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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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Presentation Transcript
index
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
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
Physical examples of spiral waves

Heart muscle:

Nerve cells:

Fireflies:

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introduction1
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
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
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 model1
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
Simulation
  • 100 coupled oscillators
  • Euler forward method
  • Tune and

Chimera state!

All oscillators in phase

Breathing state

Chaotic phase state

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simulation1
Simulation

Varying

Coupling constant: 4.0

Exactly!

Chimera state

1,455

2

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

/ Applied Physics

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stationary solution
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
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:

/ Applied Physics

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what did martens et al do
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
Comparisons

Comparison of the analytical and numerical solutions.

Good results for small α’s.

/ Applied Physics

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simulation2
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
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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