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

TL;DR

This paper presents a numerical approach to classify solutions on an attractor of a fourth-order nonlinear dynamical system with quadratic nonlinearities, using Poincaré recurrences and Lyapunov exponents to analyze stability.

Contribution

It introduces a method combining Poincaré recurrences and high-precision numerical techniques to study the stability and classification of attractors in complex dynamical systems.

Findings

Successful classification of attractor solutions

Calculation of Lyapunov exponents confirming stability

Identification of solution regimes in a fourth-order system

Abstract

This article discusses the search procedure for the Poincar\'e 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.