Dynamical systems with multiple long-delayed feedbacks: Multiscale analysis and spatiotemporal equivalence

Dynamical systems with multiple, hierarchically long-delayed feedback are introduced and studied extending our previous work [Yanchuk and Giacomelli, Phys. Rev. Lett. 112, 174103 (2014)10.1103/PhysRevLett.112.174103]. Focusing on the phenomenological model of a Stuart-Landau oscillator with two feedbacks, we show the multiscale properties of its dynamics and demonstrate them by means of a space-time representation. For sufficiently long delays, we derive a normal form describing the system close to the destabilization. The space and temporal variables, which are involved in the space-time representation, correspond to suitable time scales of the original system. The physical meaning of the results, together with the interpretation of the description at different scales, is presented and discussed. In particular, it is shown how this representation uncovers hidden multiscale patterns such as spirals or spatiotemporal chaos. The effect of the delay size and the features of the transition between small and large delays is also analyzed. Finally, we comment on the application of the method and on its extension to an arbitrary, but finite, number of delayed feedback terms.