Staff member

We numerically study a network of two identical populations of identical real-valued quadratic maps. Upon variation of the coupling strengths within and across populations, the network exhibits a rich variety of distinct dynamics. The maps in individual populations can be synchronized or desynchronized. Their temporal evolution can be periodic or aperiodic. Furthermore, one can find blends of synchronized with desynchronized states and periodic with aperiodic motions. We show symmetric patterns for which both populations have the same type of dynamics as well as chimera states of a broken symmetry. The network can furthermore show multistability by settling to distinct dynamics for different realizations of random initial conditions or by switching intermittently between distinct dynamics for the same realization. We conclude that our system of two populations of a particularly simple map is the most simple system that can show this highly diverse and complex behavior, which includes but is not limited to chimera states. As an outlook to future studies, we explore the stability of two populations of quadratic maps with a complex-valued control parameter. We show that bounded and diverging dynamics are separated by fractal boundaries in the complex plane of this control parameter.

JTD

We propose a method to control chimera states in a ring-shaped network of nonlocally coupled phase oscillators. This method acts exclusively on the network’s connectivity. Using the idea of a pacemaker oscillator, we investigate which is the minimal action needed to control chimeras. We implement the pacemaker choosing one oscillator and making its links unidirectional. Our results show that a pacemaker induces chimeras for parameters and initial conditions for which they do not form spontaneously. Furthermore, the pacemaker attracts the incoherent part of the chimera state, thus controlling its position. Beyond that, we find that these control effects can be achieved with modifications of the network’s connectivity that are less invasive than a pacemaker, namely, the minimal action of just modifying the strength of one connection allows one to control chimeras.

JTD Keywords: Complex networks, Oscillators, Spatiotemporal phenomena

We study two-layer networks of identical phase oscillators. Each individual layer is a ring network for which a non-local intra-layer coupling leads to the formation of a chimera state. The number of oscillators and their natural frequencies is in general different across the layers. We couple the phases of individual oscillators in one layer to the phase of the mean field of the other layer. This coupling from the mean field to individual oscillators is done in both directions. For a sufficient strength of this interlayer coupling, the phases of the mean fields lock across the two layers. In contrast, both layers continue to exhibit chimera states with no locking between the phases of individual oscillators across layers, and the two mean field amplitudes remain uncorrelated. Hence, the networks’ mean fields show phase synchronization which is analogous to the one between low-dimensional chaotic oscillators. The required coupling strength to achieve this mean field phase synchronization increases with the mismatches in the network sizes and the oscillators’ natural frequencies.

JTD Keywords: Chaos, Complex networks, Oscillators, Synchronisation

Networks of coupled oscillators in chimera states are characterized by an intriguing interplay of synchronous and asynchronous motion. While chimera states were initially discovered in mathematical model systems, there is growing experimental and conceptual evidence that they manifest themselves also in natural and man-made networks. In real-world systems, however, synchronization and desynchronization are not only important within individual networks but also across different interacting networks. It is therefore essential to investigate if chimera states can be synchronized across networks. To address this open problem, we use the classical setting of ring networks of non-locally coupled identical phase oscillators. We apply diffusive drive-response couplings between pairs of such networks that individually show chimera states when there is no coupling between them. The drive and response networks are either identical or they differ by a variable mismatch in their phase lag parameters. In both cases, already for weak couplings, the coherent domain of the response network aligns its position to the one of the driver networks. For identical networks, a sufficiently strong coupling leads to identical synchronization between the drive and response. For non-identical networks, we use the auxiliary system approach to demonstrate that generalized synchronization is established instead. In this case, the response network continues to show a chimera dynamics which however remains distinct from the one of the driver. Hence, segregated synchronized and desynchronized domains in individual networks congregate in generalized synchronization across networks.

JTD Keywords: Oscillators, Synchronisation