Chaos course presentation: Solvable model of spiral wave chimeras

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Chaos course presentation: Solvable model of spiral wave chimeras. Kees Hermans Remy Kusters. Index. Introduction Goal of the project Kuramoto’s model (1-dimensional) Theory Simulation Spiral wave chimeras (2-dimensional) Theory Conclusions Conclusions and Outlook. Introduction.

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### Chaos course presentation:Solvable model of spiral wave chimeras

Kees Hermans

Remy Kusters

Index
• Introduction
• Goal of the project
• Kuramoto’s model (1-dimensional)
• Theory
• Simulation
• Spiral wave chimeras (2-dimensional)
• Theory
• Conclusions
• Conclusions and Outlook

/ Applied Physics

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:

: Coupling strength

: Natural frequency

: Tunable parameter

: modulus

: phase

• Using this, Kuramoto’s problem reduces to:
• When is above a certain value we expect a certain synchronization
• Phase transition for a certain value of and

CHIMERA STATE !

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Simulation
• 100 coupled oscillators
• Euler forward method
• Tune and

Chimera state!

All oscillators in phase

Breathing state

Chaotic phase state

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Simulation

Varying

Coupling constant: 4.0

Exactly!

Chimera state

1,455

2

Back to the two dim. model
• Model:
• Local mean field:
• Using:

/ Applied Physics

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Stationary solution
• Rotating frame:
• Time-independent mean field:
• The model is now:

When : stationary solution

When : drifting oscillators

/ Applied Physics

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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*:

/ Applied Physics

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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

/ Applied Physics

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Comparisons

Comparison of the analytical and numerical solutions.

Good results for small α’s.

/ Applied Physics

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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

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