Yuri Maistrenko

Laboratory of Mathematical Modeling of Nonlinear ProcessesInstitute of Mathematics and Centre for Medical and

Biotechnical Research NANUand in the last years

Potsdam University, Technical University Berlin, Swiss Federal Institute of Technology in Lausanne (EPFL)

Research Centre Juelich in Germany

E-mail: y.maistrenko@biomed.kiev.ua

Dynamical Systems and

Chaos

Synchronization in

Networks of Oscillators

COLLABORATION IN EUROPE

Germany• WIAS, Berlin: “Laser Dynamics and Coupled Oscillators”

• Technical University Berlin: “Nonlinear Dynamics and Control”

• Humboldt University Berlin: “Dynamics and Synchronization of Complex Systems”

• Potsdam Universtity: "Statistical Physics and Theory of Chaos"

• Research Centre Juelich (beyond river Rein): “Function of Neuronal Microcircuits” “Complex Systems in Medical Electronics” Switzerland• EPFL, Lausanne: “Dynamical Networks in Electrical Engineering and Neuroscience” France• Université Paris 7 “Theory of Complex Systems”

UK• University of Exeter and Universtity of Plymouse (English Riviera) “Large networks of coupled dynamical systems” “Brain models of attention and memory”

Germany• WIAS, Berlin: “Laser Dynamics and Coupled Oscillators”

• Technical University Berlin: “Nonlinear Dynamics and Control”

• Humboldt University Berlin: “Dynamics and Synchronization of Complex Systems”

• Potsdam Universtity: "Statistical Physics and Theory of Chaos"

• Research Centre Juelich (beyond river Rein): “Function of Neuronal Microcircuits” “Complex Systems in Medical Electronics” Switzerland• EPFL, Lausanne: “Dynamical Networks in Electrical Engineering and Neuroscience” France• Université Paris 7 “Theory of Complex Systems”

UK• University of Exeter and Universtity of Plymouse (English Riviera) “Large networks of coupled dynamical systems” “Brain models of attention and memory”

BOOKS

4

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram? Are there any unifying principles underlying their topology?

From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture.

Researchers are only now beginning to unravel the structure and dynamics of complex networks.

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Examples of network topologylocal global (all-to-all)

scale-free

random

6

small world

To understand how the networks behave collectively,

we need mathematical modelling.

Let’s try to model neuronal networks

human brain: a network of 100 000 000 000 neurons

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- how complicated are neuronal networks in the brain?

- are they locally or globally (i.e. mean field) coupled?

- strength of coupling between individual neurons?

- excitatory and inhibitory neurons, why so?

BrancusiRodin

How to model neuron network?

Kiss - detailed Kiss – reduced

Alan Lloyd Hodgkin

Andrew Fielding Huxley

The H&H model; (1) Biophysical, (2) Compact, (3) Predictive

Hodgkin-Huxley model (1952)

Networks of coupled maps

Spatially continuous model as N

Pi

Pij

ni

nj

ni

ni zfzf

Pzfz )]()([

2)(1

Discrete model (our network of N oscillators) :

rx

rx

nnnn dyxzfyzfr

xzfxz ))](())(([2

))(()(1

Let N ntegral operator is obtained :

Change of the variable :)( nn zfw

rx

rx

nnn dyywr

xwfxw )(2

)()1()(1

where r=P/N - radius of coupling.

Nonlinear integral operator to study

rx

rx

dyyzr

xzfxz )(2

)()1()(F

Theorem 1. If , then every stationary state z(x) of F is a continuous function (coherent state). This state is unique with respect to shift of x.

If , then every stationary state z(x) of F is a discontinuous function (partially or fully incoherent state).

b

b

,1

1kb

).0('fk

Bifurcation value:

16

Kuramoto model (1984)

Network of N globally coupled phase oscillators:

(mathematically: system of N ordinary differential equations on torus TN )

ii

Niij

N

jii ,...,1 ),(

1ij

- coupling function)( ijijij

- phases of individual oscillators

- frequencies of individual oscillators (=Const.)

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“Standard” Kuramoto model

cKK cKK

: synchronization (phase - locking) : desynchronization and clustering

Critical Kuramoto bifurcation value :0cK

,,...,1 ),(sin1

NiNK

ij

N

jii

All-to-all sinusoidal coupling : sin(.)(.)N

Kij

Two simple properties: 1)first integral and 2)reducing to system in differences (dim=N-1)

Desynchronization transition in the Kuramoto model

Simplest example: N=2jiK

ji

ijK

Bifurcation value

New variable

Kuramoto-Sakaguchi model:

NiN

Kij

N

ji ,...,1 ), (sin

1

NiN

Kiij

N

ji ,...,1 ))],sin(2(r ) ([sin j

1

Hansel model:

parameters 0r ,20

]2sin)[sin()( rN

K

)][sin()( N

K

Kuramoto modet with time delay

.,...,1 )],()([sin1

NittN

Kij

N

jii

Chaos actually … is everywhere

CHAOS in DYNAMICAL SYSTEMS

Dynamical system: a system of one or more variables which evolve in time according to a given rule

Two types of dynamical systems:• Differential equations: time is continuous (called flow)

• Difference equations (iterated maps): time is discrete (called cascade)

R ,)( N txfdt

dx

2,... 1, 0, ),(1 nxfx nn

CHAOS = BUTTERFLY EFFECT

Henri Poincaré (1880) “ It so happens that small

differences in the initial state of the

system can lead to very large differences in its final state.

A small error in the former could then produce an enormous one in the latter. Prediction becomes impossible, and the system appears to behave randomly.”

Ray Bradbury “A Sound of Thunder “ (1952)

THE ESSENCE OF CHAOS

• processes deterministic fully determined by initial state

• long-term behavior unpredictable butterfly effect

PHYSICAL “DEFINITION “ OF CHAOS

Predrag Cvitanovich . Appl.Chaos 1992

“To say that a certain system exhibits chaos means that the system obeys deterministic law of evolution but that the outcome is highly sensitive to small uncertainties in the specification of the initial state. In chaotic system any open ball of initial conditions, no matter how small, will in finite time spread over the extent of the entire asymptotically admissible phase space”

Web Book

EXAMPLES OF CHAOTIC SYSTEMS

• the solar system (Poincare)• the weather (Lorenz)• turbulence in fluids • population growth • lots and lots of other systems…

• neuronal networks of the brain• genetic networks

“HOT” APPLICATIONS

MATHEMATICAL DEFINITION OF CHAOS

Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions

Let A be a set. The mapping f : A → A is said to be chaotic on A if: 1. f has sensitive dependence on initial conditions2. f is topologically transitive 3. periodic points are dense in A

Attractor is a closed set with the following properties:

S. Strogatz

Strange attractor (1971)

An attractor A is a set in phase space, towards which a dynamical system evolves over time. This limiting set A can be:

1) point (equilibrium)2) curve (periodic orbit)3) manifold (quasiperiodic orbit - torus)4) fractal set (chaos - strange attractor)

Up to the beginning of 60th of the last century people believe that nothing else is possible in deterministic system

“…I'm strangely attracted to you” Cole Porter (1953)

It's the wrong time, and the wrong place Though your face is charming, it's the wrong face It's not her face, but such a charming face that it's all right with me. It's the wrong song, in the wrong style Though your smile is lovely, it's the wrong smile It's not her smile, but such a lovely smile that it's all right with me.

You can't know how happy I am that we met I'm strangely attracted to you There's someone I'm trying so hard to forget ... (Don't you want to forget someone, too?) It's the wrong game, with the wrong chips Though your lips are tempting, they're the wrong lips They're not her lips, but they're such tempting lips that, if some night, you're free ... Then it's all right, yes, it's all right with me.

It's all right with meCole Porter (1953)

Edward Lorenz (1963)

Difficulties in predicting the weather are not relatedto the complexity of the Earths’ climate but to CHAOS in the climate equations!

UNPREDICTIBILITY OF THE WEATHER

LORENTZ ATTRACTOR (1963)

butterfly effect a trajectory in the phase space

The Lorenz attractor is generated by the system of 3 differential equations

dx/dt=

-10x +10y

dy/dt=

28x -y -xz

dz/dt= -8/3z +xy.

ROSSLER ATTRACTOR (1976)

A trajectory of the Rossler system, t=250

To see what solutions looks like in general, we need to perform numerical integration.One can observe that trajectories looks like behave chaotically and converge to a strange attractor.But, there exists no mathematical proof that such attractor is asymptotically aperiodic. It might well be that what we see is but a long transient on a way to an attractive periodic orbit

Reducing to discrete dynamics. Lorenz map

x

Lorenz attractorContinues dynamics . Variable z(t)

Lorenz one-dimensional map

Poincare section and Poincare return map

Rossler attractor

Rossler one-dimensional map

Tent map and logistic map

Strange attractor in Henon map (1976)

How common is chaos in dynamical systems?

To answer the question, we need discrete dynamical systems given by one-dimensional maps

Bifurcation diagram for one-dimensional logistic map. Regular and chaotic dynamics

system parameter

x

Lyapunov exponent for logistic map.

Bifurcation diagram

Lyapunov exponent λ

is positive on a nowhere dense, Cantor-like set of parameter a

parameter a

Cascade of period-doubling bifurcation. Feigenbaum (1978).

Sharkovsky ordering (1964)

For any continuous 1-Dim map, periods of cycles (periodic orbits) are ordered as:

Cascade of homoclinic bifurcations

Period three implies chaos (Li,Yorke1975)

“Period three implies chaos” (Li, Yorke 1975)

Let’s try to find chaos in the Kuramoto modelSimplest example: N=2 jiK

ji

ijK

Bifurcation value

New variable

No chaos!

Simplest non-trivial example: N=3 oscillators.

New variables:

Still no chaos!

The dynamics on 2Dim torus is given by the reduced model in phase differences

No flow Cherry flow

Identical oscillators Non-identical but symmetric

How one can define chaos and estimate its magnitude?

A notion of Lyapunov exponent is required!

Chaos in the Kuramoto model. N=4 and more

Average frequencies

Lyapunov exponents

N=4 Chaos N=7 Hyperchaos

Lyapunov exponents

Lyapunov spectrum Maximal Lyapunov exponent

Hyperchaos in the Kuramoto model: N=20 oscillators

Bifurcation diagram for the Kuramoto model

10 coupled Stuart-Landauoscillators

7 coupled Rossler oscillators

Phase chaos in other networks of coupled oscillators

Lyapunov exponents

Chimera states in the Kuramoto model

In Greek mythology, the chimera was a fire-breathing monster having a lion’s head, a goat’s body, and a serpent’s tail. Today the word refers to anything composed of incongruent parts, or anything that seems fantastic.

Chimera states in the Kuramoto-Sakaguchi model

The oscillators uniformly distributed over the interval [-1, +1]

Coupling function:

with periodic boundary conditions.

Parameters : - radius of coupling, - phase shift

Snapshots of chimera state

X(Abram, Strogatz 2004. N=256 oscillators)

(Kuramoto Battogtohk 2002. N=512 oscillators)

Average frequencies

Asymmetric chimera Symmetric chimera

Chimera state =partial frequency synchronization!

Phase-locked oscillators co-exist with drifting oscillators

Average frequencies

Chaotic wandering of the chimera state

Color code represents time-averaged frequencies of individual oscillators

Parameters: N=100, = 1.46, r = 0.7

Compare two chimera states

Chimeras are extreme sensitive to initial conditions: two chimera trajectories with initial conditions that differ by 0.001, in one oscillator only .

ВСІМ ЩИРО ДЯКУЮ,

ЩО ПРИЙШЛИ

Yuri Maistrenko

Laboratory of Mathematical Modeling of Nonlinear Processes

Institute of Mathematics and Centre for Medical and Biotechnical Research NANU

E-mail: y.maistrenko@biomed.kiev.ua