Synchronization in complex network topologies Ljupco Kocarev Institute for Nonlinear Science, University of California, San Diego In this talk I will present a couple examples of chaos synchronization which we studied in physical and biological systems. Using these examples I will illustrate the elements of chaos synchronization theory.

Outlook Chaotic oscillations and types of synchrony observed between chaotic oscillators Experimental and theoretical analysis of chaos synchronization; Stability of the synchronization manifold Synchronization in networks First, I will discuss the features of chaotic oscillations, which influence the notion of synchronization developed for chaotic systems. Then I will concider the regimes of identical chaotic synchronization in nonlinear circuits and biological neurons. And at the end of this talk I will discuss more general notion of chaos synchronization.

Periodic and Chaotic Oscillations Power Spectrum Waveform x(t) Power Spectrum Waveform x(t) Chaotic Attractor The differences between periodic and chaotic oscillations can be seen from this slide which presents the data acquired from a nonlinear circuit. The regime of periodic oscillations corresponds to the motions on a stable limit cycle. The spectrum of this oscillation contains discrete harmonics. Since the period of oscillations is well defined one can introduce a phase. Now let us consider two such periodic oscillators coupled together. The synchronization in this case means the onset of precise stable relation between the phases of oscillations in each circuit. The onset of this relation is a result of a bifurcation in joint phase space of coupled oscillators. Due to this bifurcation the attracting set changes from a ergodic torus to a limit cycle. The regime of chaotic oscillations corresponds to the motions on a chaotic attractor. Chaotic oscillations have a continues power spectrum. As the result of it there is no direct way to introduce phase for such oscillations and the original notion of chaos synchronization cannot be applied here. However it is known that chaotic oscillators can synchronize. Lyapunov exponents:

Types of chaos synchronization Complete Synchronization Partial Synchronization Identical synchronous chaotic oscillations. Generalized synchronized chaos. Threshold synchronization of chaotic pulses. Phase Synchronization. Synchronization of switching. Others time

Synchronization of chaos in electrical circuits. -2 -1 1 2 -0.5 0.0 0.5 3.0 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 2.1 -2.1 PHASE PORTRAIT Unidirectional coupling N a R C’ C r L Driving Oscillator Response Oscillator Coupling Let us consider an example of chaos synchronization in two identical chaotic circuits with unidirectional coupling. The parameters of the circuits are set to generate chaos which corresponds to this chaotic attractor. The unidirectional dissipative coupling between the circuits is implemented with the unity gain amplifier and coupling resistor Rc. Changing this resistor we can control the strength of the coupling between the circuits.

Synchronization Manifold The model: The coupling parameter: The dynamic of these circuits is described by the system of six differential equations. Variables X describe the behavior of the driving circuit. Variables Y stand for variable of the response circuit. This term describe the coupling between the circuits. g is the coupling parameter. Since the parameters of the circuits are tuned to be the same this six-dimensional system has an invariant manifold. The trajectories on this manifold correspond to the identical synchronous oscillations. Therefore if the circuits have a chaotic attractor entirely located in this manifold they will be able generate identical chaotic oscillations. There exits a 3-dimensional invariant manifold:

Synchronization of chaos: Experiment Driving Oscillator Response Oscillator Uncoupled Oscillators Coupling below the threshold of synchronization This slide presents the results of experimental studies. Here are the oscillations measured in the driving circuit. The behavior of the response circuit depends on the strength of the coupling. Without coupling the response circuits has the chaotic attractor which is identical to the attractor in the driving circuit, but the trajectories of these circuits are not correlated. It can be clearly seen from the figure where one variable of response circuit is plotted versus the same variable of the driving circuit. When we introduce coupling the oscillations remain uncorrelated until the strength of the becomes larger than some threshold value. After that the chaotic attractor entirely located in the synchronization manifold. The projection. Coupling above the threshold of synchronization

Stability of the Synchronization Manifold: Identical Synchronization Driving System: Response System: Synchronization Manifold: Perturbations transversal to the Synchronization Manifold: Linearized Equations for the transversal perturbations:

Chaos Synchronization Regime A regime of dynamical behavior should have a qualitative feature that is an invariant for this regime. Consider dynamics in the phase space No Synchronization The parameter space Synchronization p2 Synch No Synch p1 - Projection of chaotic limiting set Transient trajectories - Limit cycles

Synchronization of Chaos in Numerical Simulations Coupling: g=1.1 Transversal Lyapunov exponent evaluated for the chaotic trajectory x(t) equals Attractor in the Driving Circuit Simulation without noise and parameter mismatch Simulation with 0.4% of parameter mismatch

Network with N nodes m - dimensional vector - real matrix Assumptions: - real matrix - real number Synchronization manifold: Connectivity matrix: - real matrix

Master Stability Function Variation equation: - eigenvalue of the connectivity matrix Master Stability Function Connectivity Matrix

Properties of the master stability function Empty set Ellipsoid Half plane The master stability function for x coupling in the Rossler circuit. The dashed lines show contours in the unstable region. The solid lines are contours in the stable region.

versus Stable region:

Laplacian matrix Class-A oscillators Class-B oscillators B>1

Erdős-Rényi Random Graph (also called the binomial random graph) Consider N nodes (dots); Take every pair (i,j) of nodes and connect them with an edge with probability p

Power-law distribution Power-law networks Power-law distribution =<k> Power-law graphs with prescribed degree sequence (configuration model, 1978) Evolution models (BA model, 1999; Cooper and Frieze model, 2001) Power-law models with given expected degree sequence (Chung and Lu, 2001)

Hybrid Graphs Hybrid graph is a union of global graph (consisting of “long edges” providing small distances) and a local graph (consisting of “short edges” representing local connections). The edge set of of the hybrid graph is a disjoint union of the edge set of the global graph G and that of the local graph L. G: classical random model power-law model L: grid graph

Theorem 1. Let G(N,p) be a random graph on N vertices Theorem 1. Let G(N,p) be a random graph on N vertices. For sufficiently large N, the class-A network G(N,q) almost surely synchronize for arbitrary small coupling. For sufficiently large N, almost every class-B network G(N,p) with B>1 is synchronizable.

Theorem 2. Let M(N, , d, m) be a random power-law graph on N vertices Theorem 2. Let M(N, , d, m) be a random power-law graph on N vertices. For sufficiently large N, the class-A network M(N, , d, m) almost surely synchronize for arbitrary small coupling. For sufficiently large N, almost every class-B network M(N, , d, m) is synchronizable only if power of the power-law d expected average degree m expected maximum degree

Consider a hybrid graph for which L is a circle with N=128. Consider class-A oscillators for which A=1 and Consider class-B oscillators for which B=40 pNG number of global edges p=0.005 p=0.01

Oscillators do not synchronize Local networks Oscillators do not synchronize Hybrid networks Random networks Power-law Oscillators may or may not synchronize Binomial Oscillators synchronize Power-law Oscillators synchronize Binomial

Conclusions Two oscillators may have different synchronous behavior Synchronization of identical chaotic oscillations are found in the oscillators of various nature (including biological neurons) Global edges improve synchronization