Shrimp-shape domains in a dissipative kicked rotator

Abstract

Some dynamical properties for a dissipative kicked rotator are studied. Our results show that when dissipation is taken into account a drastic change happens in the structure of the phase space in the sense that the mixed structure is modified and attracting fixed points and chaotic attractors are observed. A detailed numerical investigation in a two-dimensional parameter space based on the behavior of the Lyapunov exponent is considered. Our results show the existence of infinite self-similar shrimp-shaped structures corresponding to periodic attractors, embedded in a large region corresponding to the chaotic regime. A strongly dissipative kicked rotator is studied. The investigation is based mainly on the asymptotic behavior of the Lyapunov exponents. Our results show that by reducing the two-dimensional map to a one-dimensional map in the limit of infinite kicks, the Feigenbaum's d is recovered. An investigation in the parameter space (the dissipation parameter and the kicking parameter) reveals the existence of infinite families of self-similar structures of shrimp-shape embedded in a large region corresponding to the chaotic regime. The organization of stability shrimps reported here for the discrete-time kicked rotator agrees well with the parameter space organization reported recently in the literature for several flows (continuous-time systems).

Statistical and dynamical properties of a dissipative kicked rotator

“…(39) and (29) allow us to conclude the α 3 = 2 and α 2 = −1 which is in agreement with our 7 numerical data shown in Fig. 9.…”

The effect of temperature on generic stable periodic structures in the parameter space of dissipative relativistic standard map

“…As confirmed by a set of numerical experiments (magnifications not shown here), in some specific portions of parameter space of DRSM there are a large number of them. As a matter of fact, as we enlarged small portion of chaotic regions, further smaller typical shrimp-shaped domains are found [26,30]. As far as we know, the shrimp-shaped domain also is known as fishhook, since 1982 by Fraser and Kapral [31], for discrete maps and flows.…”

Classical to quantum correspondence in dissipative directed transport

“…It is worth noticing that these ISSs are a common feature, found in generic dissipative dynamical systems. In fact, some of their properties have recently been studied in the dissipative kicked rotator model [19].…”