Eur. Phys. J. B 71, 111-119 (2009)
https://doi.org/10.1140/epjb/e2009-00262-8

Complex behavior of simple maps with fluctuating delay times

Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany

Corresponding authors: ^a radons@physik.tu-chemnitz.de - ^b hongliu.yang@physik.tu-chemnitz.de

Received: 18 February 2009
Revised: 15 May 2009
Published online: 21 July 2009

Abstract

Delay systems used to model retarded actions are relevant in many fields such as optics, mechanical machining, biology or physiology. A frequently encountered situation is that the length of the delay time changes with time. In this study, we use a simple map system to investigate the influence of the fluctuating delay time on the system dynamics. For simplicity, we start from a case with the delay time taking only the value of zero or one discrete time steps, where the system dynamics reduces to one- and two-dimensional map, respectively. We study two situations, periodic or random variation of the delay. Rich dynamics including coexisting multiple attractors, strange nonchaotic attractors and on-off intermittency are observed. For a special case we can show analytically the existence of a dense set of singularities of the Lyapunov exponent. Finally we present results for higher dimensional generalizations to show the relevance of our findings to more general situations.