Phase-locking swallows in coupled oscillators with delayed feedback

Exploring quasi-periodic shrimps in the parameter space of a discrete-time food chain model

Shrimps are islands of regularity within chaotic regimes in bi-parameter spaces of nonlinear dynamical systems. While the presence of periodic shrimps has been extensively reported, recent research has uncovered the existence of quasi-periodic shrimps. Compared to their periodic counterparts, quasi-periodic shrimps require a relatively higher-dimensional phase-space to come into existence and are also quite uncommon to observe. This Focus Issue contribution delves into the existence and intricate dynamics of quasi-periodic shrimps within the parameter space of a discrete-time, three-species food chain model. Through high-resolution stability charts, we unveil the prevalence of quasi-periodic shrimps in the system's unsteady regime. We extensively study the bifurcation characteristics along the two borders of the quasi-periodic shrimp. Our analysis reveals that along the outer border, the system exhibits transition to chaos via intermittency, whereas along the inner border, torus-doubling and torus-bubbling phenomena, accompanied by finite doubling and bubbling cascades, are observed. Another salient aspect of this work is the identification of quasi-periodic accumulation horizon and different quasi-periodic (torus) adding sequences for the self-distribution of infinite cascades of self-similar quasi-periodic shrimps along the horizon in certain parameter space of the system.

Sensitivity to external signals and synchronization properties of a non-isochronous auto-oscillator with delayed feedback

For auto-oscillators of different nature (e.g. active cells in a human heart under the action of a pacemaker, neurons in brain, spin-torque nano-oscillators, micro and nano-mechanical oscillators, or generating Josephson junctions) a critically important property is their ability to synchronize with each other. The synchronization properties of an auto oscillator are directly related to its sensitivity to external signals. Here we demonstrate that a non-isochronous (having generation frequency dependent on the amplitude) auto-oscillator with delayed feedback can have an extremely high sensitivity to external signals and unusually large width of the phase-locking band near the boundary of the stable auto-oscillation regime. This property could be used for the development of synchronized arrays of non-isochronous auto-oscillators in physics and engineering, and, for instance, might bring a better fundamental understanding of ways to control a heart arrythmia in medicine.

Amplitude and phase dynamics in oscillators with distributed-delay coupling

This paper studies the effects of distributed-delay coupling on the dynamics in a system of non-identical coupled Stuart-Landau oscillators. For uniform and gamma delay distribution kernels, the conditions for amplitude death are obtained in terms of average frequency, frequency detuning and the parameters of the coupling, including coupling strength and phase, as well as the mean time delay and the width of the delay distribution. To gain further insights into the dynamics inside amplitude death regions, the eigenvalues of the corresponding characteristic equations are computed numerically. Oscillatory dynamics of the system is also investigated, using amplitude and phase representation. Various branches of phase-locked solutions are identified, and their stability is analysed for different types of delay distributions.