Bifurcation from family of periodic orbits in discontinuous autonomous systems
type: Item
instance of: scholarly article
title: Bifurcation from family of periodic orbits in discontinuous autonomous systems
zbMATH Open document ID: 1258.34076
author: Michal Fečkan
author: Michal Pospíšil
published in: Differential Equations and Dynamical Systems
publication date: 2012-11-30T00:00:00Z
review text: The authors consider perturbed piecewise-smooth nonlinear differential systems, i.e., systems of the form \[ \dot{x}=F(t,x, \chi), \tag{1} \] where \(F(t,x, \chi)\) is a smooth function on \(\mathbb{R} \times (\mathbb{R}^n \setminus \mathbb{S}) \times \mathbb{R}^n\), periodic in \(t \in \mathbb{R}\). \(\mathbb{S}\) is called the ``discontinuity set'', and in the paper it is a hypersurface of \(\mathbb{R}^n\). Moreover, it is supposed that at \(\chi = 0\) the function \(F(t,x,0)=F(x)\) is independent of \(t\) and the associated autonomous nonlinear differential system \[ \dot{x}=F(x), \tag{2} \] has a periodic orbit \(\gamma(t)\). The first natural case is when \(\gamma\) crosses \(\mathbb{S}\) transversally. This case was studied by the authors in a previous work and also by other authors. In the present paper, the authors consider the case when \(F\) is independent of \(t\) and \(\gamma\) crosses \(\mathbb{S}\) transversally and call this case ``transverse autonomous''. It is also supposed that the system (2) possesses a nondegenerate family of periodic solutions. Then, the authors determine bifurcation conditions in terms of Poincaré-Andronov-Melnikov functions that ensure the bifurcation of certain periodic solutions from the family. The local asymptotic properties of the bifurcating periodic solutions such as their stability, instability or hyperbolicity are also investigated. Finally, the paper present examples of 3- and 4-dimensional discontinuous ordinary differential systems to illustrate the theory.
reviewed by: Jaume Giné
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34A36
zbMATH DE Number: 6111600
zbMATH Keywords: periodic orbit
zbMATH Keywords: discontinuous systems
zbMATH Keywords: bifurcation
MaRDI profile type: Publication
full work available at URL: s12591-011-0094-2
OpenAlex ID: W2083958116
cites work: Continuation of the periodic orbits for the differential equation with discontinuous right hand side
cites work: On the smoothness of solutions of differential equations with a discontinuous right-hand side
cites work: Periodic solutions of strongly nonlinear systems with nonclassical right- hand side in the case of a family of generating solutions
cites work: Q3596245
cites work: Some remarks on the Melnikov function
cites work: Exponential dichotomies, heteroclinic orbits, and Melnikov functions
cites work: Q5472808
cites work: Q3948194
cites work: Q5185568
cites work: Piecewise-smooth dynamical systems. Theory and applications
cites work: Q3653803
cites work: Periodic motions
cites work: On the bifurcation of periodic orbits in discontinuous systems
cites work: Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift–Hohenberg equation as an example
cites work: Q4046802
cites work: Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations
cites work: Q5330830
cites work: Jumps of the fundamental solution matrix in discontinuous systems and applications
cites work: Q5386147
cites work: Stick and non-stick periodic motions in periodically forced oscillators with dry friction
cites work: Motion switching and chaos of a particle in a generalized Fermi-acceleration oscillator
cites work: AN ANALYTICAL PREDICTION OF PERIODIC FLOWS IN THE CHUA CIRCUIT SYSTEM
cites work: Regularization of kepler's problem and the averaging method on a manifold
cites work: Chaotic transport in dynamical systems
DOI: 10.1007/S12591-011-0094-2
description: scientific article; zbMATH DE number 6111600