Dynamical Statistical Properties of Piecewise Linearly Periodically Modulated Quartic Oscillator

Abstract

Following recent results in a series of papers by Robnik and Romanovski on general time-dependent linear oscillator and by Papamikos and Robnik on time-dependent nonlinear oscillators, we study the dynamical and statistical properties of piecewise linearly periodically modulated quartic oscillator. By means of careful numerical analysis we show the phase portrait for a number of characteristic values of the system parameter, study the evolution of the average energy of a certain microcanonical ensemble of initial conditions, and look at the distribution function of final energies. This distribution function might be similar to the arcsine density (found rigorously in the linear oscillator), especially in the adiabatic regime, but generally can be entirely different, nonuniversal, especially in a chaotic regime, as we show in this work. In cases where we see a large region of hard chaos connected to infinity, we indeed find escape of orbits to infinity, meaning that the energy growth can be unbounded, which is the analogue of Fermi acceleration in time-dependent billiards. For smaller energies the growth of the average energy of a microcanonical ensemble of initial conditions is exponential, for larger energies well inside the hard-chaos region, the growth is approximately linear.