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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? 19-11-2014 PAGE 2
Physical examples of spiral waves Heart muscle: Nerve cells: Fireflies: 19-11-2014 PAGE 3
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 19-11-2014 PAGE 4
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. 19-11-2014 PAGE 5
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! 19-11-2014 PAGE 6
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 ! 19-11-2014 PAGE 7
Simulation • 100 coupled oscillators • Euler forward method • Tune and Chimera state! All oscillators in phase Breathing state Chaotic phase state 19-11-2014 PAGE 8
Simulation Varying Coupling constant: 4.0 Exactly! Chimera state 1,455 2
Back to the two dim. model • Model: • Local mean field: • Using: • This leads to: / Applied Physics 19-11-2014 PAGE 10
Stationary solution • Rotating frame: • Time-independent mean field: • The model is now: When : stationary solution When : drifting oscillators / Applied Physics 19-11-2014 PAGE 11
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 19-11-2014 PAGE 12
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 19-11-2014 PAGE 13
Comparisons Comparison of the analytical and numerical solutions. Good results for small α’s. / Applied Physics 19-11-2014 PAGE 14
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 19-11-2014 PAGE 15
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 19-11-2014 PAGE 16