Conversion of stability in systems close to a Hopf bifurcation by time-delayed coupling

Delay- and topology-independent design for inducing amplitude death on networks with time-varying delay connections

We present a procedure to systematically design the connection parameters that will induce amplitude death in oscillator networks with time-varying delay connections. The parameters designed by the procedure are valid in oscillator networks with any network topology and with any connection delay. The validity of the design procedure is confirmed by numerical simulation. We also consider a partial time-varying delay connection, which has both time-invariant and time-varying delays. The effectiveness of the partial connection is shown theoretically and numerically.

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.

Amplitude death in complex networks induced by environment

We present a mechanism for amplitude death in coupled nonlinear dynamical systems on a complex network having interactions with a common environment like external system. We develop a general stability analysis that is valid for any network topology and obtain the threshold values of coupling constants for the onset of amplitude death. An important outcome of our study is a universal relation between the critical coupling strength and the largest nonzero eigenvalue of the coupling matrix. Our results are fully supported by the detailed numerical analysis for different network topologies.

Control of delay-induced oscillation death by coupling phase in coupled oscillators

A coupling phase is deemed to be crucial in stabilizing behavior in nonlinear systems. In this paper, we study how the coupling phase influences the delay-induced oscillation death (OD) in coupled oscillators. The OD boundaries are identified analytically even in the presence of the coupling phase. We find that OD only occurs for a coupling phase belonging to a certain interval. The optimal coupling phase, under which the largest OD island forms, is characterized well by a power law scaling with respect to the frequency. The coupling phase turns out to be a key parameter that determines a delay-induced OD. Furthermore, the controlling role of the coupling phase generally is proved to hold fairly for networked delay-coupled oscillators.

General mechanism for amplitude death in coupled systems

We introduce a general mechanism for amplitude death in coupled synchronizable dynamical systems. It is known that when two systems are coupled directly, they can synchronize under suitable conditions. When an indirect feedback coupling through an environment or an external system is introduced in them, it is found to induce a tendency for antisynchronization. We show that, for sufficient strengths, these two competing effects can lead to amplitude death. We provide a general stability analysis that gives the threshold values for onset of amplitude death. We study in detail the nature of the transition to death in several specific cases and find that the transitions can be of two types--continuous and discontinuous. By choosing a variety of dynamics, for example, periodic, chaotic, hyperchaotic, and time-delay systems, we illustrate that this mechanism is quite general and works for different types of direct coupling, such as diffusive, replacement, and synaptic couplings, and for different damped dynamics of the environment.

Towards easier realization of time-delayed feedback control of odd-number orbits

We develop generalized time-delayed feedback schemes for the stabilization of periodic orbits with an odd number of positive Floquet exponents, which are particularly well suited for experimental realization. We construct the parameter regimes of successful control and validate these by numerical simulations and numerical continuation methods. In particular, it is shown how periodic orbits can be stabilized with symmetric feedback matrices by introducing an additional latency time in the control loop. Finally, we show using normal form analysis and numerical simulations how our results could be implemented in a laser setup using optoelectronic feedback.

Amplitude death in time-delay nonlinear oscillators coupled by diffusive connections

This paper analyzes the stability of the amplitude death phenomenon that occurs in a pair of scalar time-delay nonlinear oscillators coupled by static, dynamic, and delayed connections. Stability analysis shows that static connections never induce death in time-delay oscillators. Further, for the case of dynamic and delayed connections, a simple instability condition under which death never occurs is derived. A systematic procedure for estimating the boundary curves of death regions is also provided. These analytical results are then verified by electronic circuit experiments.