## Periodic Reporting for period 1 - Dynamics (Contributions to codimension k bifurcations in dynamical systems theory)

Reporting period: 2018-04-01 to 2020-03-31

The project is concerned with fundamental research in mathematics. More exactly, the project aims to create new theory, methods and tools in dynamical systems theory, which can be used to a better understanding of real-life phenomena, such as, interactions between adverse populations in biology (predator-prey models used to understand how viruses spread in a body), ecology (predator-prey models for studying endangered species), economics (models for studying economic indicators such as inflation rates, interest rates and investment targets), astrophysics (studying the motion under attractive or repulsive gravitational and intermolecular forces in diverse potentials) and so on, all these topics being important for the society.

The overall project objectives are to produce new knowledge in the area of codim k bifurcations for continuous and discrete (smooth and non-smooth) dynamical systems and provide training in this area of research to early stage researchers. More exactly, we plan firstly to study degenerate two-dimensional Bautin bifurcation for the case when the second Lyapunov coefficient equals zero. Secondly, we aim to study degenerate four-dimensional Hopf-Hopf bifurcations. The third and fourth objectives are to study other codim k bifurcations in smooth and non-smooth dynamical systems arising from other bifurcations which bear or not a known name in the literature. In particular, we will focus on discontinuous piecewise differential systems, respectively, continuous and discrete non-smooth dynamical systems resulting from modeling oscillators with impacts.

The overall project objectives are to produce new knowledge in the area of codim k bifurcations for continuous and discrete (smooth and non-smooth) dynamical systems and provide training in this area of research to early stage researchers. More exactly, we plan firstly to study degenerate two-dimensional Bautin bifurcation for the case when the second Lyapunov coefficient equals zero. Secondly, we aim to study degenerate four-dimensional Hopf-Hopf bifurcations. The third and fourth objectives are to study other codim k bifurcations in smooth and non-smooth dynamical systems arising from other bifurcations which bear or not a known name in the literature. In particular, we will focus on discontinuous piecewise differential systems, respectively, continuous and discrete non-smooth dynamical systems resulting from modeling oscillators with impacts.

The project has six precise tasks: T1.1-T1.4 and T2.1-T2.2. During the first 24 months of the project we approached five of them. We present in the following the work progress to the project tasks as it follows.

T1.1 Study degenerate Bautin bifurcations. We approached this task and revised the normal forms when the first and second Lyapunov coefficients are zero. Then, we obtained the normal form which involves the third Lyapunov coefficient and a method for determining the coefficient has been obtained.

T1.2 Study degenerate Hopf-Hopf bifurcation. We started to work on this bifurcation and tried to simplify the method for obtaining the existing normal form and reduce the number of generic conditions used for the existence of normal form. The task is currently under work. We studied also a predator-prey model which is related to the normal form of Hopf-Hopf bifurcation.

T1.3 Study other codimension k bifurcations in continuous (smooth) systems.

(a) We studied a simplified multistrain/two-stream model for the tuberculosis and the Dengue fever with three compartments, one susceptible and the other two infectious. The results are reported in the paper: J. Llibre, R.D.S. Oliveira, C. Valls, Final evolutions for simplified multistrain/two-stream model for tuberculosis and dengue fever, Chaos, Solitons and Fractals 118 (2019) 181–186.

(b) We studied a system related to the two-body problem. More details at: J. Llibre, C. Valls, C. Vidal, Global dynamics of the Buckingham’s two-body problem, Astrophysics and Space Science (2018), 363:255.

(c) A finance model was studied for finding periodic oscillations. The results are published in the paper: M.R. Candido, J. Llibre, C. Valls, Invariant algebraic surfaces and Hopf bifurcation of a finance model, IJBC, Vol. 28, No. 12 (2018) 1850150.

(d) A dynamical system possessing an equilibrium point with two zero eigenvalues was studied in: C. Rocsoreanu, M. Sterpu, Approximations of the heteroclinic orbits near a double-zero bifurcation with symmetry of order two, IJBC, Vol. 29, No. 6 (2019) 1950074.

T1.4 Study other codimension k bifurcations in discrete (smooth) systems

(a) We studied the degenerate Chenciner bifurcation in a generic discrete-time dynamical system.

(b) We studied continuous self-maps defined on a closed surface in: J. L. G. Guirao, J. Llibre, W. Gao, Topological entropy of continuous self-maps on closed surfaces, J. Difference Equ. Appl., 26(2), 203-208, 2020.

T2.1 Study codim k bifurcations in piecewise and other non-smooth systems

(a) In the paper J. Llibre, X. Zhang, Limit cycles created by piecewise linear centers, Chaos, 29(5), 053116:1-14, 2019, we study the limit cycles of a discontinuous piecewise linear differential system in the plane.

(b) In the paper C.A. Buzzi, J. C. Medrado, J. Torregrosa, Limit cycles in 4-star-symmetric planar piecewise linear systems, J. Differential Equations, 268(5), 2414-2434, 2020, we studied the number of limit cycles for nonsmooth piecewise linear vector fields on the plane when the switching curve is xy=0.

(c) We studied limit cycles in a class of piecewise differential systems in: J. Llibre, R.D.S. Oliveira, C.A.B. Rodrigues, Limit cycles for two classes of control piecewise linear differential systems, Sao Paulo J. Math. Sci., 1-17, 2020.

T1.1 Study degenerate Bautin bifurcations. We approached this task and revised the normal forms when the first and second Lyapunov coefficients are zero. Then, we obtained the normal form which involves the third Lyapunov coefficient and a method for determining the coefficient has been obtained.

T1.2 Study degenerate Hopf-Hopf bifurcation. We started to work on this bifurcation and tried to simplify the method for obtaining the existing normal form and reduce the number of generic conditions used for the existence of normal form. The task is currently under work. We studied also a predator-prey model which is related to the normal form of Hopf-Hopf bifurcation.

T1.3 Study other codimension k bifurcations in continuous (smooth) systems.

(a) We studied a simplified multistrain/two-stream model for the tuberculosis and the Dengue fever with three compartments, one susceptible and the other two infectious. The results are reported in the paper: J. Llibre, R.D.S. Oliveira, C. Valls, Final evolutions for simplified multistrain/two-stream model for tuberculosis and dengue fever, Chaos, Solitons and Fractals 118 (2019) 181–186.

(b) We studied a system related to the two-body problem. More details at: J. Llibre, C. Valls, C. Vidal, Global dynamics of the Buckingham’s two-body problem, Astrophysics and Space Science (2018), 363:255.

(c) A finance model was studied for finding periodic oscillations. The results are published in the paper: M.R. Candido, J. Llibre, C. Valls, Invariant algebraic surfaces and Hopf bifurcation of a finance model, IJBC, Vol. 28, No. 12 (2018) 1850150.

(d) A dynamical system possessing an equilibrium point with two zero eigenvalues was studied in: C. Rocsoreanu, M. Sterpu, Approximations of the heteroclinic orbits near a double-zero bifurcation with symmetry of order two, IJBC, Vol. 29, No. 6 (2019) 1950074.

T1.4 Study other codimension k bifurcations in discrete (smooth) systems

(a) We studied the degenerate Chenciner bifurcation in a generic discrete-time dynamical system.

(b) We studied continuous self-maps defined on a closed surface in: J. L. G. Guirao, J. Llibre, W. Gao, Topological entropy of continuous self-maps on closed surfaces, J. Difference Equ. Appl., 26(2), 203-208, 2020.

T2.1 Study codim k bifurcations in piecewise and other non-smooth systems

(a) In the paper J. Llibre, X. Zhang, Limit cycles created by piecewise linear centers, Chaos, 29(5), 053116:1-14, 2019, we study the limit cycles of a discontinuous piecewise linear differential system in the plane.

(b) In the paper C.A. Buzzi, J. C. Medrado, J. Torregrosa, Limit cycles in 4-star-symmetric planar piecewise linear systems, J. Differential Equations, 268(5), 2414-2434, 2020, we studied the number of limit cycles for nonsmooth piecewise linear vector fields on the plane when the switching curve is xy=0.

(c) We studied limit cycles in a class of piecewise differential systems in: J. Llibre, R.D.S. Oliveira, C.A.B. Rodrigues, Limit cycles for two classes of control piecewise linear differential systems, Sao Paulo J. Math. Sci., 1-17, 2020.

We obtained an analytic expression of the third Lyapunov coefficient to T1.1. We proposed a method for determining easier the normal form of Hopf-Hopf bifurcation and to reduce the number of generic conditions which are necessary for obtaining a normal form to T1.2. The method is presently under review. We described all the final evolution of the model described at T1.3 (a) under generic assumptions. We characterized the global dynamics of the Buckingham system describing the foliation of its phase space by some invariant sets, T1.3 (b). We pointed out small-amplitude periodic solutions arising from a non-degenerate codim 1 Hopf bifurcation and via averaging theory for the case when the bifurcation becomes degenerate with codim k, k≥2, T1.3 (c). Using a blow-up transformation and a perturbation method, we obtained second order approximations both for the heteroclinic orbits and the curve of heteroclinic bifurcation values in the model described at T1.3 (d). We obtained new bifurcation diagrams for degenerate Chenciner bifurcation T1.4 (a). The work is under progress. We obtained sufficient conditions for having positive topological entropy for continuous self-maps defined on a closed surface by using the action of this map on the homological groups of the closed surface T1.4 (b).

We proved that discontinuous piecewise linear differential systems with a straight line of separation can have 1, 2 or 3 limit cycles T2.1 (a). Also in the 2 zones but when the separation curve is a cross, we provided the center classification at infinity and showed that the maximum order of a weak focus is five, and showed the existence of systems exhibiting five limit cycles bifurcating from infinity, T2.1 (b). We obtained results on bifurcation of limit cycles from the periodic orbits of 2n–dimensional linear centers when they are perturbed inside classes of continuous and classes of discontinuous piecewise linear differential systems T2.1 (c).

We proved that discontinuous piecewise linear differential systems with a straight line of separation can have 1, 2 or 3 limit cycles T2.1 (a). Also in the 2 zones but when the separation curve is a cross, we provided the center classification at infinity and showed that the maximum order of a weak focus is five, and showed the existence of systems exhibiting five limit cycles bifurcating from infinity, T2.1 (b). We obtained results on bifurcation of limit cycles from the periodic orbits of 2n–dimensional linear centers when they are perturbed inside classes of continuous and classes of discontinuous piecewise linear differential systems T2.1 (c).