Power spectra and dynamical invariants for delay-differential and difference equations

Abstract

Listen

Translate article

The Mackey-Glass and Ikeda delay-differential equations (DDEs) are models of feedback systems in which attractor dimension is proportional to the delay. Power spectra from these models are investigated analytically and numerically. As the delay increases much beyond the system response time, an exponentially decaying spectrum with a near-periodic superimposed modulation appears. It is shown that each peak in this modulation is associated with a mode of oscillation predicted by linear stability analysis around the fixed point. The number of such modes within a characteristic autocorrelation time of the solution agrees with the Lyapunov dimension of the attractor. The disappearance of this modulation at higher delay values is also explained. The decay rate of the spectrum in the mid-to-high frequency range is found to agree with the sum of the positive Lyapunov exponents, i.e. with the metric entropy. Realistic models with a distribution of delays display similar spectra as their memory kernel narrows into a delta-function. For very large delay, the DDE approaches an infinite-dimensional continuous-time difference equation (CTDE), with each point on a delay interval evolving independently and according to a chaotic discrete-time map. CTDE spectra, which have many features of finite-delay DDE spectra, can be computed analytically if the spectrum of this map is known. For constant initial functions, they have a 1 f 2 form at low frequencies. Our analysis suggests that the spectrum of the chaotic map is at the origin of the peak shapes in DDE spectra at low frequency. Our findings highlight the relevance of the linear prrperties of DDEs to the characterization of their nonlinear properties.