Local and global bifurcations in 3D piecewise smooth discontinuous maps.

This paper approaches the problem of analyzing the bifurcation phenomena in three-dimensional discontinuous maps, using a piecewise linear approximation in the neighborhood of a border. The existence conditions of periodic orbits are analytically calculated and bifurcations of different periodic orbits are illustrated through numerical simulations. We have illustrated the peculiar features of discontinuous bifurcations involving a stable fixed point, a period-2 cycle, a saddle fixed point, etc. The occurrence of multiple attractor bifurcation and hyperchaos are also demonstrated.

Robust chaos in 3-D piecewise linear maps.

A chaotic attractor is called robust if there is no periodic window or any coexisting attractor in some open subset of the parameter space. Such a chaotic attractor cannot be destroyed by a small change in parameter values since a small change in the parameter value can only make small changes in the Lyapunov exponents. Earlier investigations have calculated the existence and the stability conditions of robust chaos in 1D and 2D piecewise linear maps. In this work, we demonstrate the occurrence of robust chaos in 3D piecewise linear maps and derive the conditions of its occurrence by analyzing the interplay between the stable and unstable manifolds.