Efficient Computation of Periodic Orbits of Forced Rayleigh Equation in the Framework of Novel Asymptotic Structures

TL;DR

This paper introduces a novel asymptotic analysis framework for accurately computing periodic orbits in forced Rayleigh equations, surpassing traditional methods in precision and capturing complex dynamical phenomena.

Contribution

It develops a new nonlinear asymptotic approach incorporating SL(2,R) invariance, enabling high-accuracy computation of relaxation oscillations in strongly nonlinear systems.

Findings

Traditional RGM methods are inadequate for precise orbit computation.

The proposed nonlinear asymptotic method achieves over 98% accuracy.

Numerical verification reveals condensation and rarefaction phenomena in the system.

Abstract

Higher precision efficient computation of period 1 relaxation oscillations of strongly nonlinear and singularly perturbed Rayleigh equations with external periodic forcing is presented. The computations are performed in the context of conventional renormalization group method (RGM). We demonstrate that although a slight homotopically modified RGM could generate approximate periodic orbits that agree qualitatively with the exact orbits, the method, nevertheless, fails miserably to reduce the large quantitative disagreement between the theoretically computed results with that of exact numerical orbits. In the second part of the work we present a novel asymptotic analysis incorporating SL(2,R) invariant nonlinear deformation of slower time scales, $t_{n} =\varepsilon^{n}t, \ n\rightarrow\infty, \ \varepsilon<1$, for asymptotic late time $t$, to a nonlinear time $T_{n}=t_{n}\sigma(t_{n})$, where the deformation factor $\sigma(t_{n})>0$ respects some well defined SL(2,R) constraints. Motivations and detailed applications of such nonlinear asymptotic structures are explained in performing very high accuracy ($> 98\%$) computations of relaxation orbits. Existence of an interesting condensation and rarefaction phenomenon in connection with dynamically adjustable scales in the context of a slow-fast dynamical system is explained and verified numerically.