Existence of perpetual points in nonlinear dynamical systems and its applications

TL;DR

This paper introduces the concept of perpetual points in nonlinear dynamical systems, which are critical for understanding transient dynamics, locating hidden attractors, and determining system dissipativity.

Contribution

It defines and analyzes perpetual points, a new class of critical points, and demonstrates their applications in identifying hidden and co-existing attractors in nonlinear systems.

Findings

Perpetual points exist in various nonlinear systems.

They help locate hidden oscillating attractors.

Perpetual points indicate whether a system is dissipative.

Abstract

A new class of critical points, termed as perpetual points, where acceleration becomes zero but the velocity remains non-zero, are observed in dynamical systems. The velocity at these points is either maximum or minimum or of inflection behavior.These points also show the bifurcation behavior as parameters of the system vary. These perpetual points are useful for locating the hidden oscillating attractors as well as co-existing attractors. Results show that these points are important for better understanding of transient dynamics in the phase space. The existence of these points confirms whether a system is dissipative or not. Various examples are presented, and results are discussed analytically as well as numerically.