Criterion of stochastic layer near a planar homoclinic orbit of nonlinear dynamical system
type: Item
instance of: scholarly article
title: Criterion of stochastic layer near a planar homoclinic orbit of nonlinear dynamical system
zbMATH Open document ID: 0898.70017
author: Albert C. J. Luo
published in: Journal of Hydrodynamics. Ser. B
publication date: 1998-11-01T00:00:00Z
review text: The examined dynamical system is described by the equation \(\ddot x= f(\dot x)+ g(x, t)\), where \(g\) has the time period \(T\), and \(f\) is a Hamiltonian vector field defined on \(\mathbb{R}^2\) and satisfying certain smoothness and boundedness conditions. Besides the author imposes some essential specific assumptions on the system. The headline criterion is based on the subharmonic resonance condition and on the energy increment along the separatrix of the system. First, the author enumerates difficulties which occur by the application of existing methods to this problem. Then he formulates three theorems (proved in an appendix) which give the foundations of a new approach to the problem. With the paper's method, the disappearence stochastic layers can be detected, and critical parameters can be calculated. Two examples are shown: the Duffing-Holmes equation, and the forced planar pendulum. The critical conditions for these systems are presented as their stochastic layers begin to disappear and global chaos ensues. The approximate critical conditions for global chaotic structure are also obtained. Finally, the author calculates the critical amplitude for the given excitation frequency. He promises to show the results of numerical simulations via the Runge-Kutta technique, but figures 14 (a)--(d) and 15 (a)--(d) mentioned in the text could not be found by the reviewer.
Mathematics Subject Classification ID: 70L05
Mathematics Subject Classification ID: 70K40
Mathematics Subject Classification ID: 70K50
zbMATH DE Number: 880684
zbMATH Keywords: Hamiltonian vector field
zbMATH Keywords: smoothness
zbMATH Keywords: boundedness
zbMATH Keywords: subharmonic resonance condition
zbMATH Keywords: energy increment
zbMATH Keywords: separatrix
zbMATH Keywords: disappearence stochastic layers
zbMATH Keywords: critical parameters
zbMATH Keywords: Duffing-Holmes equation
zbMATH Keywords: forced planar pendulum
zbMATH Keywords: global chaos
zbMATH Keywords: critical amplitude
zbMATH Keywords: Runge-Kutta technique
reviewed by: Á. P. Bosznay
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 880684