Abstract:
Dynamical systems with complex delayed interactions arise commonly when propagation times are significant, yielding complicated oscillatory instabilities. In this Letter, we introduce a class of systems with multiple, hierarchically long time delays, and using a suitable space-time representation we uncover features otherwise hidden in their temporal dynamics. The behaviour in the case of two delays is shown to ''encode'' two-dimensional spiral defects and defects turbulence. A multiple scale analysis sets the equivalence to a complex Ginzburg-Landau equation, and a novel criterium for the attainment of the long-delay regime is introduced. We also demonstrate this phenomenon for a semiconductor laser with two delayed optical feedbacks.

Abstract:
We study properties of basic solutions in systems with dime delays and $S^1$-symmetry. Such basic solutions are relative equilibria (CW solutions) and relative periodic solutions (MW solutions). It follows from the previous theory that the number of CW solutions grows generically linearly with time delay $\tau$. Here we show, in particular, that the number of relative periodic solutions grows generically as $\tau^2$ when delay increases. Thus, in such systems, the relative periodic solutions are more abundant than relative equilibria. The results are directly applicable to, e.g., Lang-Kobayashi model for the lasers with delayed feedback. We also study stability properties of the solutions for large delays.

Abstract:
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.

Abstract:
The subject of this paper is a system of phase-oscillators, which are globally pulse-coupled via excitatory interaction. The appearance and stability of one- and two-cluster-states is investigated for a family of unimodal phase-response-curves (PRC). The PRCs and their derivatives are assumed to be zero at the spiking point. We show that there exist stable homoclinic connections of the one-cluster state for PRCs with the maximum located shortly before the spiking point and coexisting stable two-clusters states when the maximum of the PRC is located shortly after the spike.

Abstract:
We show how a variety of stable spatio-temporal periodic patterns can be created in 2D-lattices of coupled oscillators with non-homogeneous coupling delays. A "hybrid dispersion relation" is introduced, which allows studying the stability of time-periodic patterns analytically in the limit of large delay. The results are illustrated using the FitzHugh-Nagumo coupled neurons as well as coupled limit cycle (Stuart-Landau) oscillators.

Abstract:
We show that a ring of phase oscillators coupled with transmission delays can be used as a pattern recognition system. The introduced model encodes patterns as stable periodic orbits. We present a detailed analysis of the underlying dynamics. In particular, we show that the system possesses a multitude of periodic solutions, prove stability results and present a bifurcation analysis. Furthermore, we show successful recognition results using artificial patterns and speech recordings.

Abstract:
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.

Abstract:
We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.

Abstract:
We investigate the dynamics of the pendulum suspended on the forced Duffing oscillator. The detailed bifurcation analysis in two parameter space (amplitude and frequency of excitation) which presents both oscillating and rotating periodic solutions of the pendulum has been performed. We identify the areas with low number of coexisting attractors in the parameter space as the coexistence of different attractors has a significant impact on the practical usage of the proposed system as a tuned mass absorber.

Abstract:
Delayed interactions are a common property of coupled natural systems and therefore arise in a variety of different applications. For instance, signals in neural or laser networks propagate at finite speed giving rise to delayed connections. Such systems are often modeled by delay differential equations with discrete delays. In realistic situations, these delays are not identical on different connections. We show that by a componentwise timeshift transformation it is often possible to reduce the number of different delays and simplify the models without loss of information. We identify dynamic invariants of this transformation, determine its capabilities to reduce the number of delays and interpret these findings in terms of the topology of the underlying graph. In particular, we show that networks with identical sums of delay times along the fundamental semicycles are dynamically equivalent and we provide a normal form for these systems. We illustrate the theory using a network motif of coupled Mackey-Glass systems with 8 different time delays, which can be reduced to an equivalent motif with three delays.