Crisis and topological entropy

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Abstract
Topological entropy characterizes the complexity of a dissipative system. Crisis means a sudden collapse in the size of a chaotic attractor or sudden destruction of a chaotic attractor. In this paper, we illustrate that at some interior crises of a dissipative system topological entropy makes a discontinuous change. This intrinsic feature indicates the onset of a crisis in dissipative systems. Using examples of excitable cell models, we estimated topological entropy in terms of the associated Poincaré maps and showed that the topological entropy changes discontinuously when an interior crisis occurs. We also show that at this crisis two opposite bifurcation processes, with very different dynamical complexities, collide with each other in these dissipative systems, and that the collision gives rise to the occurrence of the crisis in a continuous dynamical system.