We show that in integro-differential delayed dynamical systems, an hybrid state of simultaneous fast-scale hyperchaos and slow-scale periodicity can emerge subsequently
to a sequence of Hopf bifurcations. The resulting timetrace thereby consists in hyperchaotic oscillations
breathing periodically at a significantly lower frequency. We present experimental evidence of this counterintuitive type of dynamics
in a Mach-Zehnder modulator optically fed by a semiconductor laser and subjected to nonlinear electro-optical feedback.
We provide a theoretical understanding of this phenomenon, and we discuss its similarity with some dynamical behaviors observed in thermo-optical Fabry-Perot cavities
and in neurons.
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