Statistical and dynamical properties of time-dependent systems

Abstract

We study some dynamical and statistical properties of classical time-dependent systems. It is well known that the structure of the phase space of Hamiltonian systems depends on the individual characteristics of each system. Basically they can be divided in three groups: (i) integrable, (ii) ergodic and (iii) mixed. In case (i) the phase space consists of invariant tori filling the entire phase space. In case (ii) the time evolution of a single initial condition is enough to fill the phase space. In case (iii), one important property in the mixed phase space is that chaotic seas are generally surrounding Kolmogorov-Arnold-Moser (KAM) islands which are confined by invariant spanning curves. In particular such curves can cross the phase plane and partition it into several separated and disconnected portions of the phase space. Sometimes, the existence of invariant spanning curves can also prevent the unlimited energy growth of a particle. One of the main point of our work is to study some statistical properties of systems with integrable, ergodic and mixed phase space close to the transition from integrability to nonintegrability and unlimited to limited energy growth using scaling arguments. Once the critical exponents are known, classes of universality can be defined. On the other hand, the introduction of inelastic collisions on this model is enough to destroy such a mixed structure and the system exhibits attractors. Depending on the initial conditions and control parameters, one can observe a chaotic attractor characterized by a positive Lyapunov exponent. By a suitable control parameter variation, the chaotic attractor might be destroyed via a crisis event. After the destruction, the chaotic attractor is replaced by a chaotic transient. Additionally, by changing simultaneously the dissipation parameter and the parameter which controls the transition from integrability to non-integrability and by using the Lyapunov exponents in order to classify regions with chaotic and regular behavior we have also studied the structure of the parameter-space. (AU)