Crisis induced by an escape from a fat strange set

This article reports a characteristic crisis observed in a two-dimensional discontinuous and noninvertible map. The discontinuity border in the definition range of the mapping oscillates as the discrete time progresses so that the forward images of the border form a fat fractal. By choosing particular parameters the iterations on the fat fractal display chaotic motion, and the transient iterations from the initial values in a certain region of the phase space are attracted to the fat fractal. At a threshold of a control parameter an elliptic periodic orbit and the elliptic islands around it suddenly appear inside the fat strange set so that the iterations on the set escape to the islands. The fat chaotic attractor thus suddenly transfers to a fat transient set. The effect of the feature of the crisis on the rule of the lifetime in the transient set is discussed. It shows that the dependence of the lifetime on the control parameter follows a universal scaling law suggested by Grebogy, Ott, and Yorke [Phys. Rev. Lett. 57, 1284 (1986)], and the scaling exponent can be approximated according to the variation rules of the elliptic islands and the measure of the fat fractal. The strange repeller, which appears after the crisis, is also a fat fractal.