Bifurcation and Chaos in Time-Delayed Piecewise Linear Dynamical System
The phenomena of bifurcations and chaos have been well studied in nonlinear dynamical systems without delay and described by nonlinear difference, differential, difference-differential, etc. equations. Routes to bifurcations, onset of chaos, nature of the chaotic attractors and their characterizations in such systems have all been analyzed extensively [1, 2]. However, such analyses have not been carried out in any greater detail in the case of nonlinear dynamical systems with delay even in the scalar systems. In this chapter we shall introduce a prototypical delay system, which is a piecewise linear one, in order to appreciate the nature of bifurcations and chaos phenomena underlying nonlinear time-delay systems, and to understand clearly the nature of transients and difficulties in numerical analysis as well as the frequent existence of hyperchaotic attractors with multiple positive Lyapunov exponents. The dynamics of other nonlinear time-delay systems will be taken up in the next chapter.
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