Velocity-dependent KAM islands

We revisit the non-dissipative time-dependent annular billiard and we consider the chaotic dynamics in two planes of conjugate variables in order to describe the behavior of the growth, or saturation, of the mean velocity of an ensemble of particles. We observed that the changes in the 4-d phase space occur without changing any parameter. They occur depending on where the initial conditions start. The emerging KAM islands interfere in the behavior of the particle dynamics especially in the Fermi acceleration mechanism. We show that Fermi acceleration can be suppressed, without dissipation, even considering the non-dissipative energy context.

► We consider the time-dependent annular billiard with elastic collisions. ► Two phase planes are considered, the geometric and the energy phase planes. ► Any initial condition has four components. ► For high velocities, some islands of the static case are recovered in the geometric phase plane. ► Initial conditions with components in places where an island will appear, do not present Fermi acceleration.