on the number of limit cycles [31], with a periodic orbit at infinity [34] and related to the center-focus problem [36], results on the Markus and Yamabe conjecture [33], a new contribution to the Melnikov function [35].
T2.2. The results presented for this task go beyond the state of the art, such as: a model to study the interactions between human immune system and a pathogenic virus [38], new results on a SIR model [40], a SEIR model [41], respectively a predator–prey model [42], contributions to computer-assisted surgery with results demonstrating the feasibility of automatically identifying energy events during surgical incisions [39], two new NZP models [43], new results in modeling cancerous tumors [44] and [45].
T1.2. The results we have obtained in the six degenerate cases of the double-Hopf bifurcation go beyond the state of the art. New bifurcation diagrams and properties have emerged due to the degeneracies.
T1.3. New results are reported in the studied systems: four new theorems [5], a new method related to limit cycles [6], new results on a 3D Lotka–Volterra system [7] and Muthuswamy-Chua system [8], all the phase portraits of a class of systems [9], the existence of at least one limit cycle in a Belousov–Zhabotinsky model [10], a method to study the simultaneity [11], new results on the maximum number of critical periods [12], the center-focus and cyclicity problems [13], a new system for modeling inflation rate [14], a compact expression for the first-order Taylor series of the Melnikov function [15], a quintic system with 92 limit cycles [16], new results on boundary value [17], the existence of an infinite set of limit cycles in a Langford system [18], new results for a semi linear sixth-order ODE [19], a method for finding traveling wave solutions [20], two new models in economy [21,22], and new results on the existence of first integrals in a system [23]. The results go beyond the state of the art.
T1.4. The results for this task go beyond the state of the art, such as: new results on the degenerate Chenciner bifurcation [24,27,28], new insights on the limit cycles using return map [25], new results for self-maps on topological graphs [26], and the Morse–Smale diffeomorphisms [29].
T2.1. New results which go beyond the state of the art are reported for this topic, such as: contributions to the 16th Hilbert problem for piecewise differential systems [30], [32], [37], new insights on the number of limit cycles [31], with a periodic orbit at infinity [34] and related to the center-focus problem [36], results on the Markus and Yamabe conjecture [33], a new contribution to the Melnikov function [35].
T2.2. The results presented for this task go beyond the state of the art, such as: a model to study the interactions between human immune system and a pathogenic virus [38], new results on a SIR model [40], a SEIR model [41], respectively a predator–prey model [42], contributions to computer-assisted surgery with results demonstrating the feasibility of automatically identifying energy events during surgical incisions [39], two new NZP models [43], new results in modeling cancerous tumors [44] and [45].