No Arabic abstract
Coupling of chaotic oscillators has evidenced conditions where synchronization is possible, therefore a nonlinear system can be driven to a particular state through input from a similar oscillator. Here we expand this concept of control of the state of a nonlinear system by showing that it is possible to induce it to follow a textit{linear} superposition of signals from multiple equivalent systems, using only partial information from them, through one- or more variable-signal. Moreover, we show that the larger the number of trajectories added to the input signal, the better the convergence of the system trajectory to the sum input.
We consider an approach to the analysis of nonstationary processes based on the application of wavelet basis sets constructed using segments of the analyzed time series. The proposed method is applied to the analysis of time series generated by a nonlinear system with and without noise
We study chaotic systems with multiple time delays that range over several orders of magnitude. We show that the spectrum of Lyapunov exponents (LE) in such systems possesses a hierarchical structure, with different parts scaling with the different delays. This leads to different types of chaos, depending on the scaling of the maximal LE. Our results are relevant, in particular, for the synchronization properties of hierarchical networks (networks of networks) where the nodes of subnetworks are coupled with shorter delays and couplings between different subnetworks are realized with longer delay times. Units within a subnetwork can synchronize if the maximal exponent scales with the shorter delay, long range synchronization between different subnetworks is only possible if the maximal exponent scales with the long delay. The results are illustrated analytically for Bernoulli maps and numerically for tent maps.
The paper investigates a new hybrid synchronization called modified hybrid synchronization (MHS) via the active control technique. Using the active control technique, stable controllers which enable the realization of the coexistence of complete synchronization, anti-synchronization and project synchronization in four identical fractional order chaotic systems were derived. Numerical simulations were presented to confirm the effectiveness of the analytical technique.
The synchronization of the motion of microresonators has attracted considerable attention. Here we present theoretical methods to synchronize the chaotic motion of two optical cavity modes in an optomechanical system, in which one of the optical modes is strongly driven into chaotic motion and is coupled to another weakly-driven optical mode mediated by a mechanical resonator. In these optomechanical systems, we can obtain both complete and phase synchronization of the optical cavity modes in chaotic motion, starting from different initial states. We find that complete synchronization of chaos can be achieved in two identical cavity modes. In the strong-coupling small-detuning regime, we also {produce} phase synchronization of chaos between two nonidentical cavity modes.
Several complex systems can be modeled as large networks in which the state of the nodes continuously evolves through interactions among neighboring nodes, forming a high-dimensional nonlinear dynamical system. One of the main challenges of Network Science consists in predicting the impact of network topology and dynamics on the evolution of the states and, especially, on the emergence of collective phenomena, such as synchronization. We address this problem by proposing a Dynamics Approximate Reduction Technique (DART) that maps high-dimensional (complete) dynamics unto low-dimensional (reduced) dynamics while preserving the most salient features, both topological and dynamical, of the original system. DART generalizes recent approaches for dimension reduction by allowing the treatment of complex-valued dynamical variables, heterogeneities in the intrinsic properties of the nodes as well as modular networks with strongly interacting communities. Most importantly, we identify three major reduction procedures whose relative accuracy depends on whether the evolution of the states is mainly determined by the intrinsic dynamics, the degree sequence, or the adjacency matrix. We use phase synchronization of oscillator networks as a benchmark for our threefold method. We successfully predict the synchronization curves for three phase dynamics (Winfree, Kuramoto, theta) on the stochastic block model. Moreover, we obtain the bifurcations of the Kuramoto-Sakaguchi model on the mean stochastic block model with asymmetric blocks and we show numerically the existence of periphery chimera state on the two-star graph. This allows us to highlight the critical role played by the asymmetry of community sizes on the existence of chimera states. Finally, we systematically recover well-known analytical results on explosive synchronization by using DART for the Kuramoto-Sakaguchi model on the star graph.