Controlling synchrony by delay coupling in networks: from in-phase to splay and cluster states

We study synchronization in delay-coupled oscillator networks, using a master stability function approach. Within a generic model of Stuart-Landau oscillators (normal form of super- or subcritical Hopf bifurcation) we derive analytical stability conditions and demonstrate that by tuning the coupling phase one can easily control the stability of synchronous periodic states. We propose the coupling phase as a crucial control parameter to switch between in-phase synchronization or desynchronization for general network topologies, or between in-phase, cluster, or splay states in unidirectional rings. Our results are robust even for slightly nonidentical elements of the network.Comment: 4 pages, 4 figure

Chaos suppression in the parametrically driven Lorenz system

We predict theoretically and verify experimentally the suppression of chaos in the Lorenz system driven by a high-frequency periodic or stochastic parametric force. We derive the theoretical criteria for chaos suppression and verify that they are in a good agreement with the results of numerical simulations and the experimental data obtained for an analog electronic circuit

Phase transition to chimera state in two populations of oscillators interacting via a common external environment

We consider two populations of coupled oscillators, interacting with each other through a common external environment. The external environment is synthesized by the contributions from all oscillators of both populations. Such indirect coupling via an external medium arises naturally in many fields, e.g., dynamical quorum sensing in coupled biological and chemical systems. We analyze the existence and stability of a variety of stationary states on the basis of the Ott-Antonsen reduction method, which reveals that the interaction via an external environment gives rise to unusual collective behaviors such as the uniform drifting, non-uniform drifting and chimera states. We present a complete bifurcation diagram, which provides the underlying mechanism of the phase transition towards chimera state with the route of incoherence ${}\rightarrow{}$ uniform drift ${}\rightarrow{}$ non-uniform drift ${}\rightarrow{}$ chimera

Stabilization of periodic orbits near a subcritical Hopf bifurcation in delay-coupled networks

We study networks of delay-coupled oscillators with the aim to extend time-delayed feedback control to networks. We show that unstable periodic orbits of a network can be stabilized by a noninvasive, delayed coupling. We state criteria for stabilizing the orbits by delay-coupling in networks and apply these to the case where the local dynamics is close to a subcritical Hopf bifurcation, which is representative of systems with torsion-free unstable periodic orbits. Using the multiple scale method and the master stability function approach, the network system is reduced to the normal form, and the characteristic equations for Floquet exponents are derived in an analytical form, which reveals the coupling parameters for successful stabilization. Finally, we illustrate the results by numerical simulations of the Lorenz system close to a subcritical Hopf bifurcation. The unstable periodic orbits in this system have no torsion, and hence cannot be stabilized by the conventional time delayed-feedback technique.CUC acknowledges support from TWAS with the code-number 09-138 RG/PHYS/AS_SI. PH acknowledges support by the BMBF under the grant no. 01GQ1001B (Förderkennzeichen). VF and PH acknowledge financial support from the German Academic Exchange Service (DAAD). This work was also supported by DFG in the framework of SFB 910.Peer Reviewe

Simultaneous stabilization of periodic orbits and fixed points in delay-coupled Lorenz systems

We study two delay-coupled Lorenz systems and demonstrate unified chaos control by noninvasive time- delayed coupling. Both an unstable periodic orbit and an unstable fixed point of the system can be stabilized close to a subcritical Hopf bifurcation. Using a multiple scales method, the systems are reduced to Hopf normal forms, and an analytical approach for stabilizing a periodic orbit as well as a fixed point of the system is developed. As a result, the equations for the characteristic exponents are derived in an analytical form, re- vealing the range of coupling parameters for successful stabilization. Finally, we illustrate the results with numerical simulations, which show good agreement with the theory.Peer reviewe