On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities

Abstract

This article discusses the search procedure for Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system, using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated, using the modified Benettin’s algorithm to study the stability of the found regime and confirm the type of attractor.

On a High-Precision Method for Studying Attractors of Dynamical Systems and Systems of Explosive Type

“…In the papers [12,13], the numerical method based on the representations ( 5)-( 7) is called the FGBFI method. Next, we describe the algorithm underlying it.…”

“…and for the study of the limit points of dynamic systems of the form (1) for Poisson stability and the calculation of the Lyapunov exponents by the modified Benettin's algorithm [12,13], a software in the C++ language for Linux was developed. The source codes are available on GitHub [14].…”

“…There is the FGBFI numerical method (the firmly grounded backward-forward integration method) described in [10][11][12][13]. In this articles, the recurrence relations for calculating of the coefficients of expansion of local solutions in a power series are received for any dynamic system with quadratic nonlinearities in the vector form (it distinguishes the FGBFI-method from automatic differentiation and CNS).…”

“…The criteria for checking the accuracy of the resulting approximate solution are described in [10,11,13]. Note that the backward time pass is used in some of them.…”

Modulo periodic Poisson stable solutions of dynamic equations on a time scale

“…Poisson stable motions, which were first introduced by Poincaré [29], include the cases of oscillations such as periodic, quasi-periodic, almost periodic, almost automorphic, recurrent, and pseudo-recurrent ones [8,9,14,21,37]. Results on Poisson stable solutions for stochastic differential equations and a class of fourth-order dynamical systems can be found in the studies [13,27,28]. Recently, a new type of flow called modulo periodic Poisson stable (MPPS) was introduced in paper [5], where the authors also considered the presence of MPPS trajectories in quasilinear systems of ordinary differential equations.…”