Besides the alternately excited oscillators, the principle of manipulation by phases in the course of the excitation transfer may be applied to systems with delayed feedback. In this case, it is sufficient to have a single self-oscillator manifesting successive stages of activity and suppression, while the excitation transfer accompanied with appropriate phase transformation is carried out through the delayed feedback loop, from one stage of activity to another. In practical implementation, these systems can be even simpler than the alternately excited oscillators. From a mathematical point of view, they arc more complex, because presence of the delay implies formally infinite dimension of the state space. Careful mathematical analysis of attractors in such systems, including rigorous foundation of hyperbolicity, is a challenging problem that goes far beyond the scope of this book. In the present chapter, we start with some necessary material concerning the mathematical nature of the differential equations with delay. The next two sections consider models of systems with delay, generating sequences of pulses with the phases of the carrier transformed from pulse to pulse in accordance with expanding circle maps. First, the non-autonomous systems are discussed that operate due to the externally imposed parameter modulation with or without an auxiliary reference signal. Afterwards, we consider an autonomous system with time delay. On the physical level of reasoning it seems confident that chaotic attractors in these systems are of the same nature as those in the above discussed alternately excited oscillatory systems being associated with expanding circle maps for some cyclic variable. Particularly, it is believed that the generated chaos is structurally stable. As one can hypothesize, attractors in these systems as they are treated in a framework of discrete-time description (like the Poincaré maps) belong to the class of solenoids of Smale-Williams embedded in the infinite-dimensional phase space.
KeywordsLyapunov Exponent Parameter Modulation Chaotic Attractor Iteration Diagram Large Lyapunov Exponent
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