Lyapunov Exponents for a Family of Dissipative Two-dimensional Mappings

Abstract

In this work we consider a family of dissipative two-dimensional mappings and the Lyapunov exponents to characterize the chaos. The mapping is defined as: \( T : \{ I_{n+1} = |\delta I_n - (1 + \delta)\epsilon \sin(2\pi\theta_n)|, \theta_{n+1} = \theta_n + I_{n+1}^\gamma \mod(1) \} \), where \(\theta\) and \(I\) are angle-action variables, \(\epsilon\) controls the nonlinearity, \(\delta\) controls the dissipation magnitude and \(\gamma\) is a dynamical exponent that recovers several dynamical systems known in the literature. For \(\delta = 1\) the conservative case is recovered such as area-preserving in the phase space is observed. Our main goal of investigation is to characterize the chaotic attractors using the Lyapunov exponents.