Step-barrier dynamical system

Abstract

The step-barrier dynamical system is discussed in detail. The behavior of particles captured in the barrier bucket remarkably varies depending on their initial conditions. The intrinsic nature of motion observed in the fully normalized area-preserving mapping simulations is simply clarified by a value of energy or momentum of particle at the initial stage. A rational initial value of y in the normalized phase space (x, y) at beginning leads to the periodic solution. Meanwhile, an irrational value of y gives the oscillation in amplitude with multi-frequency, resulting in broad tracks in the phase space. Effectively, the barrier bucket size is reduced due to these broad tracks. We arrive at the concept of effective stable region, which is defined by the contour of the averaged step-barrier Hamiltonian and useful in actual applications.