1. Dynamical Systems and Chaos Synchronization in Networks of Oscillators
Yuri Maistrenko
Laboratory of Mathematical Modeling of Nonlinear Processes Institute of Mathematics and Centre for Medical and Biotechnical Research NANU and in the last years Potsdam University, Technical University Berlin, Swiss Federal Institute of Technology in Lausanne (EPFL) Research Centre Juelich in Germany

2. 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”

3. 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.

5. Examples of network topology
local global (all-to-all)
Examples of network topology
random
scale-free

6. small world

8. To understand how the networks behave collectively, we need mathematical modelling.

9. Let’s try to model neuronal networks human brain: a network of 100 000 000 000 neurons
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?

10. How to model neuron network?
Kiss - detailed
Kiss – reduced
Rodin
Brancusi 10

11. The H&H model; (1) Biophysical, (2) Compact, (3) Predictive
Hodgkin-Huxley model (1952)
Alan Lloyd Hodgkin
Andrew Fielding Huxley
The H&H model; (1) Biophysical, (2) Compact, (3) Predictive 11

12. Networks of coupled maps

14. Spatially continuous model as N  ∞.
Discrete model (our network of N oscillators) :
Let N  ∞ thermodynamic limit . Then, an integral operator is obtained :
where r=P/N - radius of coupling.
Change of the variable :

15. Nonlinear integral operator to study
Bifurcation value:
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).

16. Kuramoto model (1984)
Network of N globally coupled phase oscillators:
(mathematically: system of N ordinary differential equations on torus TN )
- phases of individual oscillators
- frequencies of individual oscillators (=Const.)
- coupling function

17. “Standard” Kuramoto model
All-to-all sinusoidal coupling
Critical Kuramoto bifurcation value :
: synchronization (phase - locking)
: desynchronization and clustering
Two simple properties: 1)first integral and 2)reducing to system in differences (dim=N-1)

18. Desynchronization transition in the Kuramoto modelj i
Simplest example: N=2
New variable
Bifurcation value

19. Kuramoto-Sakaguchi model:
Hansel model:

20. Kuramoto modet with time delay

21. Chaos actually … is everywhere   

22. 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)

23. 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)

24. THE ESSENCE OF CHAOS processes deterministic
fully determined by initial state
long-term behavior unpredictable
butterfly effect

25. PHYSICAL “DEFINITION “ OF CHAOS
“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”
Predrag Cvitanovich . Appl.Chaos 1992

26. Web Book

27. 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

28. 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 conditions 2. f is topologically transitive 3. periodic points are dense in A

29. is a closed set with the following properties:
Attractor
is a closed set with the following properties:
S. Strogatz

30. Strange attractor (1971)
“…I'm strangely attracted to you”
Cole Porter (1953)
An attractor A is a set in phase space, towards which a dynamical system evolves over time. This limiting set A can be:
point (equilibrium)
curve (periodic orbit)
manifold (quasiperiodic orbit - torus)
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

31. It's all right with me 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 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.

32. UNPREDICTIBILITY OF THE WEATHER
Edward Lorenz (1963)
Difficulties in predicting
the weather are not related
to the complexity of the
Earths’ climate but to
CHAOS
in the climate equations!

33. 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.

34. 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

35. Reducing to discrete dynamics. Lorenz map
Lorenz attractor
Continues dynamics . Variable z(t) x
Lorenz one-dimensional map

36. Poincare section and Poincare return map
Rossler attractor
Rossler one-dimensional map

37. Tent map and logistic map

38. Strange attractor in Henon map (1976)

39. How common is chaos in dynamical systems? To answer the question,
we need discrete dynamical systems given by
one-dimensional maps

40. Bifurcation diagram for one-dimensional logistic map.
Regular and chaotic dynamics x
system parameter 𝑎

41. Lyapunov exponent for logistic map.
Bifurcation diagram
Lyapunov exponent λ
is positive on a nowhere dense,
Cantor-like set of parameter a
parameter a

42. Cascade of period-doubling bifurcation. Feigenbaum (1978).
Sharkovsky ordering (1964)
For any continuous 1-Dim map, periods of cycles (periodic orbits) are ordered as:

43. Cascade of homoclinic bifurcations

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

45. Let’s try to find chaos in the Kuramoto model
Simplest example: N=2 j i
New variable
Bifurcation value
No chaos!

46. Still no chaos! Simplest non-trivial example: N=3 oscillators.
New variables:
Still no chaos!

47. The dynamics on 2Dim torus is given by the reduced model in phase differences
Identical oscillators Non-identical but symmetric
No flow Cherry flow

48. How one can define chaos and estimate its magnitude?
A notion of Lyapunov exponent
is required!

49. Chaos in the Kuramoto model. N=4 and more
N=4 Chaos N=7 Hyperchaos
Average frequencies
Lyapunov exponents

50. Hyperchaos in the Kuramoto model: N=20 oscillators
Lyapunov exponents
Lyapunov spectrum
Maximal Lyapunov exponent

51. Bifurcation diagram for the Kuramoto model

52. Phase chaos in other networks of coupled oscillators
Lyapunov exponents
10 coupled Stuart-Landau
oscillators
7 coupled Rossler

53. Chimera states in the Kuramoto model

54. 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.

55. Chimera states in the Kuramoto-Sakaguchi model
The oscillators uniformly distributed over the interval [-1, +1]
with periodic boundary conditions.
Coupling function:
Parameters : radius of coupling, phase shift

56. Snapshots of chimera state
Phase-locked oscillators co-exist with drifting oscillators
Asymmetric chimera
Symmetric chimera X
Average frequencies
(Abram, Strogatz N=256 oscillators)
Average frequencies
Chimera state =
partial frequency synchronization!
(Kuramoto Battogtohk N=512 oscillators)

57. Chaotic wandering of the chimera state
Color code represents time-averaged frequencies of individual oscillators
Parameters: N=100, = 1.46, r = 0.7

58. 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 .

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