Qualitative analysis of certain generalized classes of quadratic oscillator systems

TL;DR

This paper provides a qualitative analysis of two generalized quadratic oscillator systems, exploring their equilibrium points, stability, and transition to chaos under periodic forcing, extending previous work by including dissipation effects.

Contribution

It offers a detailed local stability analysis of two quadratic oscillator models and investigates the onset of chaos with dissipation and periodic forcing, which was not previously studied.

Findings

First potential has a pair of equilibrium points, with stability depending on coupling sign.

Second potential exhibits only a center for all coupling signs.

Introduction of periodic forcing leads to chaos in the system.

Abstract

We carry out a systematic qualitative analysis of the two quadratic schemes of generalized oscillators recently proposed by C. Quesne [J.Math.Phys.\textbf{56},012903 (2015)]. By performing a local analysis of the governing potentials we demonstrate that while the first potential admits a pair of equilibrium points one of which is typically a center for both signs of the coupling strength $\lambda$, the other points to a centre for $\lambda < 0$ but a saddle $\lambda > 0$. On the other hand, the second potential reveals only a center for both the signs of $\lambda$ from a linear stability analysis. We carry out our study by extending Quesne's scheme to include the effects of a linear dissipative term. An important outcome is that we run into a remarkable transition to chaos in the presence of a periodic force term $f\cos \omega t$.