Dynamical properties of a particle in a wave packet: scaling invariance and boundary crisis

TL;DR

This paper investigates the dynamical behavior of particles in wave packets, revealing scaling laws, phase space structure, and the effects of dissipation leading to boundary crises in a two-dimensional area-preserving map model.

Contribution

It introduces a scaling formalism for mixed phase space systems and analyzes the impact of dissipation on chaotic attractors and boundary crises.

Findings

The first invariant spanning curve scales as a power law with exponent 2/3.

The kinetic energy's standard deviation follows a power law over time and saturates.

Dissipation causes destruction of chaotic attractors and induces boundary crises.

Abstract

Some dynamical properties present in a problem concerning the acceleration of particles in a wave packet are studied. The dynamics of the model is described in terms of a two-dimensional area preserving map. We show that the phase space is mixed in the sense that there are regular and chaotic regions coexisting. We use a connection with the standard map in order to find the position of the first invariant spanning curve which borders the chaotic sea. We find that the position of the first invariant spanning curve increases as a power of the control parameter with the exponent 2/3. The standard deviation of the kinetic energy of an ensemble of initial conditions obeys a power law as a function of time, and saturates after some crossover. Scaling formalism is used in order to characterize the chaotic region close to the transition from integrability to nonintegrability and a relationship between the power law exponents is derived. The formalism can be applied in many different systems with mixed phase space. Then, dissipation is introduced into the model and therefore the property of area preservation is broken, and consequently attractors are observed. We show that after a small change of the dissipation, the chaotic attractor as well as its basin of attraction are destroyed, thus leading the system to experience a boundary crisis. The transient after the crisis follows a power law with exponent -2.