Nonlinear dynamics theory of stochastic layers in Hamiltonian systems

Abstract

This article reviews the history and recent development of the nonlinear dynamics theory of the stochastic layer in nonlinear Hamiltonian systems. The exponentially small splitting of separatrix, standard and whisker map approaches, Chirikov overlap criterion, renormalization group technique, and incremental energy method are presented herein for analytic predictions of the onset of resonance in stochastic layers. An energy spectrum technique is also presented for the numerical prediction of such an onset, and the numerical computation of stochastic layer widths is introduced as well. Some technical problems in this area are pointed out. The objective of this review is to excite more research in nonlinear dynamic systems. There are 47 references cited in this review article.