Abstract
In this paper, boundary and interior crises in a fractional-order piecewise system are studied using the extended generalized cell mapping (EGCM) method as a system control parameter varies. A new development of the EGCM method is presented to deal with the non-smooth characteristics of a fractional-order piecewise system. A boundary crisis occurs when a chaotic attractor collides with a regular saddle on the basin boundary leaving behind a chaotic saddle in the place of the original attractor and saddle. With an increase of the system parameter, both the chaotic saddle and the chaotic attractor become larger and finally touch each other in the basin of attraction, which causes the size change of the chaotic attractor suddenly, namely, an interior crisis occurs. Additionally, the routes to chaos and out of chaos for the system are explored by the EGCM method. It is found that the route to chaos is a period-doubling bifurcation, and out of chaos is a saddle-node bifurcation. These results further explain the dynamics evolution of the fractional-order piecewise system from a global perspective. Based on this, it can be demonstrated that the EGCM method is also effective for the global dynamics of fractional-order piecewise systems.
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