On the dynamics of two-dimensional dissipative discontinuous maps

Abstract

• In this work we consider a dissipative two-dimensional discontinuous mapping. • We show that from a large initial action, the dynamics converges exponentially to chaotic attractors. • The maximum of the chaotic attractors as a function of the control parameters provides power laws. • We also investigate the positive Lyapunov exponents as a function of the control parameters. Some dynamical properties for a dissipative two-dimensional discontinuous standard mapping are considered. The mapping, in action-angle variables, is parameterized by two control parameters; namely, k ≥ 0 controlling the intensity of the nonlinearity and γ ∈ [0, 1] representing the dissipation. The case of γ = 0 recovers the non-dissipative model while any γ ≠ 0 yields to the breaking of area preservation; hence leading to the existence of attractors, including chaotic ones. We show that when starting from a large initial action, the dynamics converges to chaotic attractors through an exponential decay in time, while the speed of the decay depends on the dissipation intensity. We also investigate the positive Lyapunov exponents and describe their behavior as a function of the control parameters.