The Physics of the Logistic Map in the Continuous-Time Limit

M. Duong-Van

Physics Letters A , vol. 151, no.9, 493-499 (31 December, 1990)

Abstract

We propose an equation of Brownian motion with intrinsic fluctuations that manifests both a 1/f power spectrum and white-noise stochasticity. The solution x(t) of this equation, when evaluated at multiples of the time interval dt=tau, where 1/tau is the diffusion constant, generates the random sequences xn of the logistic map xn+1 = lambda xn (1-xn) for lambda=4. The corresponding power spectrum is consistent with a white-noise power spectrum. When time is treated as continuous, the power spectrum of x(t) is asymptotically 1/f. This equation correctly represents frequency-dependent ionic conductivities without requiring a random forcing function.

math model of 1/f section