Memory function approach to a nonlinear oscillator

Abstract

An exact linear equation of motion of the form x ̈ (t) + ω 2x(t) + ∫ t 0 Λ(t − t′)x(t′) dt′ = 0 proposed for an undamped anharmonic oscillator. A renormalized frequency ω and a memory function Λ( t) reflect the nonlinearity. The laplace transform Λ( z) of the memory function is given by a combination of infinite continued fractions in z 2. With a cubic anharmonic oscillator as an example, we show that higher-order memory functions Λ n ( t), which are associated with Λ( t), oscillate rapidly so that Λ n ( t) ≡ 0 is a good approximation (cf. the instantaneous decay approximation Λ n ( t) ∝ δ ( t) in dissipative systems).