A chaotic system with only one stable equilibrium

TL;DR

This paper reports a surprising discovery of a three-dimensional autonomous quadratic system with only one stable equilibrium that exhibits chaotic behavior, challenging prior expectations in chaos theory.

Contribution

The paper introduces a novel three-dimensional quadratic system with a single stable equilibrium that demonstrates chaos, expanding understanding of possible system dynamics.

Findings

System exhibits chaos with positive Lyapunov exponent

System has fractional dimension and broad frequency spectrum

Chaotic behavior follows a period-doubling route

Abstract

If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the new system is not of saddle-focus type, therefore the familiar \v{S}i'lnikov homoclinic criterion is not applicable, it is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous broad frequency spectrum, and a period-doubling route to chaos.