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Intricate routes to chaos in the Mackey-Glass delayed feedback system. (English) Zbl 1266.93057

Summary: We describe some remarkable continuous deformations which create and destroy peaks in periodic oscillations of the Mackey-Glass equation, a paradigmatic example of a delayed feedback system. Peak creation and destruction results in richer bifurcation diagrams which, in addition to the familiar branches arising from period-doubling and peak-adding bifurcations, may also display arbitrary combinations of doubling and adding, leading to highly complex mosaics of stability domains in control parameter space. In addition, we show that the onset of higher dimensionality does not alter the prevailing dynamics instantaneously and, remarkably, even may have no effect at all, a result that cannot be predicted analytically with standard methods.

MSC:

93B52 Feedback control
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
34K11 Oscillation theory of functional-differential equations
34H10 Chaos control for problems involving ordinary differential equations
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