1 / 17

Chaos course presentation: Solvable model of spiral wave chimeras

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.

asher-ellison
Download Presentation

Chaos course presentation: Solvable model of spiral wave chimeras

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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

  3. 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

  4. Physical examples of spiral waves Heart muscle: Nerve cells: Fireflies: 19-11-2014 PAGE 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 19-11-2014 PAGE 4

  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. 19-11-2014 PAGE 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! 19-11-2014 PAGE 6

  8. 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

  9. 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

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

  11. Back to the two dim. model • Model: • Local mean field: • Using: • This leads to: / Applied Physics 19-11-2014 PAGE 10

  12. 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

  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 19-11-2014 PAGE 12

  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 19-11-2014 PAGE 13

  15. Comparisons Comparison of the analytical and numerical solutions. Good results for small α’s. / Applied Physics 19-11-2014 PAGE 14

  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 19-11-2014 PAGE 15

  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 19-11-2014 PAGE 16

More Related