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Frequency-locking and pattern formation in the Kuramoto model with Manhattan delay

Frequency-locking and pattern formation in the Kuramoto model with Manhattan delay Karol Trojanowski, Lech Longa Jagiellonian University Institute of Physics , Dept. Stat . Phys . CompPhys11, 24-27.11.2011 Leipzig

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Frequency-locking and pattern formation in the Kuramoto model with Manhattan delay

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  1. Frequency-locking and patternformationintheKuramoto model with Manhattan delay Karol Trojanowski, Lech Longa JagiellonianUniversityInstitute of Physics, Dept. Stat. Phys. CompPhys11, 24-27.11.2011 Leipzig Project carried out within the MPD programme "Physics of Complex Systems" of the Foundation for Polish Science and cofinanced by the European Regional Development Fund in the framework of the Innovative Economy Programme

  2. Outline • Examples of synchronizationincomplex systems. • TheKuramoto model. • Previous 2D latticemodels. • Our model. • Results, summary.

  3. 1. Examples of synchronizationincomplex systems.

  4. More examples: • clappingaudiences • theheart • thebrain • many more… • SYNCHRONIZATION:Self-organizationphenomenon • systems of many (possiblydifferent) oscillators • mutual interaction • emergent, system-sizecollectivebehavior • frequency-locking and phase-locking • Google: „Steven Strogatz TED talk”

  5. 2. TheKuramoto model. • YoshikiKuramoto (1984): • N self-drivenoscillatorswithdifferentintrinsicfrequencies • everyoscillatoriscoupled to allothers • interactionwhichspeedsuposcillatorswhicharebehind and slows downtheoneswhichareahead

  6. We can decouple: Using: r - macroscopic order parameter measuring phase-locking

  7. 3. Previous 2D latticemodels.

  8. 3. Our model. The connection graph does not change, however it gains a spatial, lattice structure. Delay between nodes, proportional to distance „Manhattan” (AKA „taxi-driver’s”) metric (on a periodiclattice) Thedistanceaffectsonlythedelay, thecouplingremains uniform!

  9. 4. Results

  10. Non-trivialfrequency-locking Numerical integration with random initial conditions Pattern states with different sizes (r=0) Phase-locked states (r>0) The patterns change wavelength/size seemingly when the mean frequency jumps from one jag to another (no „frequency-locking” since oscillators are identical) phase-locking threshhold

  11. Solution for phase-lockedstates Rewritethe sum as a sum overdistance Apply to both sides This sum has a limit as N→∞

  12. Summary • We haveproposed a model whichproducesphasepatternswithoutexplicitlyincludinglocalinteractionsorcouplingdecreasingwithdistance and is(at leastpartially) mathematicallytractable

  13. Development outlook • stability analysis (difference-differential equations, v.v. tedious) • provide an ansatz for other solutions, test against exp. data e.g. plane-wave ansatz: Furtherreading • Kuramoto, Y., Cooperative Dynamics of Oscillator Community: A Study Based on Lattice of Rings, Prog. Theor. Phys. Sup. 79, pp. 223-240 (1984) • Acebron et al., The Kuramoto model: A simple paradigm for synchronizationPhenomena, Rev. Mod. Phys. 77, pp. 137-185 (2005)

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