Stability of fixed points placed on the border in the piecewise linear systems

In this paper we consider two-dimensional piecewise linear maps characterized by nondifferentiability on a curve in the phase space. According to the stability of the fixed point without having its Jacobian information, recently found dangerous border-collision bifurcations could happen. It is thus important to determine the stability of the nondifferential fixed point. We investigate the global behavior of trajectories near the fixed point, which can be characterized by the dynamics of a map defined on the unit circle with the assigned dilation ratios, and then introduce a novel method to determine the stability of nondifferential fixed points of piecewise linear systems. We also present a special bifurcation phenomenon exhibiting the unbounded behavior of orbits before and after the critical bifurcation value, but the stable fixed point at the critical bifurcation value, which is one of unexpected phenomena in smooth bifurcation theory.