Criterion of stochastic layer near a planar homoclinic orbit of nonlinear dynamical system
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.
scientific article; zbMATH DE number 880684