Intricate routes to chaos in the Mackey–Glass delayed feedback system

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. -- Highlights: ► We report a new class of bifurcations: waveform deformations. ► We describe new characteristics in phase diagrams of delayed feedback. ► We explain how waveform deformations appear and disappear. ► Deformations imply a complex mosaic of phases in control space. Waveforms may be efficiently encoded using a small set of sub-patterns.