Hyperchaos in 3-D piecewise smooth maps

In this paper, we show various ways of the occurrence of a hyperchaotic orbit in 3D piecewise linear normal form maps. We show that hyperchaotic orbit can be born from a periodic orbit or a quasiperiodic orbit in various ways like-(a) a direct transition to a hyperchaotic orbit from a periodic orbit or a from a quasiperiodic orbit through border collision bifurcation; (b) a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic and chaotic orbit; (c) a transition from a mode-locked periodic orbit to a hyperchaotic orbit via higher dimensional torus. We also show bifurcations where a hyperchaotic orbit bifurcates to a different hyperchaotic orbit or a three-piece hyperchaotic orbit. We further show period increment with the coexistence of hyperchaotic attractors. Moreover, we numerically calculate the existence region of a hyperchaotic orbit in the parameter space region.