Yuri Maistrenko

Academy of Sciences of Ukraine, Kiev, Ukraine E-mail: y.maistrenko@biomed.kiev.ua

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