Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations

Abstract

A comparative study of the homotopy analysis method and an improved renormalization group method is presented in the context of the Rayleigh and the Van der Pol equations. Efficient approximate formulae as functions of the nonlinearity parameter ε for the amplitudes a(ε) of the limit cycles for both these oscillators are derived. The improvement in the renormalization group analysis is achieved by invoking the idea of nonlinear time that should have significance in a nonlinear system. Good approximate plots of limit cycles of the concerned oscillators are also presented within this framework.

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

“…The importance of high precision computation in applied mathematics and science need not be overemphasized [5]. The higher precision quantitatively accurate computation of periodic orbits is facilitated in the framework of a novel asymptotic analysis [6][7][8][9], so as to allow significant numerical improvements in the computations of periodic orbits by the conventional asymptotic techniques such as renormalization group method(RGM) [10], multiple scale method (MSM) [1], homotopy analysis method [11,12] etc. As a prototype of strongly nonlinear oscillator, we consider here singularly perturbed Rayleigh Equation (SRLE) [1,4] equation with an external periodic excitation…”

“…where dots are used to designate the derivatives with respect to time. Rayleigh Equation, either regular or singularly perturbed, and a close cousin of Van der Pol equation [7], is one of the extensively studied nonlinear oscillatory systems because of its wide applications in acoustics, physiology and cardiac cycles, solid mechanics, electronics and nonlinear electrical circuits, musical instruments and many other different fields [1,2,13,14]. Singularly perturbed Rayleigh equation is equivalently related closely to the regularly perturbed Rayleigh equation (RLE)…”

“…In the present work, we investigate a modified and improved renormalization group (RG) method (IRGM) incorporating a novel asymptotic structure, called SL(2, R) duality structure, that is introduced and is being investigated previously by Datta et al [6][7][8][9]23] in various nonlinear applications. An application of IRGM and duality structure was presented in [7] in the context of Rayleigh equation (1.2) and Van der Pol equation (without forcing) ẍ + ε ẋ…”

“…To recall, the improvement of RG Method was made possible in [7] by introduction of SL(2, R) invariant, renormalized (control) variables, which can be used to ensure an improved convergence of approximate solutions through continuous deformations to the exact solution of the nonlinear oscillation problem concerned, with any desired accuracy level. The introduction of continuous deformation is based on the novel idea of nonlinear time [6,8,9,24], incorporating ideas of asymptotic duality structure and dynamically adustable time scales [8,9,24].…”

The renormalization method based on the Taylor expansion and applications for asymptotic analysis

“…Kunihiro [28][29][30][31][32][33][34][35][36] tried to give the RG method a geometrical interpretation based on the classical theory of envelop in differential geometry. Furthermore the standard RG method and its geometrical formulas have been developed and applied to many problems [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] such as center manifolds [39], quantum kinetics [41,47], normal form theory [42], invariant manifold and reduction equation [43][44][45]52], variation of parameters [48] and resonance of quantum [51] and so forth. In addition, the error estimate of approximate solutions obtained by the RG method are given in [37,39,42], and comparison of RG method and homotopy analysis is discussed in [55].…”