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Register a new userMaster Stability Functions for Coupled Near-Identical Dynamical Systems
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We derive a master stability function (MSF) for synchronization in networks of coupled dynamical systems with small but arbitrary parametric variations. Analogous to the MSF for identical systems, our generalized MSF simultaneously solves the linear stability problem for near-synchronous states (NSS) for all possible connectivity structures. We also derive a general sufficient condition for stable near-synchronization and show that the synchronization error scales linearly with the magnitude of parameter variations.Our analysis underlines significant roles played by the Laplacian eigenvectors in the study of network synchronization of near-identical systems.
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We derive variational equations to analyze the stability of synchronization for coupled near-identical oscillators. To study the effect of parameter mismatch on the stability in a general fashion, we define master stability equations and associated master stability functions, which are independent of the network structure. In particular, we present several examples of coupled near-identical Lorenz systems configured in small networks (a ring graph and sequence networks) with a fixed parameter mismatch and a large Barabasi-Albert scale-free network with random parameter mismatch. We find that several different network architectures permit similar results despite various mismatch patterns.
This report investigates the dynamical stability conjectures of Palis and Smale, and Pugh and Shub from the standpoint of numerical observation and lays the foundation for a stability conjecture. As the dimension of a dissipative dynamical system is increased, it is observed that the number of positive Lyapunov exponents increases monotonically, the Lyapunov exponents tend towards continuous change with respect to parameter variation, the number of observable periodic windows decreases (at least below numerical precision), and a subset of parameter space exists such that topological change is very common with small parameter perturbation. However, this seemingly inevitable topological variation is never catastrophic (the dynamic type is preserved) if the dimension of the system is high enough.
Providing efficient and accurate parametrizations for model reduction is a key goal in many areas of science and technology. Here we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parametrizations of weakly coupled dynamical systems. Such parametrizations yield a set of stochastic integro-differential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integro-differential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equations-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings support, on the one hand, the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parametrizations.
We study the Kuramoto-Sakaguchi (KS) model composed by any N identical phase oscillators symmetrically coupled. Ranging from local (one-to-one, R = 1) to global (all-to-all, R = N/2) couplings, we derive the general solution that describes the network dynamics next to an equilibrium. Therewith we build stability diagrams according to N and R bringing to the light a rich scenery of attractors, repellers, saddles, and non-hyperbolic equilibriums. Our result also uncovers the obscure repulsive regime of the KS model through bifurcation analysis. Moreover, we present numerical evolutions of the network showing the great accordance with our analytical one. The exact knowledge of the behavior close to equilibriums is a fundamental step to investigate phenomena about synchronization in networks. As an example, at the end we discuss the dynamics behind chimera states from the point of view of our results.
In this article we synchronize by active control method all 19 identical Sprott systems provided in paper [10]. Particularly, we find the corresponding active controllers as well as we perform (as an example) the numerical synchronization of two Sprott-A models.
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