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Yuri Maistrenko
Academy of Sciences of Ukraine, Kiev, Ukraine E-mail: [email protected]
Lecture 14
Synchronization and Chimera States
in Complex Networks
You can find me in room ER 222 10.02.2016
Lorentz and Rössler systems
Are there chimera states?
Lorenz system (1963)
butterfly effect a trajectory in phase space
The Lorenz attractor is generated by a system of three differential equations
28 ,3/8b 10,
)(
rbzxyz
xzyrxy
xyx
Lorenz system: Reduction to discrete dynamics
x
Lorenz attractor Continues dynamics . Variable z(t)
Lorenz map
Lecture 16, from 55 min. to the end, at: https://www.youtube.com/watch?v=U-bWDtbB4qY&index=16&list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V Lecture 17, from 22 min to the end, at: https://www.youtube.com/watch?v=gscKcPAm-H0&index=17&list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V Lecture 18, from the beginning to the end, at: https://www.youtube.com/watch?v=ERzcine5Mqc&list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V&index=18
Nonlinear Dynamics and Chaos - Lecture Course Steven Strogatz, Cornell University Spring 2014
Bifurcation diagram of Lorenz system r vary and 3/8b 10, parametersfix
Subcritical transition to turbulence
Subcritical transition to turbulence
Network of Lorenz systems: Standing chimeras
Network of Lorenz systems: Traveling waves
),...,1 (
)(2
)(2
Niczyxz
yyP
xzybxy
xxP
ayaxx
iiii
ij
Pi
Pij
iiii
ij
Pi
Pij
iii
𝑎 = 10, 𝑏 = 28, 𝑐 = 8/3
Lorenz attractor
Chaotic
synchronization
Space-temporal chaos
𝜎 = 16, 𝑟 = 0.1, 𝑁 = 100
𝜎 = 13.3 , 𝑟 = 0.1, 𝑁 = 100
𝜎 = 13.8 , 𝑟 = 0.05, 𝑁 = 300
Rössler system : Poincare return map
Rössler attractor
Rössler one-dimensional map
Chimera state for coupled Rössler systems
I.Omelchenko, Riemenschneider, Hövel, Maistrenko, Schöll (PRE 2012)
)(2
ij
P
Pij
iii xxP
zyx
),...,1( )( Nicxzbz
ayxy
iii
iii
Riemenschneider
2-Dim models with limit cycles
Are there chimera states?
)(2
iy xzzzidt
dz
Stuart-Landau oscillator
)y(
)y(
22
22
xyyxy
xxyxx
0rLimit cycle with radius and frequency 𝜔
Harmonic (sinusoidal) oscillations:
Van der Pol oscillator (1926)
0 )1( 2 xxxx
with Lienard transformation
)3
(
1
3
xy
yx
xx
3
3
xxxy
If large parameter 1
relaxation oscillations
5
If small parameter 1 harmonic (sinusoidal) oscillations
S. Strogatz’s Lecture 10
Van der Pol oscillator: Slow-fast Dynamics
)3
(
1
3
xy
yx
xx
)3
(3
2
xy
yx
xx
tt
3
3
xy
yx
xx
2
1
From Van der Pol to FitzHugh-Nagumo (FHN) model
3
3
xy
yx
xx
Van der Pol:
3
ext
3
byaxy
Iyx
xx
FitzHugh-Nagumo:
often: 𝐼𝑒𝑥𝑡 = 0 and 𝑏 = 0
FitzHugh-Nagumo model (1961)
Oscllating versus excitable dynamics
3
ext
3
byaxy
Iyx
xx
zkykz
ykyCxky
xzkyCxkx
32
21
01
)(
)(
Belousov-Zhabotinsky chemical reaction (1951)
BZ reaction is one of a class of reactions that serve as a classical example nonlinear chemical oscillator. The mechanism for the reaction is very complex and involve around 18 different steps. It acts for a significant length of time as an example of non-equilibrium biological phenomena.
Zhabotinsky@Korzuhin (1967) Lengyel et al (1990)
)1
1(
1
4
2
2
yx
ybxy
x
xyxax
Limit cycle exists for 𝑏 <3𝑎
5−
25
𝑎
Simplified models for BZ reaction
video at: https://www.youtube.com/watch?v=8R33KWPmqlo
Belousov-Zhabotinsky reaction (experiment)
N=20 N=20
𝟐-Dim system for Belousov-Zhabotinsky reaction
Chimera state in coupled Belousov-Zhabotinsky reactions
with two-group topology
Experiment: time trace and snapshots
Experiment: bifurcation diagram
Chimera state
𝑘𝐴𝐵
N=20 N=20
Chimera state
𝑘𝐴𝐵
Experiment Numerical simulation
Chimera state
To compare experiment and simulations
To compare experiment and simulations
Simulations Experiment
s
snapshots space-time plot
local order parameter
Spiral chimera states in simulations. 𝟓𝟎𝒙𝟓𝟎 oscillators
Simulations of spiral chimera states in populations of BZ oscillators. The system is composed of 50×50 oscillators in a square-lattice configuration, with a coupling radius of n=4. The top images show the phase of each oscillator in the lattice at t=3500 for values of delay τ=4.0 (a) and 3.4 (b). Each simulation is initiated with a pair of symmetric counterrotating spirals, with τ=0. The delay is switched on at t=500, and the simulation is continued to t=3500. (c), (d) Shown is the local order parameter Rat t=3500. The dark blue line shows the trajectory of the minimum in R between t=700 and 3500. Parameters: κ=0.3, K′=1.4×10-3, and ϕ0=1.1×10-4.
Spiral chimera states in experiments?
Not obtained yet..
Scroll wave chimera: Two incoherent rolls
Solitary scroll vortex