We present a detailed report [see S. Sen et al., Phys. Rev. Lett. 77, 4855 (1996)] of our numerical and analytical studies on the relaxation of a classical particle in the potentials [Formula Presented]. Both of the approaches confirm that at all temperatures, the relaxation functions (e.g., velocity relaxation function and position relaxation function) decay asymptotically in time t as [Formula Presented]. Numerically calculated power spectra of the relaxation functions show a gradual transition with increasing temperature from a single sharp peak located at the harmonic frequency [Formula Presented] to a broad continuous band. The [Formula Presented] relaxation is also found when [Formula Presented] is a polynomial in powers of [Formula Presented] with a nonvanishing coefficient accompanying the [Formula Presented] term in [Formula Presented]. Numerical calculations show that in the cases in which the leading term in [Formula Presented] behaves as [Formula Presented] with integer n, the asymptotic relaxation exhibits [Formula Presented] decay where [Formula Presented]. We briefly discuss the analytical approaches to relaxation studies in these strongly anharmonic systems using direct solution of the equation of motion and using the continued fraction formalism approach for relaxation studies. We show that the study of the dynamics of strongly anharmonic oscillators poses unique difficulties when studied via the continued fraction or any other time-series construction based approaches. We close with comments on the physical processes in which the insights presented in this work may be applicable. © 1999 The American Physical Society.