Delay-time Electronic Devices Generating Trains of Oscillations with Phases Governed by Chaotic Maps

Abstract

Here we turn to consideration of systems with time delayed feedback corresponding to models of Cha. 10, which have been implemented as electronic laboratory devices and studied in experiments described in (Kuznetsov and Ponomarenko, 2008; Baranov et al., 2010). In a frame of the hyperbolic theory the status of dynamics observed in these systems is not so well defined because the classic formulation of the theory relates to finite-dimensional systems, while for the timedelay systems the stale space is formally infinite-dimensional. Nevertheless, physically operation principle and observed dynamical phenomena look very similar to those in systems of alternately excited oscillators, for which the hyperbolic nature of the chaotic attractor is well established (see Chaps. 4, 7 and 12). On one hand, it implies a challenging problem for mathematicians to give rigorous definitions and foundation for existence of uniformly hyperbolic chaotic attractors embedded in the infinite dimensional state space of the time-delay systems. On the other hand, accounting transparent principle of operation and evident analogy with the alternately excited oscillators, on physical level of reasoning one can confidently suggest that the chaotic dynamics in time-delay systems of the considered class will have the same features relevant to technical applications like structural stability (robustness).