1. 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.

3. 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.

4. 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:

5. 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

6. Synchronization of chaos in electrical circuits.-2 -1 1 2
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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.

7. 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:

8. 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

9. 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:

10. 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

11. 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

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

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

14. 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.

15. versus
Stable region:

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

17. 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

18. 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)

19. 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

20. 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.

21. 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

22. 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

23. 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

24. 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