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.
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 the
We extend the Melnikov method to non-smooth dynamical systems to study the global behavior near a non-smooth homoclinic orbit under small time-periodic perturbations. The definition and an explicit expression for the extended Melnikov function are given and applied to determine the appearance of transversal homoclinic orbits and chaos. In addition to the standard integral part, the extended Melnikov function contains an extra term which reflects the change of the vector field at the discontinuity. An example is discussed to illustrate the results.
Consider the following 2×2 nonlinear system:where f(u): R→R is a, smooth function. Setwhere F’(u)= f(u). Then (1) can be rewritten as an equivalent Hamiltonian system:
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.
