Final Report Summary - QURIBIUS (Qualitative theory and non-degenerate and degenerate bifurcations in n-dimensional dynamical systems)
In this project we have three main objectives:
1) The first one is related to fold-Hopf degenerate bifurcations. We investigated the case when the fifth generic conditions fails: the map transforming old parameters into new ones is not regular at the origin. We studied also the Morioka-Shimizu model, that emerges as an asymptotic normal form for bifurcations of triply degenerate equilibrium states and periodic orbits in systems with certain types of symmetry.
2) The second objective is to investigate the existence of limit cycles in two-dimensional (2D) polynomial differential systems. To this aim, we investigated a concrete model: a perturbed quadratic Hamiltonian system with general quadratic polynomial perturbations.
3) The last objective of this project is to propose and investigate up to some extent two concrete models: a predator-prey one and a perturbed interpolating Hamiltonian system.
For the first objective, we started with a general three-dimensional (3D) nonlinear differential equation. We used the normal forms for the fold-Hopf non-degenerate case. Since the transforming map is not regular, we had to work with the old parameters. The degeneracy gave birth to new bifurcation curves and a wealth of bifurcation diagrams have been obtained. The studies carried out so far are summarised in the notes: Analysis of degenerate fold-Hopf bifurcations. Moreover, for the Simizu-Morioka model, we obtained an analytic (free of computer assistance) derivation of the existence of the homoclinic loops with zero saddle value in this system, which is an essential step forward towards a fully analytical proof of the existence of the Lorenz attractor in this model. The results on this model have been published in Physica D: 'Analytical search for homoclinic bifurcations in the Shimizu-Morioka model', Physica D 240 (2011) 985-989.
For the second objective, we have investigated the existence of limit cycles in a 2D (perturbed Hamiltonian) polynomial differential system. By choosing a proper decomposition for the system, we were able to use the k-Melnikov functions with k large in some degenerate cases, by means of an algorithm developed by other researchers. We have obtained the number of limit cycles for our model. The studies are summarised in the preprint: 'Application of Melnikov functions to a Hamiltonian quadratic system'.
Moreover, for this second objective, we started working on a class of discontinuous nonlinear dynamical systems. For this class of systems, we have introduced the concept of degenerate grazing orbits. This degeneracy gives rise to a complex dynamics near the grazing orbit. Partial results obtained so far on this topic have been presented at: SIAM Conference on Applications of Dynamical Systems, DS11, May 22-26, Snowbird, United States (US).
For the last objective of the project, we proposed a 2D predator-prey model and a perturbed interpolating Hamiltonian system. The models have been investigated up to some extent and the work carried out so far is summarised in the notes: 'Analysis of a predator-predator model, respectively, Bifurcation process in Hamiltonian systems associated to nontwist cubic maps'.
Brief description of the main results achieved so far.
a) The existence of various bifurcation diagrams in degenerate 3D fold-Hopf bifurcations. The analytical proof of the existence of the homoclinic loops with zero saddle value in the Shimizu-Morioka system.
b) An exhaustive study of a perturbed quadratic Hamiltonian system with general quadratic polynomial perturbations with respect to the existence of limit cycles.
c) The potential existence of degenerate periodic grazing bifurcations in one-dimensional impact oscillators.
d) Description of the reconnection process of any two neighbouring chains when two parameters are fixed and one varies.
1) The first one is related to fold-Hopf degenerate bifurcations. We investigated the case when the fifth generic conditions fails: the map transforming old parameters into new ones is not regular at the origin. We studied also the Morioka-Shimizu model, that emerges as an asymptotic normal form for bifurcations of triply degenerate equilibrium states and periodic orbits in systems with certain types of symmetry.
2) The second objective is to investigate the existence of limit cycles in two-dimensional (2D) polynomial differential systems. To this aim, we investigated a concrete model: a perturbed quadratic Hamiltonian system with general quadratic polynomial perturbations.
3) The last objective of this project is to propose and investigate up to some extent two concrete models: a predator-prey one and a perturbed interpolating Hamiltonian system.
For the first objective, we started with a general three-dimensional (3D) nonlinear differential equation. We used the normal forms for the fold-Hopf non-degenerate case. Since the transforming map is not regular, we had to work with the old parameters. The degeneracy gave birth to new bifurcation curves and a wealth of bifurcation diagrams have been obtained. The studies carried out so far are summarised in the notes: Analysis of degenerate fold-Hopf bifurcations. Moreover, for the Simizu-Morioka model, we obtained an analytic (free of computer assistance) derivation of the existence of the homoclinic loops with zero saddle value in this system, which is an essential step forward towards a fully analytical proof of the existence of the Lorenz attractor in this model. The results on this model have been published in Physica D: 'Analytical search for homoclinic bifurcations in the Shimizu-Morioka model', Physica D 240 (2011) 985-989.
For the second objective, we have investigated the existence of limit cycles in a 2D (perturbed Hamiltonian) polynomial differential system. By choosing a proper decomposition for the system, we were able to use the k-Melnikov functions with k large in some degenerate cases, by means of an algorithm developed by other researchers. We have obtained the number of limit cycles for our model. The studies are summarised in the preprint: 'Application of Melnikov functions to a Hamiltonian quadratic system'.
Moreover, for this second objective, we started working on a class of discontinuous nonlinear dynamical systems. For this class of systems, we have introduced the concept of degenerate grazing orbits. This degeneracy gives rise to a complex dynamics near the grazing orbit. Partial results obtained so far on this topic have been presented at: SIAM Conference on Applications of Dynamical Systems, DS11, May 22-26, Snowbird, United States (US).
For the last objective of the project, we proposed a 2D predator-prey model and a perturbed interpolating Hamiltonian system. The models have been investigated up to some extent and the work carried out so far is summarised in the notes: 'Analysis of a predator-predator model, respectively, Bifurcation process in Hamiltonian systems associated to nontwist cubic maps'.
Brief description of the main results achieved so far.
a) The existence of various bifurcation diagrams in degenerate 3D fold-Hopf bifurcations. The analytical proof of the existence of the homoclinic loops with zero saddle value in the Shimizu-Morioka system.
b) An exhaustive study of a perturbed quadratic Hamiltonian system with general quadratic polynomial perturbations with respect to the existence of limit cycles.
c) The potential existence of degenerate periodic grazing bifurcations in one-dimensional impact oscillators.
d) Description of the reconnection process of any two neighbouring chains when two parameters are fixed and one varies.