Memory Spectra and Lorentzian Power Spectra of the Chaotic Duffing Oscillator(General and Mathematical Physics)

概要

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We derive a non-Markovian evolution equation for the momentum p^^^(t) of the chaotic Duffing oscillator by introducing a collective frequency Ω^^^_0 and a memory function γ^^^(t) using the Mori projection operator formalism. Then, denoting the real and imaginary parts of the memory spectrum γ^^^(iw)=∫^∞_0γ^^^(t)exp[-iwt] dt by γ^^^_r(w) and γ^^^_i(w), respectively, we find that the power spectrum I_<p^^^>(w) of the momentum P^^^(t) can be written in terms of Ω^^^_0, γ^^^_r(w) and γ^^^_i(w). For a Duffing oscillator with molecular viscosity γ^0=0.5 and an external force with amplitude b=0.55 and frequency w_0=1.2, we find that γ^^^_r(w) has one sharp peak at frequency w=1.80 and a few small peaks. It is also shown that the power spectrum I_<p^^^>(w) has two sharp peaks at frequencies w_1=0.509 and W_2=1.89 and one line spectrum at w=w_0, leading to a two-peaks approximation to the time correlation function C_p(t) with correlation time T_M〜4.69T, (T=2π/w_0). Then, it is shown that the structure of the w_1-peak can be represented by an asymmetric Lorentzian peak.
2008-10-25