Dissipative solitons with extreme spikes (DSESs), previously thought to be rare solutions of the complex cubic–quintic Ginzburg–Landau equation, occupy in fact a significant region in its parameter space. The variation of any of its five parameters results in a rich structure of bifurcations. We have constructed several bifurcation diagrams that reveal periodic and chaotic dynamics of DSESs. There are various routes to the chaotic behavior of DSESs, including a sequence of period-doubling bifurcations. It is well known that the complex cubic–quintic Ginzburg–Landau equation can serve as a master equation for the description of passively mode-locked lasers. Our results may lead to the observation of DSESs in laser systems.
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