A dissipative Fermi-Ulam model under two different kinds of dissipation

Abstract

The problem of a classical particle confined to bounce between two rigid walls, where one of them is fixed and the other one moves periodically on time is considered. The collisions with the moving wall can be inelastic, where two different kinds of dissipations are introduced. Here the dissipations are given by a restitution coefficient and/or due to a quadratic frictional force. A two-dimensional mapping to describe this dynamic is obtained. The phase space for this model in the non-dissipative case is of mixed type, containing invariant spanning curves, chaotic seas and KAM islands. After introducing dissipation, sinks and chaotic attractors arise. An analytical expression for the decay of velocity is obtained, where an exponential behavior is observed after introducing the dissipations. This decay can be considered a transient time before reaching an attractor. Boundary crisis and connections with the parameter space (also called shrimp-shape domain) are done. Analytical expressions for an inverse parabolic bifurcation are obtained. Here the exact moment when the eigenvalues of the Jacobian matrix become real numbers with the same values is shown. In such a situation, the Lyapunov exponents are no longer constant values. It can be verified that the transient times apparently have a power law behavior near the criticality at the boundary crisis, but the slopes obtained are not necessarily equal to the ones obtained in the literature for the complete version of the mapping. It is also observed that the position of some periodic orbits follow a straight line in some situations, where the periods are related to the nearest large periodic region. When the two dissipations are taken into account, it is observed that periodic regions change their position in the parameter space, and the numeric coordinates of a periodic region are obtained. (C) 2014 Elsevier B.V. All rights reserved.