Critical Switching in Globally Attractive Chimeras

We report on a new type of chimera state that attracts almost all initial conditions and exhibits power-law switching behavior in networks of coupled oscillators. Such switching chimeras consist of two symmetric configurations, which we refer to as subchimeras, in which one cluster is synchronized and the other is incoherent. Despite each subchimera being linearly stable, switching chimeras are extremely sensitive to noise: arbitrarily small noise triggers and sustains persistent switching between the two symmetric subchimeras. The average switching frequency scales as a power law with the noise intensity, which is in contrast with the exponential scaling observed in typical stochastic transitions. Rigorous numerical analysis reveals that the power-law switching behavior originates from intermingled basins of attraction associated with the two subchimeras, which in turn are induced by chaos and symmetry in the system. The theoretical results are supported by experiments on coupled optoelectronic oscillators, which demonstrate the generality and robustness of switching chimeras

Parameter Identification and Hybrid Synchronization in an Array of Coupled Chaotic Systems with Ring Connection: An Adaptive Integral Sliding Mode Approach

This article presents an adaptive integral sliding mode control (SMC) design method for parameter identification and hybrid synchronization of chaotic systems connected in ring topology. To employ the adaptive integral sliding mode control, the error system is transformed into a special structure containing nominal part and some unknown terms. The unknown terms are computed adaptively. Then the error system is stabilized using integral sliding mode control. The controller of the error system is created that contains both the nominal control and the compensator control. The adapted laws and compensator controller are derived using Lyapunov stability theory. The effectiveness of the proposed technique is validated through numerical examples

Synchronization of Coupled and Periodically Forced Chemical Oscillators

Physiological rhythms are essential in all living organisms. Such rhythms are regulated through the interactions of many cells. Deviation of a biological system from its normal rhythms can lead to physiological maladies. The tremor and symptoms associated with Parkinson\u27s disease are thought to emerge from abnormal synchrony of neuronal activity within the neural network of the brain. Deep brain stimulation is a therapeutic technique that can remove this pathological synchronization by the application of a periodic desynchronizing signal. Herein, we used the photosensitive Belousov--Zhabotinsky (BZ) chemical reaction to test the mechanism of deep brain stimulation. A collection of oscillators are initially synchronized using a regular light signal. Desynchronization is then attempted using an appropriately chosen desynchronizing signal based on information found in the phase response curve.;Coupled oscillators in various network topologies form the most common prototypical systems for studying networks of dynamical elements. In the present study, we couple discrete BZ photochemical oscillators in a network configuration. Different behaviors are observed on varying the coupling strength and the frequency heterogeneity, including incoherent oscillations to partial and full frequency entrainment. Phase clusters are organized symmetrically or non-symmetrically in phase-lag synchronization structures, a novel phase wave entrainment behavior in non-continuous media. The behavior is observed over a range of moderate coupling strengths and a broad frequency distribution of the oscillators

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

Synchronous chaos and broad band gamma rhythm in a minimal multi-layer model of primary visual cortex

Visually induced neuronal activity in V1 displays a marked gamma-band component which is modulated by stimulus properties. It has been argued that synchronized oscillations contribute to these gamma-band activity [... however,] even when oscillations are observed, they undergo temporal decorrelation over very few cycles. This is not easily accounted for in previous network modeling of gamma oscillations. We argue here that interactions between cortical layers can be responsible for this fast decorrelation. We study a model of a V1 hypercolumn, embedding a simplified description of the multi-layered structure of the cortex. When the stimulus contrast is low, the induced activity is only weakly synchronous and the network resonates transiently without developing collective oscillations. When the contrast is high, on the other hand, the induced activity undergoes synchronous oscillations with an irregular spatiotemporal structure expressing a synchronous chaotic state. As a consequence the population activity undergoes fast temporal decorrelation, with concomitant rapid damping of the oscillations in LFPs autocorrelograms and peak broadening in LFPs power spectra. [...] Finally, we argue that the mechanism underlying the emergence of synchronous chaos in our model is in fact very general. It stems from the fact that gamma oscillations induced by local delayed inhibition tend to develop chaos when coupled by sufficiently strong excitation.Comment: 49 pages, 11 figures, 7 table

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com

Festschrift on the occasion of Ulrike Feudel’s 60th birthday

Peer reviewedPostprin

Dynamische Eigenschaften gekoppelter chaotischer Abbildungen auf Netzwerken

The main topic of this thesis is the investigation of dynamical properties of coupled Tchebycheff map networks. At every node of the network the dynamics is given by the iteration of a Tchebycheff map, which shows strongest possible chaotic behaviour. By applying a coupling between the various individual dynamics along the links of the network, a rich structure of complex dynamical patterns emerges. Accordingly, coupled chaotic map networks provide prototypical models for studying the interplay between local dynamics, network structure, and the emergent global dynamics. An exciting application of coupled Tchebycheff map lattices in quantum field theory has been proposed Beck in Spatio-temporal chaos and vacuum fluctuations of quantized fields' (2002). In this so-called chaotic string model, the coupled map lattice dynamics generates the noise needed for the Parisi-Wu approach of stochastic quantization. The remarkable obversation is that the respective dynamics seems to reproduce distinguished numerical values of coupling constants that coincide with those observed in the standard model of particle physic. The results of this thesis give insights into the chaotic string model and its network generalization from a dynamical point of view. This leads to a deeper understanding of the dynamics, which is essential for a critical discussion of possible physical embeddings. Apart from this specific application to particle physics, the investigated concepts like synchronization or a most random behaviour of the dynamics are of general interest for dynamical system theory and the science of complex networks. As a first approach, discrete symmetry transformations of the model are studied. These transformations are formulated in a general way in order to be also applicable to similar dynamics on bipartite network structures. An observable of main interest in the chaotic string model is the interaction energy. In Spatio-temporal chaos and vacuum fluctuations of quantized fields' (2002) it has been observed that certain chaotic string couplings, corresponding to a vanishing interaction energy, coincide with coupling constants of the standard model of elementary particle physics. Since the interaction energy is basically a spatial correlation measure, an interpretation of the respective dynamical states in terms of a most random behaviour is tempting. In order to distinguish certain states as most random', or evoke another dynamical principle, a deeper understanding of the dynamics essential. In the present thesis the dynamics is studied numerically via Lyapunov measures, spatial correlations, and ergodic properties. It is shown that the zeros of the interaction energy are distinguished only with respect to this specific observable, but not by a more general dynamical principle. The original chaotic string model is defined on a one-dimensional lattice (ring-network) as the underlying network topology. This thesis studies a modification of the model based on the introduction of tunable disorder. The effects of inhomogeneous coupling weights as well as small-world perturbations of the ring-network structure on the interaction energy are discussed. Synchronization properties of the chaotic string model and its network generalization are studied in later chapters of this thesis. The analysis is based on the master stability formalism, which relates the stability of the synchronized state to the spectral properties of the network. Apart from complete synchronization, where the dynamics at all nodes of the network coincide, also two-cluster synchronization on bipartite networks is studied. For both types of synchronization it is shown that depending on the type of coupling the synchronized dynamics can display chaotic as well as periodic or quasi-periodic behaviour. The semi-analytical calculations reveal that the respective synchronized states are often stable for a wide range of coupling values even for the ring-network, although the respective basins of attraction may inhabit only a small fraction of the phase space. To provide analytical results in closed form, for complete synchronization the stability of all fixed points and period-2 orbits of all chaotic string networks are determined analytically. The master stability formalism allows to treat the ring-network of the chaotic string model as a special case, but the results are valid for coupled Tchebycheff maps on arbitrary networks. For two-cluster synchronization on bipartite networks, selected fixed points and period-2 orbits are analyzed.In der vorliegenden Dissertation werden dynamische Eigenschaften von gekoppelten chaotischen Abbildungen auf Netzwerken untersucht. Eine besondere Beachtung findet dabei die Untersuchung in Hinblick auf die Anwendung dieser dynamischen Systeme als Modelle für Vakuumfluktuationen in stochastisch quantisierten Feldtheorien. Gekoppelte chaotische Abbildungen als Modelle für raum-zeitliches Chaos, Strukturbildung und andere emergente Phänomene stellen bereits unabhängig von möglichen konkreten Anwendungen ein intensiv studiertes Forschungsgebiet dar. Ist die Topologie der Kopplungsstruktur durch ein komplexes Netzwerk gegeben, so lässt sich anhand dieser Systeme exemplarisch die Verbindung zwischen Netzwerkstruktur und globaler Dynamik studieren. In einer von Beck in Spatio-temporal chaos and vacuum fluctuations of quantized fields' (2002) vorgeschlagenen Anwendung gekoppelter chaotischer Tchebycheff-Abbildungen auf Ring-Netzwerken dient die raum-zeitlich chaotische Dynamik als stochastisches Feld für die stochastische Quantisierung von Feldtheorien. Bemerkenswerterweise zeichnen sich in diesem als Chaotic-Strings' bezeichneten Modell bestimmte numerische Werte der Kopplungsstärke der Dynamik aus, die mit Kopplungskonstanten des Standardmodells der Elementarteilchenphysik assoziiert werden können. In der vorliegenden Arbeit werden die im genannten Modell betrachteten Dynamiken unter unterschiedlichen Gesichtspunkten untersucht, einschließlich der Berücksichtigung der Bedeutung dieser Ergebnisse für eine mögliche physikalische Interpretation. Zuerst erfolgt eine ausführliche Untersuchung diskreter Symmetrien in Systemen gekoppelter chaotischer Abbildungen auf bipartiten Netzwerken. Mit Hilfe der erhaltenen Ergebnisse wird die Äquivalenz scheinbar unterschiedlicher Dynamiken aufgezeigt. Eine der grundlegenden Observablen des Chaotic-String-Modells ist die Wechselwirkungsenergie. Von besonderem Interesse sind bestimmte zu einer verschwindenen Wechselwirkungsenergie führende Kopplungswerte der chaotischen Dynamik, da für diese eine numerische Übereinstimmung mit Kopplungskonstanten des Standardmodells beobachtet wird. Da die Wechselwirkungsenergie im Wesentlichen ein räumliches Korrelationsmaß für die Dynamik auf den Knoten des Rings darstellt, wurde im Modell bisher eine Interpretation der entsprechenden dynamischen Zustände als maximal chaotisch' oder maximal stochastisch' nahegelegt. In der vorliegenden Arbeit wird die Hypothese eines solchen maximal stochastischen Verhaltens der Dynamiken numerisch anhand von Lyapunovmaßen, räumlichen Korrelationen sowie Ergodizitätseigenschaften untersucht. Es wird gezeigt, dass die mit Kopplungskonstanten des Standardmodells assoziierten Nullstellen der Wechselwirkungsenergie nur durch diese Observable, jedoch nicht durch ein allgemeineres dynamisches Prinzip ausgezeichnet sind. Es wird weiterhin die Auswirkung unterschiedlicher Formen von Unordnung auf die Dynamik betrachtet. Zum einen wird diese Unordnung mittels des Übergangs von einer regulären Ringstruktur zu allgemeinen Netzwerkstrukturen wie z.B. Small-World' Netzwerken eingeführt. Zum anderen wird anstatt einer homogenen eine heterogene, d.h. ungeordnete Verteilung von Kopplungswerten zugrunde gelegt. Auch hier erfolgt die Anwendung auf die im Chaotic-String-Modell betrachteten dynamischen Systeme. Während die Untersuchung von Korrelationen sowie die Auswirkung von Unordnung auf die Dynamiken fast ausschließlich numerisch möglich ist, kann die Betrachtung von Synchronisationeigenschaften gekoppelter chaotischer Abbildungen auch analytisch erfolgen. Die entsprechenden in der Arbeit vorgestellten Rechnungen basieren auf dem sogenannten Master-Stability-Formalismus', mit dessen Hilfe die Stabilität des synchronisierten Zustands zu den spektralen Eigenschaften des Netzwerks in Beziehung gesetzt werden kann. Die vorliegende Arbeit behandelt zuerst vollständige Synchronisation, das heißt, alle Knoten des Netzwerks zeigen exakt identisches Verhalten. Zwei-Gruppen-Synchronisation auf bipartiten Netzwerken wird daran anschließend untersucht. Es wird beobachtet, dass für gekoppelte Tchebycheff-Abbildungen die synchronisierte Dynamik für vollständige wie auch für Zwei-Gruppen-Synchronisation sowohl chaotisches, wie auch periodisches Verhalten oder sogar einen stabilen Fixpunkt aufweisen kann. Semi-analytische Berechnungen zeigen, dass für einen großen Bereich der Kopplungswerte die Synchronisation einen stabilen Zustand darstellt. Neben diesen semi-analytischen Berechnungen werden auch analytische Resultate für vollständig sowie zwei-Gruppen-synchronisierte Fixpunkte und Periode-2-Orbits hergeleitet. Aufgrund der Verwendung des Master-Stability-Formalismus können die Ergebnisse auf die im Rahmen des Chaotic-String-Modells betrachteten Ring-Netzwerke angewendet werden, sind aber allgemein für beliebige (für Zwei-Gruppen-Synchronisation bipartite) Netzwerkstrukturen gültig