From Generalized Controlled Nonlinear Oscillators to Lorenz-like Systems

Year: 2026

Authors: Ginoux J.M., Meucci R., Llibre J., Chen G.R., Chua L.O.

Autors Affiliation: Univ Toulon & Var, Aix Marseille Univ, CNRS, LIS, Toulon, France; CNR, Natl Inst Opt, Florence, Italy; Univ Autonoma Barcelona, Dept Matemat, Barcelona 08193, Spain; City Univ Hong Kong, Dept Elect Engn, Hong Kong, Peoples R China; Univ Calif Berkeley, Dept EECS, Berkeley, CA 94720 USA.

Abstract: Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes-Rand equation. These equations have been extensively studied during the last century and the last two decades. In the early 1990s, Abarbanel, Rabinovich and Sushchik proposed replacing the sinusoidal forcing term with a force controlled by the movements of the oscillator itself, i.e. by the product of two variables: the first being the solution of the oscillator itself, while the second is the solution of a first-order nonlinear ordinary differential equation. They referred to the resulting autonomous dynamical system of two coupled nonlinear ordinary differential equations as a controlled nonlinear oscillator. To that end, they introduced a change of variables and parameters to transform the controlled nonlinear oscillator that corresponds to a particular case of the forced Duffing equation into the Lorenz system. The aim of this work is to show that their idea can be further generalized and applied to many other dynamical systems, including the forced Holmes-Rand equation, Chua’s cubic circuit, Chen’s system and the forced Helmholtz oscillator. It is proved that a certain class of three-dimensional dynamical systems can be rewritten into the form of generalized controlled nonlinear oscillators, which can then be transformed into various Lorenz-like systems. Such a transformation could be very useful for the study of intermittent chaos.

Journal/Review: INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS

Volume: 36 (10)      Pages from:   to:

More Information: The authors would like to thank very much Pr. Paul Manneville for his fruitful advice. Jaume Llibre is partially supported by the Agencia Estatal de Investigacion of Spain Grant PID2022-136613NB-100.
KeyWords: Chua´s cubic circuit; Chen system; Lorenz-like system; controlled nonlinear oscillator; nonlinear dynamics; chaotic system; intermittent chaos
DOI: 10.1142/S0218127426300223