The Hamiltonian Mean Field Model: Effect of Network Structure on Synchronization Dynamics. Yogesh Virkar University of Colorado, Boulder.

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1 The Hamiltonian Mean Field Model: Effect of Network Structure on Synchronization Dynamics Yogesh Virkar University of Colorado, Boulder. 1

2 Collaborators Prof. Juan G. Restrepo Prof. James D. Meiss Department of Applied Mathematics, University of Colorado, Boulder. 2

3 Kuramoto Model 1 Mutual synchronization: Coupled oscillators characterized by their phases All-to-all case: n =! n + K NX sin ( m n ) N m=1 K: coupling strength N: number of oscillators θ: phase ω: randomly chosen intrinsic frequency 1 Y. Kuramoto. Intl. Symp. on Math. Probs., Lecture Notes in Phys., 39,

4 Hamiltonian Mean Field Model 2 Version of Kuramoto model that conserves energy H = 1 X p 2 K X n A nm cos ( m n ) A nm = A mn 2 2N n Kinetic Energy n = p n n,m Potential Energy ṗ n = K N NX A nm sin ( m n ) m=1 2 Antoni, M. and S. Ruffo. Phys. Rev. E. 52,

5 Hamiltonian Mean Field Model (2) Link to video This video shows the evolution of phase-space for the HMF model for a high coupling constant. 5

6 Hamiltonian Mean Field Model (3) Recent work has focused on partial extensions to HMF for the network case: Chavanis, P., J. Vatteville and F. Bouchet. Eur. Phys. J. B., 46, Ciani, A., D. Fanelli and S. Ruffo. Long-range Interactions, Stochasticity and Fractional Dynamics. Springer, Nigris, S. and X. Leoncini. Phys. Rev. E, 88, In this work, we develop a more general framework. Virkar, Y. S., J. G. Restrepo and J. D. Meiss. Hamiltonian mean field model: Effect of network structure on synchronization dynamics. Phys. Rev. E 92, (2015). 6

7 Global order parameter Desynchronized state (small K) Synchronized state (large K) R n e i n = 1 N NX m=1 A nm e i m R = 1 hdi NX n=1 R n 7

8 Global order parameter as a function of coupling strength R K c K 8

9 Critical coupling constant K c For initial conditions: θ n is drawn from a uniform distribution. p n is drawn from a Gaussian distribution with mean μ and standard deviation σ, we get from theory (linear stability analysis 3 ), K c = 2 2 N λ is the principal eigenvalue of the adjacency matrix. 3 Strogatz S. and R. Mirollo, J. Stat. Phys., 63,

10 Numerical example in simulated network We construct networks with degree distribution P(d) = C d -α. R t α Scale-free K 10

11 Numerical example in simulated network (2) R t α Scale-free? λk 11

12 Synchronized solutions Assuming that in the synchronized state all rotors rotate with a common frequency, and letting r n denote the time average of R n, we obtain the following equations which can be solved numerically, r n = 1 N NX m=1 2 = K N Kr m2 I 1 A nm Kr I m2, 0 NX n=1 r n I 1 I 0 Kr n 2 Kr n 2 I 1 and I 0 are bessel functions. 12

13 Result: Theoretical vs Simulated (R vs K) Erdös-Renyi random network: N nodes with q the probability of connecting any two nodes. R t simulated data theory Erdös-Renyi K 13

14 Result: Theoretical vs Simulated (R vs K) R t simulated data theory Scale-free K 14

15 Result: Maximum synchrony R t simulation theory increasing homogeneity! 15

16 Conclusions Network heterogeneity impacts both the onset of synchronization (K c ) and the path to synchrony. We developed a theory for quantifying the onset of synchronization and the synchronized state for the network version of HMF. The maximum possible synchrony is however fairly independent of network heterogeneity. 16

17 Thanks! J References: Antoni, M. and S. Ruffo. Phys. Rev. E. 52, Barre, J. et. al. Physica A. 365, Capma, A., A. Giansanti and G. Morelli. Phys. Rev. E. 76, Dauxios, T., et. al. Dynamics and thermodynamics of Systems with Long-Range Interactions, Springer,