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 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.
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 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).