A chaotic crisis between chaotic saddle and attractor in forced Duffing oscillators

We investigate crises in forced Duffing oscillators by the generalized cell mapping digraph method to efficiently complete the global analysis of non-linear systems, which includes global transient analysis through digraphic algorithms based on a strictly theoretical correspondence between generalized cell mappings and digraphs. A process of generalized cell mapping method is developed to refine persistent and transient self-cycling sets. The refining procedures of persistent and transient self-cycling sets are respectively given on the basis of their definitions in the cell state space. A chaotic boundary crisis and a chaotic interior crisis are discovered. A chaotic boundary crisis owing to a collision between chaotic attractor and saddle occurs in its basin boundary possessing a fractal structure. In such a case the chaotic attractor together with its basin of attraction is suddenly destroyed as the parameter passes through the critical value, and the chaotic saddle also undergoes an abrupt enlargement in its size. Namely, the chaotic attractor is converted into an incremental portion of the chaotic saddle after the collision. For a chaotic interior crisis, there is a sudden increase in the size of a chaotic attractor as the parameter passes through the critical value. For the chaotic interior crisis, it is demonstrated that the chaotic attractor collides with a chaotic saddle in its basin interior when the crisis occurs. This chaotic saddle is an invariant and non-attracting set. The origin and evolution of the chaotic saddle are investigated as well.