Oscillation death in asymmetrically delay-coupled oscillators

Analysis of bifurcation and explosive amplitude death in a pair of oscillators coupled via time-delay connection

Delay-induced amplitude death (AD) has received considerable research interest. Most studies on delay-induced AD investigated the local stability of equilibrium points. The present study examines the global dynamics of delay-induced AD in a pair of identical Stuart-Landau oscillators. Bifurcation diagrams consisting of synchronized periodic orbits and an equilibrium point are used to determine the mechanism of the emergence of delay-induced AD. It is shown that explosive delay-induced AD can occur via a Hopf bifurcation at the equilibrium point and a saddle-node bifurcation of synchronized periodic orbits when the delay time for the connection is continuously varied. The Hopf and saddle-node bifurcation curves in the coupling parameter space clarify the dependence of the coupling parameters on the global dynamics.

Delay-induced amplitude death in multiplex oscillator network with frequency-mismatched layers

The present paper analytically investigates the stability of amplitude death in a multiplex Stuart-Landau oscillator network with a delayed interlayer connection. The network consists of two frequency-mismatched layers, and all oscillators in each layer have identical frequencies. We show that, if the matrices describing the network topologies of each layer commute, then the characteristic equation governing the stability can be reduced to a simple form. This form reveals that the stability of amplitude death in the multiplex network is equally or more conservative than that in a pair of frequency-mismatched oscillators coupled by a delayed connection. In addition, we provide a procedure for designing the delayed interlayer connection such that amplitude death is stable for any commuting matrices and for any intralayer coupling strength. These analytical results are verified through numerical examples. Moreover, we numerically discuss the results for the case in which the commutative property does not hold.

Effects of frequency mismatch on amplitude death in delay-coupled oscillators

The present paper analytically reveals the effects of frequency mismatch on the stability of an equilibrium point within a pair of Stuart-Landau oscillators coupled by a delay connection. By analyzing the roots of the characteristic function governing the stability, we find that there exist four types of boundary curves of stability in a coupling parameters space. These four types depend only on the frequency mismatch. The analytical results allow us to design coupling parameters and frequency mismatch such that the equilibrium point is locally stable. We show that, if we choose appropriate frequency mismatches and delay times, then it is possible to induce amplitude death with strong stability, even by weak coupling. In addition, we show that parts of these analytical results are valid for oscillator networks with complete bipartite topologies.

Mathematical Modeling for Description of Oscillation Suppression Induced by Deep Brain Stimulation

A mathematical modeling for description of oscillation suppression by deep brain stimulation (DBS) is explored in this paper. High-frequency DBS introduced to the basal ganglia network can suppress pathological neural oscillations that occur in the Parkinsonian state. However, selecting appropriate stimulation parameters remains a challenging issue due to the limited understanding of the underlying mechanisms of the Parkinsonian state and its control. In this paper, we use a describing function analysis to provide an intuitive way to select the optimal stimulation parameters based on a biologically plausible computational model of the Parkinsonian neural network. By the stability analysis using the describing function method, effective DBS parameter regions for inhibiting the pathological oscillations can be predicted. Additionally, it is also found that a novel sinusoidal-shaped DBS may become an alternative stimulation pattern and expends less energy, but with a different mechanism. This paper provides new insight into the possible mechanisms underlying DBS and a prediction of optimal DBS parameter settings, and even suggests how to select novel DBS wave patterns for the treatment of movement disorders, such as Parkinson's disease.

The impact of propagation and processing delays on amplitude and oscillation deaths in the presence of symmetry-breaking coupling

We numerically investigate the impacts of both propagation and processing delays on the emergences of amplitude death (AD) and oscillation death (OD) in one system of two Stuart-Landau oscillators with symmetry-breaking coupling. In either the absence of or the presence of propagation delay, the processing delay destabilizes both AD and OD by revoking the stability of the stable homogenous and inhomogenous steady states. In the AD to OD transition, the processing delay destabilizes first OD from large values of coupling strength until its stable regime completely disappears and then AD from both the upper and lower bounds of the stable coupling interval. Our numerical study sheds new insight lights on the understanding of nontrivial effects of time delays on dynamic activity of coupled nonlinear systems.

Master stability islands for amplitude death in networks of delay-coupled oscillators

This paper presents a master stability function (MSF) approach for analyzing the stability of amplitude death (AD) in networks of delay-coupled oscillators. Unlike the familiar MSFs for instantaneously coupled networks, which typically have a single input encoding for the effects of the eigenvalues of the network Laplacian matrix, for delay-coupled networks we show that such MSFs generally require two additional inputs: the time delay and the coupling strength. To utilize the MSF for determining the stability of AD of general networks for a chosen nonlinear system (node dynamics) and coupling function, we introduce the concept of master stability islands (MSIs), which are two-dimensional stability islands of the delay-coupling parameter space together with a third dimension ("altitude") encoding for eigenvalues that result in stable AD. We numerically compute the MSFs and visualize the corresponding MSIs for several common chaotic systems including the Rössler, the Lorenz, and Chen's system and find that it is generally possible to achieve AD and that a nonzero time delay is necessary for the stabilization of the AD states.

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.

Mobility and density induced amplitude death in metapopulation networks of coupled oscillators

We investigate the effects of mobility and density on the amplitude death of coupled Landau-Stuart oscillators and Brusselators in metapopulation networks, wherein each node represents a subpopulation occupied any number of mobile individuals. By numerical simulations in scale-free topology, we find that the systems undergo phase transitions from incoherent state to amplitude death, and then to frequency synchronization with increasing the mobility rate or density of oscillators. Especially, there exists an extent of intermediate mobility rate and density that can lead to global oscillator death. Furthermore, we show that such nontrivial phenomena are robust to diverse network topologies. Our findings may invoke further efforts and attentions to explore the underlying mechanism of collective behaviors in coupled metapopulation systems.

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.

Oscillation death in diffusively coupled oscillators by local repulsive link

A death of oscillation is reported in a network of coupled synchronized oscillators in the presence of additional repulsive coupling. The repulsive link evolves as an averaging effect of mutual interaction between two neighboring oscillators due to a local fault and the number of repulsive links grows in time when the death scenario emerges. Analytical condition for oscillation death is derived for two coupled Landau-Stuart systems. Numerical results also confirm oscillation death in chaotic systems such as a Sprott system and the Rössler oscillator. We explore the effect in large networks of globally coupled oscillators and find that the number of repulsive links is always fewer than the size of the network.

Amplitude death in nonlinear oscillators with mixed time-delayed coupling

Amplitude death (AD) is an emergent phenomenon whereby two or more autonomously oscillating systems completely lose their oscillations due to coupling. In this work, we study AD in nonlinear oscillators with mixed time-delayed coupling, which is a combination of instantaneous and time-delayed couplings. We find that the mixed time-delayed coupling favors the onset of AD for a larger set of parameters than in the limiting cases of purely instantaneous or completely time-delayed coupling. Coupled identical oscillators experience AD under instantaneous coupling mixed with a small proportion of time-delayed coupling. Our work gives a deeper understanding of delay-induced AD in coupled nonlinear oscillators.

Pinning noise-induced stochastic resonance

This paper proposes the concept of pinning noise and then investigates the phenomenon of stochastic resonance of coupled complex systems driven by pinning noise, where the noise has an α-stable distribution. Two kinds of pinning noise are taken into account: partial noise and switching noise. In particular, we establish a connection between switching noise and global noise when Gaussian noise is considered. It is shown that switching noise can not only achieve a stronger resonance effect, but it is also more robust to induce the resonance effect than partial noise.

Amplitude death phenomena in delay-coupled Hamiltonian systems

Hamiltonian systems, when coupled via time-delayed interactions, do not remain conservative. In the uncoupled system, the motion can typically be periodic, quasiperiodic, or chaotic. This changes drastically when delay coupling is introduced since now attractors can be created in the phase space. In particular, for sufficiently strong coupling there can be amplitude death (AD), namely, the stabilization of point attractors and the cessation of oscillatory motion. The approach to the state of AD or oscillation death is also accompanied by a phase flip in the transient dynamics. A discussion and analysis of the phenomenology is made through an application to the specific cases of harmonic as well as anharmonic coupled oscillators, in particular the Hénon-Heiles system.

Stabilizing oscillation death by multicomponent coupling with mismatched delays

The dynamics of a symmetric network of oscillators that are mutually coupled via multiple dynamical components with mismatched delays is studied. We find that networked oscillators experience oscillation death (OD) over a much larger domain of parameters when their different dynamical components are linked with mismatched delays than with only one delay. In particular, if the delays are mismatched by retaining a certain bias, OD is proved to be linearly stable even for very large delays for an arbitrary symmetric network. Further, we show that the minimal value of the intrinsic frequency necessary to induce OD decreases as the degree of mismatch in the coupling delays increases. The stabilizing effect of multicomponent coupling with mismatched delays is shown to be valid in networked chaotic oscillators also. The proposed coupling strategy can possibly be applied in controlling several pathological activities in neuronal systems and in engineering applications.