Characteristics of a piecewise smooth area-preserving map

We are reporting a study carried out in a system concatenated by two area-preserving maps. The system can be viewed as a model of an electronic relaxation oscillator with over-voltage protection. We found that a border-collision bifurcation may interrupt a period-doubling bifurcation cascade, and that some special features, such as “quasicoexisting periodic orbits crossing border” as well as the transition between “quasitransience” and chaotic orbits, accompany the process. These features belong to the so-called “quasidissipative” properties. Here “quasitransience” denotes the behavior of iterations outside elliptic islands. They are “attracted” to the islands. As soon as it reaches the islands, the iteration follows the conservative regulations exactly. This induces a kind of escaping from strange sets. The scaling behavior of the escaping rate is obtained numerically.