On the stability of unstable states, bifurcation, chaos of nonlinear dynamic systems

TL;DR

This paper investigates the stability, bifurcation, and chaos phenomena in nonlinear dynamical systems without small parameters, highlighting their significance in physics and mechanics, especially in trapping atomic particles.

Contribution

It provides new insights into the stability and chaotic behavior of nonlinear systems lacking small parameters, with applications to atomic particle trapping.

Findings

Unstable states can exhibit bifurcations and chaos without small parameters.

The stability analysis applies to systems in physics and mechanics.

Results have implications for controlling atomic particles in traps.

Abstract

The question of the stability of unstable states of dynamical systems that do not explicitly contain a small parameter, chaos and bifurcations in them has attracted attention ever since [1-14]. This is due to the fact that this problem often arises not only in mathematics, but also in various fields of mechanics and physics. In particular, the task of retaining atomic particles in electrodynamic traps has recently become of special interest [14]...