Chaos course presentation: Solvable model of spiral wave chimeras Kees Hermans Remy Kusters.
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Transcript of Chaos course presentation: Solvable model of spiral wave chimeras Kees Hermans Remy Kusters.
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?
04/18/23 PAGE 3
Physical examples of spiral waves
Heart muscle:
Nerve cells:
Fireflies:
04/18/23 PAGE 4
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
PAGE 504/18/23
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.
04/18/23 PAGE 6
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!
04/18/23 PAGE 7
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
04/18/23 PAGE 8
• 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
04/18/23PAGE 9
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
/ Applied Physics PAGE 1204/18/23
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 PAGE 1304/18/23
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 PAGE 1404/18/23
Comparisons
• Comparison of the analytical and numerical solutions.• Good results for small α’s.
/ Applied Physics PAGE 1504/18/23
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
PAGE 1604/18/23
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