Study of the Wada fractal boundary and indeterminate crisis

Abstract

By using the generalized cell mapping digraph (GCMD) method, we study bifurcations governing the escape of periodically forced oscillators in a potential well, in which a chaotic saddle plays an extremely important role. In this paper, we find the chaotic saddle and we demonstrate that the chaotic saddle is embedded in a strange fractal boundary which has the Wada property, that any point on the boundary of that basin is also simultaneously on the boundary of at least two other basins. The chaotic saddle in the Wada fractal boundary, by colliding with a chaotic attractor, leads to a chaotic boundary crisis with a global indeterminate outcome which presents an extreme form of indeterminacy in a dynamical system. We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the chaotic saddle in the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal boundaries. After a final escape bifurcation, there only exists the attractor at infinity; a chaotic saddle with a beautiful pattern is left behind in phase space.