## Final Report Summary - DYNSYSAPLL (DYNAMICAL SYSTEMS AND THEIR APPLICATIONS)

The main objective of this project is to create fundamental understanding in dynamical systems theory and to apply this theory in formulating and analyzing real world models met especially in Neuroscience, Plasma Physics and Medicine. The specific objectives are contained in the 5 WPs of the project. In WP1 we want to develop new methods for the center and isochronicity problems, study bifurcations of limit cycles and critical periods. In WP2 we deal with the problem of integrability for some differential systems with invariant algebraic curves and classification of cubic systems with a given number of invariant lines. The main objective of WP3 is to study dynamics of some classes of continuous and discontinuous vector fields. WP4 deals with Hamiltonian systems, discrete maps, numerical methods, and the study of symmetries of certain kinds of k-cosymplectic Hamiltonians. The last WP tackles neuronal and bones models.

WP1. Task1.1. “Center and isochronicity problems.” We have developed new methods based on generalized symmetries and the elimination theory to study isochronicity and linearizability. Task1.2. “The center problem for ODEs.” We proved a theorem which supports that it does not exist planar polynomial Hamiltonian systems of even degree having an isochronous center. We generalized a (weakly) persistent center problem for complex planar differential systems. Task1.3. “Computational methods for bifurcation of limit cycles.” Computational methods for bifurcations of limit cycles in 2D systems were developed. A computational approach to find an upper bound for the cyclicity in terms of the Bautin depth of the ideal generated by the switching Lyapunov quantities was proposed. A new computational approach based on the parallelization of computing the linear parts of the Lyapunov quantities was proposed. Task1.4. “Bifurcation of critical periods.” A general approach to studying bifurcations of critical periods based on a complexification of the system and algorithms of computational algebra was developed. Computational routines to perform the calculations using the computer algebra system Singular were proposed. Task1.5. “Non-analytical and degenerate systems.“ We found 42 different phase portraits for a non-analytical (non-smooth) system. We proved that at most three limit cycles can bifurcate from a degenerate center in a cubic homogeneous polynomial differential system. As far as we know, this was the first time that a complete analysis up to second order in the small parameter of the perturbation was done by averaging method.

WP2. Task2.1. “Integrability for some ODEs.“ We proved the Chen-Wang system has neither invariant algebraic surfaces nor polynomial and Darboux first integrals. We provided a characterization of integrablity of a system of vector fields via inverse Jacobian multipliers and normalizers of smooth vector fields. Task2.2. “Configurations of invariant straight lines.” We obtained a classification theorem and bifurcation diagram in the 12-dimensional parameter space of a class of quadratic systems according to the configuration of singularities at infinity of the systems. We established a geometric classification of singularities, finite and infinite, for two subclasses of quadratic differential systems with total finite multiplicity 4 possessing exactly 3 finite singularities. Task2.3. “To classify cubic systems.” We proved a classification theorem of real planar cubic vector fields which possess 3 (or 4) distinct infinite singularities and 8 invariant straight lines, respectively. We classified cubic systems with degenerate infinity and having invariant straight lines of total parallel multiplicity five. Task2.4. “Coexistence of limit cycles with lines and centers.“ It is proved for a class of cubic differential systems that a weak focus is a center if and only if the first two Liapunov quantities vanish. Task2.5. “Global attractors.” We proved the analogues of Birkhoff’s theorem for one-sided dynamical systems (continuous and discrete times) with non-compact space having a compact global attractor. We generalized the Absil-Kurdyka theorem about stability of gradient systems with analytic potential for non-gradient systems, respectively, the Lagrange-Dirichet theorem. Task2.6. “Levitan/Bohr almost periodic motions." We obtained new results for the C-analytical version of Belitskii-Lyubich conjecture and a complete description of the structure of compact global attractors for non-autonomous perturbations of autonomous gradient-like dynamical systems.

WP3. Task3.1. “Systems with symmetries.” We obtained results on the existence conditions of families of homoclinic orbits associated to periodic orbits near an equilibrium, respectively, on bounded orbits such as closed, homoclinic and heteroclinic in a Lorenz-like 3D nonlinear system having some symmetries. Task3.2. “Systems without symmetries.” We provided a new, direct, unified and easier proof of the fact that the 6 models of Bianchi class A have no periodic solutions. We obtained an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center. We obtained new results on degenerate with respect to parameters fold-Hopf bifurcations in a class of 3D systems. Task3.3. “Bifurcation of limit cycles or invariant manifolds.” We obtained results on limit cycles for discontinuous generalized Lienard polynomial differential equations. We provided a method to deal with limit cycles in planar piecewise smooth Hamiltonian systems. We established an averaging theory for one dimensional perturbed piecewise smooth periodic differential equations. We developed the averaging theory of first and second order for studying the periodic solutions of discontinuous piecewise differential systems in arbitrary dimension and with an arbitrary number of systems. Task3.4. “Study of typical singularities.” We reported results on bifurcation diagrams for the local Hopf bifurcations of codimension one and two in the Shimizu-Morioka system. We obtained results on topological equivalence for centers (either nondegenerate or degenerate) of a planar piecewise smooth vector fields.

WP4. Task4.1. “Hamiltonian systems with one-and-a-half degrees of freedom.” We presented the behavior of Poincaré–Birkhoff or dimerised chains in their routes to reconnection in a perturbed Hamiltonian model (m1). The reconnection scenarios we obtained in our model present some similarities with the behavior of magnetic field lines in tokamaks. Task4.2. “A further study of a previous model.“ We obtained results on the Hopf bifurcation in a low dimensional model (m2) which theoretically describes the dynamics of the plasma pressure gradient and amplitude of the displacement of the magnetic field in tokamaks. Task4.3. “Numerical methods in these models.” We obtained interesting properties for the first model (m1), such as pairs of homoclinic orbits and triple reconnection. The study is highly based on numerical approaches. We have built a specific code based on matrix approach in order to study numerically the second model (m2). Task4.4. “Numerical simulations of specific models from biology, ecology and engineering.” We obtained results on models used in ecology for describing the spread of epidemics and the interactions between two, three or more species (Lotka-Volterra) as well as models describing electronic circuits (Chua model). For these models some computational studies were performed and some conservation laws were pointed out. Task4.5. “Integral curves and Hamiltonians (Lagrangians).” A class of models associated with a Hamilton function was studied from a geometrical point of view. A method for obtaining a sequence of k-semisprays and two sequences of nonlinear connections on the k-tangent bundle was obtained.

WP5. Task5.1. “ODE-based and map-based neuronal models.“ We showed the existence of canard solutions in FitzHugh-Nagumo equations and the Hodgkin-Huxley model. We obtained results on the existence of global exponential stability of antiperiodic solutions of a class of impulsive neural networks of type Cohen-Grossberg with delays. Task5.2. “Modelling osteoblastic cellular population.” We improved a human cortical bone model (SiNuPrOs) by a module which is able to find the possible architectural configurations of a human cortical bone in correlation with elementary volume. It was shown that this module reproduces the mechanical properties measured experimentally.

With few exceptions, the project results are relevant to dynamical systems theory and its applications and we do not see a direct socio-economic impact of them. The exceptions refer at the models we studied within the Tasks 4.1-4.3 which may have some relevance for physicists working in plasma physics while the model studied in the Task 5.2 for medical researchers. We do not think the results obtained so far within the project can be exploited in industry.

WP1. Task1.1. “Center and isochronicity problems.” We have developed new methods based on generalized symmetries and the elimination theory to study isochronicity and linearizability. Task1.2. “The center problem for ODEs.” We proved a theorem which supports that it does not exist planar polynomial Hamiltonian systems of even degree having an isochronous center. We generalized a (weakly) persistent center problem for complex planar differential systems. Task1.3. “Computational methods for bifurcation of limit cycles.” Computational methods for bifurcations of limit cycles in 2D systems were developed. A computational approach to find an upper bound for the cyclicity in terms of the Bautin depth of the ideal generated by the switching Lyapunov quantities was proposed. A new computational approach based on the parallelization of computing the linear parts of the Lyapunov quantities was proposed. Task1.4. “Bifurcation of critical periods.” A general approach to studying bifurcations of critical periods based on a complexification of the system and algorithms of computational algebra was developed. Computational routines to perform the calculations using the computer algebra system Singular were proposed. Task1.5. “Non-analytical and degenerate systems.“ We found 42 different phase portraits for a non-analytical (non-smooth) system. We proved that at most three limit cycles can bifurcate from a degenerate center in a cubic homogeneous polynomial differential system. As far as we know, this was the first time that a complete analysis up to second order in the small parameter of the perturbation was done by averaging method.

WP2. Task2.1. “Integrability for some ODEs.“ We proved the Chen-Wang system has neither invariant algebraic surfaces nor polynomial and Darboux first integrals. We provided a characterization of integrablity of a system of vector fields via inverse Jacobian multipliers and normalizers of smooth vector fields. Task2.2. “Configurations of invariant straight lines.” We obtained a classification theorem and bifurcation diagram in the 12-dimensional parameter space of a class of quadratic systems according to the configuration of singularities at infinity of the systems. We established a geometric classification of singularities, finite and infinite, for two subclasses of quadratic differential systems with total finite multiplicity 4 possessing exactly 3 finite singularities. Task2.3. “To classify cubic systems.” We proved a classification theorem of real planar cubic vector fields which possess 3 (or 4) distinct infinite singularities and 8 invariant straight lines, respectively. We classified cubic systems with degenerate infinity and having invariant straight lines of total parallel multiplicity five. Task2.4. “Coexistence of limit cycles with lines and centers.“ It is proved for a class of cubic differential systems that a weak focus is a center if and only if the first two Liapunov quantities vanish. Task2.5. “Global attractors.” We proved the analogues of Birkhoff’s theorem for one-sided dynamical systems (continuous and discrete times) with non-compact space having a compact global attractor. We generalized the Absil-Kurdyka theorem about stability of gradient systems with analytic potential for non-gradient systems, respectively, the Lagrange-Dirichet theorem. Task2.6. “Levitan/Bohr almost periodic motions." We obtained new results for the C-analytical version of Belitskii-Lyubich conjecture and a complete description of the structure of compact global attractors for non-autonomous perturbations of autonomous gradient-like dynamical systems.

WP3. Task3.1. “Systems with symmetries.” We obtained results on the existence conditions of families of homoclinic orbits associated to periodic orbits near an equilibrium, respectively, on bounded orbits such as closed, homoclinic and heteroclinic in a Lorenz-like 3D nonlinear system having some symmetries. Task3.2. “Systems without symmetries.” We provided a new, direct, unified and easier proof of the fact that the 6 models of Bianchi class A have no periodic solutions. We obtained an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center. We obtained new results on degenerate with respect to parameters fold-Hopf bifurcations in a class of 3D systems. Task3.3. “Bifurcation of limit cycles or invariant manifolds.” We obtained results on limit cycles for discontinuous generalized Lienard polynomial differential equations. We provided a method to deal with limit cycles in planar piecewise smooth Hamiltonian systems. We established an averaging theory for one dimensional perturbed piecewise smooth periodic differential equations. We developed the averaging theory of first and second order for studying the periodic solutions of discontinuous piecewise differential systems in arbitrary dimension and with an arbitrary number of systems. Task3.4. “Study of typical singularities.” We reported results on bifurcation diagrams for the local Hopf bifurcations of codimension one and two in the Shimizu-Morioka system. We obtained results on topological equivalence for centers (either nondegenerate or degenerate) of a planar piecewise smooth vector fields.

WP4. Task4.1. “Hamiltonian systems with one-and-a-half degrees of freedom.” We presented the behavior of Poincaré–Birkhoff or dimerised chains in their routes to reconnection in a perturbed Hamiltonian model (m1). The reconnection scenarios we obtained in our model present some similarities with the behavior of magnetic field lines in tokamaks. Task4.2. “A further study of a previous model.“ We obtained results on the Hopf bifurcation in a low dimensional model (m2) which theoretically describes the dynamics of the plasma pressure gradient and amplitude of the displacement of the magnetic field in tokamaks. Task4.3. “Numerical methods in these models.” We obtained interesting properties for the first model (m1), such as pairs of homoclinic orbits and triple reconnection. The study is highly based on numerical approaches. We have built a specific code based on matrix approach in order to study numerically the second model (m2). Task4.4. “Numerical simulations of specific models from biology, ecology and engineering.” We obtained results on models used in ecology for describing the spread of epidemics and the interactions between two, three or more species (Lotka-Volterra) as well as models describing electronic circuits (Chua model). For these models some computational studies were performed and some conservation laws were pointed out. Task4.5. “Integral curves and Hamiltonians (Lagrangians).” A class of models associated with a Hamilton function was studied from a geometrical point of view. A method for obtaining a sequence of k-semisprays and two sequences of nonlinear connections on the k-tangent bundle was obtained.

WP5. Task5.1. “ODE-based and map-based neuronal models.“ We showed the existence of canard solutions in FitzHugh-Nagumo equations and the Hodgkin-Huxley model. We obtained results on the existence of global exponential stability of antiperiodic solutions of a class of impulsive neural networks of type Cohen-Grossberg with delays. Task5.2. “Modelling osteoblastic cellular population.” We improved a human cortical bone model (SiNuPrOs) by a module which is able to find the possible architectural configurations of a human cortical bone in correlation with elementary volume. It was shown that this module reproduces the mechanical properties measured experimentally.

With few exceptions, the project results are relevant to dynamical systems theory and its applications and we do not see a direct socio-economic impact of them. The exceptions refer at the models we studied within the Tasks 4.1-4.3 which may have some relevance for physicists working in plasma physics while the model studied in the Task 5.2 for medical researchers. We do not think the results obtained so far within the project can be exploited in industry.