Temporal Dissipative Solitons in Time-Delay Feedback Systems

Abstract

Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, auto-solitons, spot or pulse solutions, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studied in spatially extended systems, temporally localized states are gaining attention only recently, driven primarily by applications from fiber or semiconductor lasers. Here we present a theory for temporal dissipative solitons (TDS) in systems with time-delayed feedback. In particular, we derive a system with an advanced argument, which determines the profile of the TDS. We also provide a complete classification of the spectrum of TDS into interface and pseudo-continuous spectrum. We illustrate our theory with two examples: a generic delayed phase oscillator, which is a reduced model for an injected laser with feedback, and the FitzHugh-Nagumo neuron with delayed feedback. Finally, we discuss possible destabilization mechanisms of TDS and show an example where the TDS delocalizes and its pseudo-continuous spectrum develops a modulational instability.

Robust spike timing in an excitable cell with delayed feedback

“…Excitable systems subject to delayed feedback are capable of regenerating their own excitable response, resulting in robust multi-pulse patterns dependent on the feedback strength and the length of the delay interval [15][16][17]. The presence of rich dynamics, including coexistence between different types of dynamic behaviour in nonlinear systems with delays [15][16][17], has sparked significant interest in investigating the interplay between excitability and delay in a variety of systems.…”

“…Excitable systems subject to delayed feedback are capable of regenerating their own excitable response, resulting in robust multi-pulse patterns dependent on the feedback strength and the length of the delay interval [15][16][17]. The presence of rich dynamics, including coexistence between different types of dynamic behaviour in nonlinear systems with delays [15][16][17], has sparked significant interest in investigating the interplay between excitability and delay in a variety of systems. A prominent example of such a system involves semiconductor micro-cavity lasers [18][19][20]; in an analogous fashion to neural systems, the number of emission pulses per delay period in the laser system is thought to be reflective of its ability to store information.…”

Spatial dissipative solitons in graphene-based active random metamaterials

“…Dissipative solitons (DSs) are self-organized localized structures arising in diverse dissipative physical and biological systems [1], playing a central role in mode-locked lasers [2,3], photonic molecules [4], time-delay feedback systems [5], Bose-Einstein condensates of exciton polaritons [6] and cold atoms [7,8]. Furthermore, DSs offer an appealing platform for cavity-free stimulated emission of radiation, as in particular in optical amplifiers [9].…”

Emulating complex networks with a single delay differential equation

“…Systems with time-delays, or delay differential equations (DDE), play an important role in modeling various natural phenomena and technological processes [1][2][3][4][5][6][7][8]. In optoelectronics, delays emerge due to finite optical or electric signal propagation time between the elements [9][10][11][12][13][14][15][16][17][18][19][20]. Similarly, in neuroscience, propagation delays of the action potentials play a crucial role in information processing in the brain [21][22][23][24][25][26][27][28].…”