Hopf Bifurcation for Non—smooth Planar Dynamical Systems

The purpose of this paper is to study the Hopf bifurcation property of the non-smooth planar dynamical systemwhere the individual functions K± R2 x I R2 at the right-hand side for some interval I containing 0 are smooth enough and given byThe parameter dependent matrices Aa(A) are assumed to be of the formwhere a 0 and w±() > 0. And the nonlinear terms in equation (??) are assumed to be smooth enough and have the formIn a standard way we extend the system (??) to a differential inclusion …     

Generalized Hopf Bifurcation for Non-smooth Planar Dynamical Systems： the Corner Case

Piece-wise smooth systems are an important class of ordinary differential equations whosedynamics are known to exhibit complex bifurcation scenarios and chaos. Broadly speaking,piece-wise smooth systems can undergo all the bifurcation that smooth ones can. Moreinterestingly, there is a whole class of bifurcation that are unique to piece-wise smoothsystems, such as the bifurcation caused by the geometric shape of the region in which thevector field is analyzed. For example (see Figure 1), the region is divided into two partsI and Ⅱ by a discontinuity boundary which contains a corner at O. When an orbit crossthe corner, border-collision bifurcation may occur (cf. [1]). The present paper deals withthe mechanics of the generalized Hopf bifurcation when the stationary point locates at thecorner.     

8-形同宿轨道附近的同宿轨分支性质

In this paper we investigate the homoclinic bifurcation properties near an eight-figure homoclinic orbit of co-dimension two of a planar dynamical system. The corresponding local bifurcation diagram is also illustrated by numerical computation.     

A Petrov－Galerkin Method with Linear Trial and Quadratic Test Spaces for Parabolic Convection－Diffusion Problems

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THE COMPUTATION OF CENTER MANIFOLDS AND HOPF TRAJECTORIES

In this paper, we use a kind of main part symmetry scheme to study the center manifolds and Hop f bifurcations for ODEs, and set up a kind of method for calculation them.     

数值离散化对环面分支结构的影响

Parameterized dynamical systems with a simple zero eigenvalue and a couple of purely imaginary eigenvalues are considered. It is proved that this type of eigen-structure leads to torus bifurcation under certain nondegenerate conditions. We show that the discrete systems, obtained by discretizing the ODEs using symmetric, eigen-structure preserving schemes, inherit the similar torus bifurcation properties. Predholm theory in Banach spaces is applied to obtain the global torus bifurcation. Our results complement those on the study of discretization effects of global bifurcation.