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For a system of globally pulse-coupled phase-oscillators, we derive conditions for stability of the completely synchronous state and all possible two-cluster states and explain how the different states are naturally connected via bifurcations. The coupling is modeled using the phase-response-curve (PRC), which measures the sensitivity of each oscillator's phase to perturbations. For large systems with a PRC, which turns to zero at the spiking threshold, we are able to find the parameter regions where multiple stable two-cluster states coexist and illustrate this by an example. In addition, we explain how a locally unstable one-cluster state may form an attractor together will its homoclinic connections. This leads to the phenomenon of intermittent, asymptotic synchronization with abating beats away from the perfect synchrony., 12 pages. 6 figures

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay., postprint version

We study locking of the modulation frequency of a relative periodic orbit in a general $S^1$-equivariant system of ordinary differential equations under an external forcing of modulated wave type. Our main result describes the shape of the locking region in the three-dimensional space of the forcing parameters: intensity, wave frequency, and modulation frequency. The difference of the wave frequencies of the relative periodic orbit and the forcing is assumed to be large and differences of modulation frequencies to be small. The intensity of the forcing is small in the generic case and can be large in the degenerate case, when the first order averaging vanishes. Applications are external electrical and/or optical forcing of selfpulsating states of lasers., Comment: 5 figures

We study chaotic synchronization in networks with time-delayed coupling. We introduce the notion of strong and weak chaos, distinguished by the scaling properties of the maximum Lyapunov exponent within the synchronization manifold for large delay times, and relate this to the condition for stable or unstable chaotic synchronization, respectively. In simulations of laser models and experiments with electronic circuits, we identify transitions from weak to strong and back to weak chaos upon monotonically increasing the coupling strength.

Stability of synchronization in delay-coupled networks of identical units generally depends in a complicated way on the coupling topology. We show that for large coupling delays synchronizability relates in a simple way to the spectral properties of the network topology. The master stability function used to determine stability of synchronous solutions has a universal structure in the limit of large delay: it is rotationally symmetric around the origin and increases monotonically with the radius in the complex plane. This allows a universal classification of networks with respect to their synchronization properties and solves the problem of complete synchronization in networks with strongly delayed coupling., Comment: 4 pages, 1 figure, 1 table. Phys. Rev. Lett. in print, v2 has minor changes in response to PRL referee comments

We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures, and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that, in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.

We describe the appearance and stability of spatiotemporal periodic patterns (rotating waves) in unidirectional rings of coupled oscillators with delayed couplings. We show how delays in the coupling lead to the splitting of each rotating wave into several new ones. The appearance of rotating waves is mediated by the Hopf bifurcations of the symmetric equilibrium. We also conclude that the coupling delays can be effectively replaced by increasing the number of oscillators in the chain. The phenomena are shown for the Stuart-Landau oscillators as well as for the coupled FitzHugh-Nagumo systems modeling an ensemble of spiking neurons interacting via excitatory chemical synapses.

Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be splitted into two parts: pseudo-continuous and strongly unstable. The pseudo-continuous part of the spectrum mediates destabilization of periodic solutions., Comment: 24 pages, 9 figures

We describe the mechanism of destabilization in a chain of identical coupled oscillators. Along with the transition from stationary to oscillatory behavior of the single oscillator, the network undergoes a complicated bifurcation scenario including the coexistence of multiple periodic orbits with different frequencies, spatial patterns, and modulation instabilities. This scenario, which is similar to the well-known Eckhaus scenario in spatially extended systems, occurs here also in the case of purely convective unidirectional coupling, and hence it cannot be explained as a simple discretization of its spatially continuous counterpart. Although the number of coexisting periodic orbits grows with the number of oscillators, we are able to treat this problem independently of the actual size of the network by investigating the limiting equations for the related spectral problems.

We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of Ginzburg-Landau type, which describes the destabilization of a uniform stationary state and close-by solutions in the limit of a large number of nodes. Studying numerically an example of unidirectionally coupled Duffing oscillators, we observe a coupling induced transition to collective spatio-temporal chaos, which can be understood using the derived amplitude equations.

We present an asymptotic analysis of time-delayed feedback control of steady states for large delay time. By scaling arguments, and a detailed comparison with exact solutions, we establish the parameter ranges for successful stabilization of an unstable fixed point of focus type. Insight into the control mechanism is gained by analyzing the eigenvalue spectrum, which consists of a pseudocontinuous spectrum and up to two strongly unstable eigenvalues. Although the standard control scheme generally fails for large delay, we find that if the uncontrolled system is sufficiently close to its instability threshold, control does work even for relatively large delay times.

The dynamical behavior of various physical and biological systems under the influence of delayed feedback or coupling can be modeled by including terms with delayed arguments in the equations of motion. In particular, the case of long delay may lead to complicated and high-dimensional dynamics. We investigate the effects of delay in systems that display an oscillatory instability (Hopf bifurcation) in the absence of delay. We show by analytical and numerical methods that the dynamical scenario includes the coexistence of multiple stable periodic solutions and can be described in terms of the Eckhaus instability, which is well known in the context of spatially extended systems.

The goal of this paper is twofold. In the first part we discuss a general approach to determine Lyapunov exponents from ensemble- rather than time-averages. The approach passes through the identification of locally stable and unstable manifolds (the Lyapunov vectors), thereby revealing an analogy with generalized synchronization. The method is then applied to a periodically forced chaotic oscillator to show that the modulus of the Lyapunov exponent associated to the phase dynamics increases quadratically with the coupling strength and it is therefore different from zero already below the onset of phase-synchronization. The analytical calculations are carried out for a model, the generalized special flow, that we construct as a simplified version of the periodically forced Rossler oscillator., Comment: Submitted to Physica D

We consider the dynamics of a linear array of coupled semiconductor lasers. Particular attention is paid to the synchronous states, which are caused by the permutation of two outer lasers. A system of three coupled lasers is studied in more details. We report different types of multistability of synchronous and asynchronous states including chaotic ones. We identify parameter values, for which a synchronous chaos can occur. Moreover, we show that transition to the synchronization occurs via blowup of the synchronous transversely unstable invariant set within the synchronization manifold. Finally, we present numerical analysis of larger arrays of coupled lasers and note some common qualitative features of the synchronization regions, which are independent of the number of lasers.

We study the dynamics of two mutually coupled chaotic oscillators with a time delayed coupling. Due to the delay, the allowed frequencies of the oscillators are shown to be discretized. The phenomenon is observed in the case when the delay is much larger than the characteristic period of the solitary uncoupled oscillator.

In this article we derive conditions for complete synchronization of two symmetrically coupled identical systems of ordinary differential equations and differential-delay equations. Using Lyapunov function approach we give an estimate of the region of attraction of the synchronized solution. We also established that complete synchronization is robust with respect to small perturbations of the identical systems.

We consider two single-mode semiconductor lasers, which are coupled face to face via the injection of the optical field. We describe the symmetries of the coupled rate-equations model, the associated stationary solutions (synchronous and antisynchronous), and the bifurcations between them for the case when the propagation delay of the injected field is small. For the coupled lasers with a detuning, we discuss the emergence of self-pulsations with specific properties (anti-phase, in-phase). We also identify parameter regions, where chaotic pulsations occur.

In this paper, we define a simple criterion of the synchronization threshold in the set of coupled chaotic systems (flows or maps ) with diagonal diffusive coupling. The condition of chaotic synchronization is determined only by two “parameters of order,” i.e., the largest Lyapunov exponent and the coupling coefficient. Our approach can be applied for both regular chaotic networks and arrays or lattices of chaotic oscillators with irregular, arbitrarily assumed structure of coupling. The main idea of the synchronization stability criterion is based on linear analysis of the ensembles of simplest dynamical systems. Numerical simulations confirm that such a linear approach approximates the synchronization threshold with high precision.

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained, providing a special kind of dynamical patterns called clusters. The simplest, coherent clusters arise when all oscillators display the same temporal behavior. Others, more complicated clusters are developed when population of the oscillators splits into subgroups such that all oscillators within a given group move in synchrony. Considering a system of mean-field coupled logistic maps, we study in details the transition from coherence to clustering and demonstrate that there are four different mechanisms of the desynchronization: riddling and blowout bifurcations, appearance of symmetric and asymmetric clusters. We also investigate the cluster-splitting bifurcation when the underlying dynamics is periodic. For the system of three and four coupled Rossler oscillators, we prove the existence of clusters and describe related bifurcations and in-cluster dynamics.

In weakly coupled systems, it is possible to observe the coexistence of the chaotic attractors which are located out of the invariant manifold and are not symmetrical in relation to this manifold. When the control parameter is changed, these attractors can undergo a chaos-hyperchaos transition. We give numerical evidence that before this transition the coexisting attractors merge together creating an attractor symmetrical with respect to the invariant manifold. We argue that the attractors that are not located at the invariant manifold can exhibit dynamical behavior similar to bubbling and on-off intermittency previously observed for the attractors located at the invariant manifold, and we describe the mechanism of these phenomena.

We study chaotic systems with multiple time delays that range over several orders of magnitude. We show that the spectrum of Lyapunov exponents (LEs) in such systems possesses a hierarchical structure, with different parts scaling with the different delays. This leads to different types of chaos, depending on the scaling of the maximal LE. Our results are relevant, in particular, for the synchronisation properties of hierarchical networks (networks of networks) where the nodes of subnetworks are coupled with shorter delays and couplings between different subnetworks are realised with longer delay times. Units within a subnetwork can synchronise if the maximal exponent scales with the shorter delay, long-range synchronisation between different subnetworks is only possible if the maximal exponent scales with the longer delay. The results are illustrated analytically for Bernoulli maps and numerically for tent maps and semiconductor lasers. © Copyright EPLA, 2013., TJ acknowledges support by FEDER (EU) under the project FISICOS (FIS2007-60327). SY acknowledges support by the German Research Foundation in the framework of the Collaborative Research Center SFB 910.

The correspondence between long-delayed systems and one-dimensional spatially extended media enables a direct interpretation of purely temporal phenomena in terms of spatio- temporal patterns. On the basis of this result, we provide the evidence of a characteristic spatio- temporal dynamics —coarsening— in a long-delayed bistable system. Nucleation, propagation and annihilation of fronts, leading eventually to a single phase, are observed in an experiment based on a laser with opto-electronic feedback. A numerical and analytical study of a general phenomenological model is also performed and compared with the experimental findings. Copyright c EPLA, 2012

We show that a ring of unidirectionally delay-coupled spiking neurons may possess a multitude of stable spiking patterns and provide a constructive algorithm for generating a desired spiking pattern. More specifically, for a given time-periodic pattern, in which each neuron fires once within the pattern period at a predefined time moment, we provide the coupling delays and/or coupling strengths leading to this particular pattern. The considered homogeneous networks demonstrate a great multistability of various travelling time- and space-periodic waves which can propagate either along the direction of coupling or in opposite direction. Such a multistability significantly enhances the variability of possible spatio-temporal patterns and potentially increases the coding capability of oscillatory neuronal loops. We illustrate our results using FitzHugh-Nagumo neurons interacting via excitatory chemical synapses as well as limit-cycle oscillators.

Mode-locked semiconductor lasers are widely used for generation of short optical pulses with high repetition rates and optical frequency combs suitable for numerous practical applications. By combining many lasers into an array one can achieve much larger output power and substantially improve the characteristics of the output radiation. In this presentation we study dynamical regimes of operation in an array of mode-locked lasers locally coupled in a ring geometry. We demonstrate that unlike a solitary mode-locked laser emitting a sequence of equidistant pulses with the pulse repetition frequency close to the inverse cavity round trip time, an array of mode-locked lasers can radiate a periodic sequence of clusters of fundamental mode-locked pulses. This regime associated with the formation of closely packed bound states of coupled mode-locked pulses due to a balance between attraction and repulsion is very different from the standard harmonic mode-locked regime where the pulses always repel each other.

Summary form only given. Cavity solitons are localized spots of light in the transverse section of passive and active optical devices: broad area lasers and semiconductor cavities with external coherent pumping. Under certain conditions a drift instability can appear in these devices leading to a transverse motion of cavity solitons. Such motion in distributed dynamical systems of different nature can be induced by various effects, e.g., walk-off, convection, phase gradient, vorticity, finite carrier relaxation times, the so-called Ising-Bloch transition, symmetry breaking due to off-axis feedback or resonator detuning. Recently it was shown within the framework of the Swift-Hohenberg equation [1] that a drift instability leading to a spontaneous motion of localized structures in arbitrary direction can be induced by a delayed feedback term. More recently the appearance of nontrivial instabilities resulting in the formation of oscillons, soliton rings, labyrinth patterns, or moving structures was demonstrated in this system [2].First, we study the effect of delayed feedback on the mobility properties of transverse cavity solitons in a broad area semiconductor microcavity. We present analytical and numerical analysis of the dependence of the drift instability threshold and on the feedback strength, feedback phase, and carrier relaxation time. In particular we demonstrate that due to finite carrier relaxation rate the delay induced drift instability can be suppressed to a certain extent. We give analytical estimation of the soliton velocity near the drift instability point which is in a good agreement with numerical results obtained using the full model equations. Next, the effect of delayed optical feedback on the dynamics of cavity solitons in a broad area semiconductor laser with a saturable absorber is investigated theoretically. It is shown analytically that the threshold of delay induced drift instability can be considerably reduced in a laser with broad spectral bandwidth of the gain medium. Furthermore, it is demonstrated that depending on the feedback phase it is possible not only to destabilize cavity solitons via the delay induced drift instability but also to suppress another drift instability [3] related to the finite relaxation rates of the gain and absorber sections.