HORSESHOES NEAR HOMOCLINIC ORBITS FOR PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ 3 (original) (raw)

Periodic orbits for perturbations of piecewise linear systems

Journal of Differential Equations, 2011

We consider the existence of periodic orbits in a class of threedimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.

On the fold-Hopf bifurcation for continuous piecewise linear differential systems with symmetry

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2010

In this paper a partial unfolding for an analog to the fold-Hopf bifurcation in three-dimensional symmetric piecewise linear differential systems is obtained. A particular biparametric family of such systems is studied starting from a very degenerate configuration of nonhyperbolic periodic orbits and looking for the possible bifurcation of limit cycles. It is proved that four limit cycles can coexist after perturbation of the original configuration, and other two limit cycles are conjectured. It is shown that the described bifurcation scenario appears for appropriate values of parameters in the celebrated Chua's oscillator.

On limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems

Applied Mathematics and Computation, 2015

In this paper we consider the linear differential center (ẋ,ẏ) = (−y, x) perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y = 0. We provide sufficient conditions to ensure the existence of a limit cycle bifurcating from the infinity. The main tools used are the Bendixson transformation and the averaging theory.

Limit Cycles of Continuous Piecewise Differential Systems Formed by Linear and Quadratic Isochronous Centers I

International Journal of Bifurcation and Chaos, 2022

First, we study the planar continuous piecewise differential systems separated by the straight line [Formula: see text] formed by a linear isochronous center in [Formula: see text] and an isochronous quadratic center in [Formula: see text]. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line [Formula: see text] having in [Formula: see text] the general quadratic isochronous center [Formula: see text], [Formula: see text] after an affine transformation, and in [Formula: see text] an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise diffe...

Homoclinic Trajectories in Discontinuous Systems

Journal of Dynamics and Differential Equations, 2008

We study bifurcations of bounded solutions from homoclinic orbits for timeperturbed discontinuous systems. Functional analytic method is used. An illustrative example of a periodically perturbed piecewise linear differential equation in R 3 is presented.

Bifurcations from families of periodic solutions in piecewise differential systems

Physica D: Nonlinear Phenomena, 2020

Consider a differential system of the form x ′ = F 0 (t, x) + k i=1 ε i F i (t, x) + ε k+1 R(t, x, ε), where F i : S 1 × D → R m and R : S 1 × D × (−ε 0 , ε 0) → R m are piecewise C k+1 functions and T-periodic in the variable t. Assuming that the unperturbed system x ′ = F 0 (t, x) has a d-dimensional submanifold of periodic solutions with d < m, we use the Lyapunov-Schmidt reduction and the averaging theory to study the existence of isolated T-periodic solutions of the above differential system.

Hopf-like bifurcations in planar piecewise linear systems

Publicacions Matemàtiques, 1997

Continuous planar piecewise linear systems with two linear zones are considered. Due to their low differentiability specific techniques of analysis must be developed. Several bifurcations giving rise to limit cycles are pointed out.

07-16 Global Analysis of Piecewise Linear Dynamical Systems

2008

In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary number of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system with k dropping sections and 2k + 1 singular points can have at most k + 2 limit cycles, k+1 of which surround the foci one by one and the last, k+2, limit cycle surrounds all of the singular points of this system.