1. Chaos course presentation: Solvable model of spiral wave chimeras Kees Hermans Remy Kusters

2. / Applied Physics PAGE 117-6-2015 Index Introduction Goal of the project Kuramoto’s model (1-dimensional) Theory Simulation Spiral wave chimeras (2-dimensional) Theory Conclusions Conclusions and Outlook

3. Introduction Title of the main article: Solvable model of spiral wave chimeras What is a spiral wave?What is a chimera? 17-6-2015PAGE 2

4. Physical examples of spiral waves Heart muscle: Nerve cells: Fireflies: 17-6-2015PAGE 3

5. 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 417-6-2015

6. 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. 17-6-2015PAGE 5

7. 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! 17-6-2015PAGE 6

8. Kuramoto’s model (complex) Order parameter: Using this, Kuramoto’s problem reduces to: When is above a certain value we expect a certain synchronization : Coupling strength : Natural frequency 17-6-2015PAGE 7 Phase transition for a certain value of and : modulus : phase : Tunable parameter CHIMERA STATE !

9. Simulation 100 coupled oscillators Euler forward method Tune and 17-6-2015 PAGE 8 Chimera state! Breathing stateAll oscillators in phase Chaotic phase state

10. Simulation Coupling constant: 4.0 1,455 2 Chimera state Varying Exactly!

11. Back to the two dim. model Model: Local mean field: Using: This leads to: / Applied Physics PAGE 1017-6-2015

12. Stationary solution Rotating frame: Time-independent mean field: The model is now: When : stationary solution When : drifting oscillators / Applied Physics PAGE 1117-6-2015

13. 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 1217-6-2015

14. 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 1317-6-2015

15. Comparisons Comparison of the analytical and numerical solutions. Good results for small α’s. / Applied Physics PAGE 1417-6-2015

16. 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 1517-6-2015

17. 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 1617-6-2015