Stochastic synchronization in blinking networks of chaotic maps

Multistable ghost attractors in a switching laser system

This paper studies the effects of a switching parameter on the dynamics of a multistable laser model. The laser model represents multistability in distinct ranges of parameters. We assume that the system's parameter switches periodically between different values. Since the system is multistable, the presence of a ghost attractor is also dependent on the initial condition. It is shown that when the composing subsystems are chaotic, a periodic ghost attractor can emerge and vice versa, depending on the initial conditions. In contrast to the previous studies in which the attractor of the fast blinking systems approximates the average attractor, here, the blinking attractor differs from the average in some cases. It is shown that when the switching parameter values are distant from their average, the blinking and the average attractors are different, and as they approach, the blinking attractor approaches the average attractor too.

Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications

Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic "Bristol book" [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.

Blinking coupling enhances network synchronization

This paper studies the synchronization of a network with linear diffusive coupling, which blinks between the variables periodically. The synchronization of the blinking network in the case of sufficiently fast blinking is analyzed by showing that the stability of the synchronous solution depends only on the averaged coupling and not on the instantaneous coupling. To illustrate the effect of the blinking period on the network synchronization, the Hindmarsh-Rose model is used as the dynamics of nodes. The synchronization is investigated by considering constant single-variable coupling, averaged coupling, and blinking coupling through a linear stability analysis. It is observed that by decreasing the blinking period, the required coupling strength for synchrony is reduced. It equals that of the averaged coupling model times the number of variables. However, in the averaged coupling, all variables participate in the coupling, while in the blinking model only one variable is coupled at any time. Therefore, the blinking coupling leads to an enhanced synchronization in comparison with the single-variable coupling. Numerical simulations of the average synchronization error of the network confirm the results obtained from the linear stability analysis.

Stochastic master stability function for noisy complex networks

In this paper, we broaden the master stability function approach to study the stability of the synchronization manifold in complex networks of stochastic dynamical systems. We provide necessary and sufficient conditions for exponential stability that allow us to discriminate the impact of noise. We observe that noise can be beneficial for synchronization when it diffuses evenly in the network. On the contrary, an excessively large amount of noise only acting on a subset of the node state variables might have disruptive effects on the network synchronizability. To demonstrate our findings, we complement our theoretical derivations with extensive simulations on paradigmatic examples of networks of noisy systems.

Random temporal connections promote network synchronization

We report a phenomenon of collective dynamics on discrete-time complex networks: a random temporal interaction matrix even of zero or/and small average is able to significantly enhance synchronization with probability one. According to current knowledge, there is no verifiably sufficient criterion for the phenomenon. We use the standard method of synchronization analytics and the theory of stochastic processes to establish a criterion, by which we rigorously and accurately depict how synchronization occurring with probability one is affected by the statistical characteristics of the random temporal connections such as the strength and topology of the connections as well as their probability distributions. We also illustrate the enhancement phenomenon using physical and biological complex dynamical networks.

Network synchronization with periodic coupling

The synchronization behavior of networked chaotic oscillators with periodic coupling is investigated. It is observed in simulations that the network synchronizability could be significantly influenced by tuning the coupling frequency, even making the network alternating between the synchronous and nonsynchronous states. Using the master stability function method, we conduct a detailed analysis of the influence of coupling frequency on network synchronizability and find that the network synchronizability is maximized at some characteristic frequencies comparable to the intrinsic frequency of the local dynamics. Moreover, it is found that as the amplitude of the coupling increases, the characteristic frequencies are gradually decreased. Using the finite-time Lyapunov exponent technique, we investigate further the mechanism for the maximized synchronizability and find that at the characteristic frequencies the power spectrum of the finite-time Lyapunov exponent is abruptly changed from the localized to broad distributions. When this feature is absent or not prominent, the network synchronizability is less influenced by the periodic coupling. Our study shows the efficiency of finite-time Lyapunov exponent in exploring the synchronization behavior of temporally coupled oscillators and sheds lights on the interplay between the system dynamics and structure in general temporal networks.

Leader-follower consensus and synchronization in numerosity-constrained networks with dynamic leadership

In this work, we study leader-follower consensus and synchronization protocols over a stochastically switching network. The agents representing the followers can communicate with any other agent, whereas the agents serving as leaders are restricted to interact only with the other leaders. The model incorporates the phenomenon of numerosity, which limits the perceptual capacity of the agents while allowing for shuffling with whom each individual interacts at each time step. We derive closed form expressions for necessary and sufficient conditions for consensus, the rate of convergence to consensus, and conditions for stochastic synchronization in terms of the asymptotic convergence factor. We provide simulation results to validate the theoretical findings and to illustrate the dependence of this factor on system parameters. The closed form results enable us to study the factors affecting the feasibility of consensus. We show that agents' traits can be chosen for an engineered system to maximize the convergence speed and that protocol speed is enhanced as the proportion of the leaders increases in certain cases. These results may find application in the design and control of an engineered leader-follower system, where consensus or synchronization at the fastest possible rate is desired.

Synchronization in slowly switching networks of coupled oscillators

Networks whose structure of connections evolves in time constitute a big challenge in the study of synchronization, in particular when the time scales for the evolution of the graph topology are comparable with (or even longer than) those pertinent to the units’ dynamics. We here focus on networks with a slow-switching structure, and show that the necessary conditions for synchronization, i.e. the conditions for which synchronization is locally stable, are determined by the time average of the largest Lyapunov exponents of transverse modes of the switching topologies. Comparison between fast- and slow-switching networks allows elucidating that slow-switching processes prompt synchronization in the cases where the Master Stability Function is concave, whereas fast-switching schemes facilitate synchronization for convex curves. Moreover, the condition of slow-switching enables the introduction of a control strategy for inducing synchronization in networks with arbitrary structure and coupling strength, which is of evident relevance for broad applications in real world systems.

Chimera states in time-varying complex networks

Chimera states have been recently found in a variety of different coupling schemes and geometries. In most cases, the underlying coupling structure is considered to be static, while many realistic systems display significant temporal changes in the pattern of connectivity. In this work we investigate a time-varying network made of two coupled populations of Kuramoto oscillators, where the links between the two groups are considered to vary over time. As a main result we find that the network may support stable, breathing, and alternating chimera states. We also find that, when the rate of connectivity changes is fast, compared to the oscillator dynamics, the network may be described by a low-dimensional system of equations. Unlike in the static heterogeneous case, the onset of alternating chimera states is due to the presence of fluctuations, which may be induced either by the finite size of the network or by large switching times.

Local and global epidemic outbreaks in populations moving in inhomogeneous environments

We study disease spreading in a system of agents moving in a space where the force of infection is not homogeneous. Agents are random walkers that additionally execute long-distance jumps, and the plane in which they move is divided into two regions where the force of infection takes different values. We show the onset of a local epidemic threshold and a global one and explain them in terms of mean-field approximations. We also elucidate the critical role of the agent velocity, jump probability, and density parameters in achieving the conditions for local and global outbreaks. Finally, we show that the results are independent of the specific microscopic rules adopted for agent motion, since a similar behavior is also observed for the distribution of agent velocity based on a truncated power law, which is a model often used to fit real data on motion patterns of animals and humans.

Criteria for stochastic pinning control of networks of chaotic maps

This paper investigates the controllability of discrete-time networks of coupled chaotic maps through stochastic pinning. In this control scheme, the network dynamics are steered towards a desired trajectory through a feedback control input that is applied stochastically to the network nodes. The network controllability is studied by analyzing the local mean square stability of the error dynamics with respect to the desired trajectory. Through the analysis of the spectral properties of salient matrices, a toolbox of conditions for controllability are obtained, in terms of the dynamics of the individual maps, algebraic properties of the network, and the probability distribution of the pinning control. We demonstrate the use of these conditions in the design of a stochastic pinning control strategy for networks of Chirikov standard maps. To elucidate the applicability of the approach, we consider different network topologies and compare five different stochastic pinning strategies through extensive numerical simulations.