Yongkui Zou

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

AbstractIn this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.

On manifolds of connecting orbits in discretizations of dynamical systems

It is shown that one-step methods, when applied to a one-parametric dynamical system with a homoclinic orbit, exhibit a closed loop of discrete homoclinic orbits. On this loop, the parameter varies periodically while the orbit shifts its index after one revolution. We show that at least two homoclinic tangencies occur on this loop. Our approach works for systems with finite smoothness and also applies to general connecting orbits. It provides an alternative to the interpolation approach by Fiedler and Scheurle (Mem. Amer. Math. Soc. 119 (570) (1996)) and it allows to recover some of their results on exponentially small splittings of separatrices by using some recent backward error analysis for the analytic case.

NUMERICAL ANALYSIS OF DEGENERATE CONNECTING ORBITS FOR MAPS

This paper contains a survey of numerical methods for connecting orbits in discrete dynamical systems. Special emphasis is put on degenerate cases where either the orbit loses transversality or one of its endpoints loses hyperbolicity. Numerical methods that approximate the connecting orbits by finite orbit sequences are described in detail and theoretical results on the error analysis are provided. For most of the degenerate cases we present examples and numerical results that illustrate the applicability of the methods and the validity of the error estimates.

On the existence of transversal heteroclinic orbits in discretized dynamical systems

In this paper we prove the existence of transversal heteroclinic orbits for maps that are obtained from one-step methods applied to a continuous dynamical system. It is assumed that the continuous system exhibits a heteroclinic orbit at a specific value of a parameter. While it is known that analytic vector fields lead to exponentially small splittings of separatrices in the discrete system, we analyse here the case of a continuous system that is smooth of finite order only. Assuming that a certain derivative has a jump discontinuity at a specific hyper-plane we show that discretized systems have transversal heteroclinic orbits. The essential step in deriving such a result is a refinement of a previously developed error analysis that applies exponential dichotomy and Fredholm techniques to the discretized system.

HOMOCLINIC BIFURCATIONS IN A PLANAR DYNAMICAL SYSTEM

The homoclinic bifurcation properties of a planar dynamical system are analyzed and the corresponding bifurcation diagram is presented. The occurrence of two Bogdanov–Takens bifurcation points provides two local existing curves of homoclinic orbits to a saddle excluding the separatrices not belonging to the homoclinic orbits. Using numerical techniques, these curves are continued in the parameter space. Two further curves of homoclinic orbits to a saddle including the separatrices not belonging to the homoclinic orbits are calculated by numerical methods. All these curves of homoclinic orbits have a unique intersection point, at which there exists a double homoclinic orbit. The local homoclinic bifurcation diagram of both the double homoclinic orbit point and the points of homoclinic orbits to a saddle-node are also gained by numerical computation and simulation.