A Boundary Crisis in High Dimensional Chaotic Systems
A crisis is investigated in high dimensional chaotic systems by means of generalized cell mapping digraph (GCMD) method. The crisis happens when a hyperchaotic attractor collides with a chaotic saddle in its fractal boundary, and is called a hyperchaotic boundary crisis. In such a case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis, namely, the hyperchaotic attractor is converted into an incremental portion of the hyperchaotic saddle after the collision. This hyperchaotic saddle is an invariant and nonattracting hyperchaotic set. In the hyperchaotic boundary crisis, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary.
This work is supported by the National Science Foundation of China under Grant Nos. 10772140 and 10872155 as well as the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
- 19.Kawakami H, Kobayashi O (1976) Computer experiment on chaotic solution. Bull Fac Eng Tokushima Univ 16:29–46Google Scholar
- 20.He DH, Xu JX, Chen YH (1999) A study on strange dynamics of a two dimensional map. Acta Physica Sin 48(9):1611–1617Google Scholar