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

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