We show that the output of systems with time-varying delay can exhibit a new kind of chaotic behavior characterized by laminar phases, which are periodically interrupted by irregular bursts. Within each laminar phase the output intensity remains almost constant, but its level varies chaotically from phase to phase. In scalar systems, the periodic dynamics of the lengths and the chaotic dynamics of the intensity levels can be understood and also tuned via two one-dimensional maps, which can be deduced from the nonlinearity of the delay equation and from the delay variation, respectively.
We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating with the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with a constant delay. The new, second class is characterized by the mode-locking behavior of their access maps and by an asymptotically linear, instead of a logarithmic, scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay.
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