  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.
