Bifurcation of Quasiperiodic Orbit in a 3D Piecewise Linear Map

Abstract

Earlier investigations have demonstrated how a quasiperiodic orbit in a three-dimensional smooth map can bifurcate into a quasiperiodic orbit with two disjoint loops or into a quasiperiodic orbit of double the length in the shape of a Möbius strip. Using a three-dimensional piecewise smooth (PWS) normal form map, we show that in a piecewise smooth system, in addition to the mechanisms reported earlier, new pathways of creation of tori with multiple loops may result from border collision bifurcations. We also illustrate the occurrence of multiple attractor bifurcations due to the interplay between the stable and the unstable manifolds. Two techniques of analyzing bifurcations of ergodic tori are available in literature: the second Poincaré section method and the Lyapunov bundle method. We have shown that these methods can explain the period-doubling and double covering bifurcations in PWS systems, but fail in some cases — especially those which result from nonsmoothness of the system. We have shown that torus bifurcations due to border collision can be explained by change in eigenvalues of the unstable fixed points.