Chaos course presentation:

Solvable model of spiral wave

chimeras

Kees Hermans

Remy Kusters

/ Applied Physics PAGE 204/18/23

Index

• Introduction• Goal of the project• Kuramoto’s model (1-dimensional)

• Theory• Simulation

• Spiral wave chimeras (2-dimensional)• Theory • Conclusions

• Conclusions and Outlook

Introduction

Title of the main article: Solvable model of spiral wave chimeras

• What is a spiral wave? • What is a chimera?

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Physical examples of spiral waves

Heart muscle:

Nerve cells:

Fireflies:

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Introduction

• System of coupled oscillators in two dimensions• Field of NxN oscillators

• Local Gaussian coupling

• Fabulous result:− Phase-randomized core of

desynchronized oscillators surrounded by phase-locked oscillators moving in spiral arms

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Goal of the project

• Article by Martens, Laing and Stogatz (2010)• They found an analytical description for

• The spiral wave arm rotation speed;• Size of its incoherent core.

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Kuramoto’s model

Let’s go eight years back in time and review Kuramoto’s article• Ring of N oscillators • Finite-range nonlocal coupling• Behavior of the array of oscillators divides into two parts:

• One with mutually synchronized oscillators• One with desynchronized oscillators

Chimera state!

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Kuramoto’s model

• (complex) Order parameter:

N

j

ii jeN

re1

1

• Using this, Kuramoto’s problem reduces to: )sin( iii Kr

• When is above a certain value we expect a certain synchronizationK

K : Coupling strength

: Natural frequency

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• Phase transition for a certain value of and K

N

jiji

i

N

K

dt

d

1

)sin(

r : modulus

: phase

: Tunable parameter

CHIMERA STATE !

Simulation

• 100 coupled oscillators • Euler forward method• Tune and

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Chimera state!

Breathing stateAll oscillators in phase Chaotic phase state

K

Simulation

Coupling constant: 4.0

1,455 2

Chimera state

Varying

Exactly!

Back to the two dim. model

• Model:

• Local mean field:

• Using:

• This leads to:

/ Applied Physics PAGE 1104/18/23

Stationary solution

• Rotating frame:

• Time-independent mean field:

• The model is now:

When : stationary solution

When : drifting oscillators

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Resulting nonlinear integral equation

• Now it is possible to get an equation that contains the time-independent values R(x) and θ(x):

• For the drifting oscillators the probability density ρ(ψ) is:

• The phases of the spiral arms approach a stable point ψ*:

• Using this leads to:

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What did Martens et al. do?

• Changing to polar coordinates (r,Θ):

Ansatz: ,• Look to small α’s and use perturbation theory:

• Conclusions after lots of mathematics:

- Spiral arms rotate at angular velocity Ω = ω - α

- Incoherent core radius is given by ρ = (2/√π) α

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Comparisons

• Comparison of the analytical and numerical solutions.• Good results for small α’s.

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Simulation

• 36X36 oscillators• Simulations took very long• Only created the state dominated by chaos

• Simulation time was to long to reach synchronized state• More than 1000 coupled oscillators

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Conclusions

• Theory• Analytical solution for small values of α.• Chimera states not yet experimentally observed

(observation of spiral wave chimeras in a neural network may be a good candidate)

• Spiral wave chimeras in 2D exist for small α’s, while in lower dimensions α should be around π/2

• Why spiral waves?

• One-dimensional simulation:• Recovered chimera state and other funny symmetries

/ Applied Physics PAGE 1704/18/23