Post on 27-Dec-2015
transcript
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”
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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.
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?
Alan Lloyd Hodgkin
Andrew Fielding Huxley
The H&H model; (1) Biophysical, (2) Compact, (3) Predictive
Hodgkin-Huxley model (1952)
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:
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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
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”
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
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
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:
Let’s try to find chaos in the Kuramoto modelSimplest example: N=2 jiK
ji
ijK
Bifurcation value
New variable
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
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
10 coupled Stuart-Landauoscillators
7 coupled Rossler oscillators
Phase chaos in other networks of coupled oscillators
Lyapunov exponents
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