Memory Spectra and Lorentzian Power Spectra of the Chaotic Duffing Oscillator

We derive a non-Markovian evolution equation for the momentum (p) over cap (t) of the chaotic Duffing oscillator by introducing a collective frequency (Omega) over cap (0) and a memory function (gamma) over cap (t) using the Mori projection operator formalism. Then, denoting the real and imaginary parts of the memory spectrum (gamma) over cap (i omega) = integral(infinity)(0) (gamma) over cap (t) exp [-i omega t] dt by (gamma) over cap (r)(omega) and (gamma) over cap (i)(omega), respectively, we find that the power spectrum I((p) over cap)(omega) of the momentum (p) over cap (t) can be written in terms of that the power spectrum In(w) of the momentum p(t) can be written in terms of (Omega) over cap (0), that the power spectrum In(w) of the momentum p(t) can be written in terms (Omega) over cap (0), (gamma) over cap (r)(omega) and (gamma) over cap (i)(omega). For a Duffing oscillator with molecular viscosity gamma(0) = 0.5 and an external force with amplitude b = 0.55 and frequency omega(0) = 1.2, we find that (gamma) over cap (r)(omega) has one sharp peak at frequency omega = 1.80 and a few small peaks. It is also shown that the power spectrum I((p) over cap)(omega) has two sharp peaks at frequencies omega(1) = 0.509 and omega(2) = 1.89 and one line spectrum at omega = omega(0), leading to a two-peaks approximation to the time correlation function C(p)(t) with correlation time tau(M) similar to 4.69T, (T = 2 pi/omega(0)). Then, it is shown that the structure of the omega(1)-peak can be represented by an asymmetric Lorentzian peak.