Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

Nonlinearity

Volumn 25, Issue 5, 2012, Pages 1247-1273

  Giacomin, Giambattista ^a   Pakdaman, Khashayar ^b   Pellegrin, Xavier ^b  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b INSTITUT JACQUES MONOD   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84859795069     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/25/5/1247     Document Type: Article

References (29)

1. Acebrón J A, Bonilla L L, Pérez Vicente C J, Ritort F and Spigler R 2005 The Kuramoto model: a simple paradigm for synchronization phenomena Rev. Mod. Phys. 77 137-85 (Pubitemid 41490657)
2. Arnold A, Bonilla L L and Markowich P A 1996 Liapunov functionals and large-time-asymptotics of mean-field nonlinear Fokker-Planck equations Transport Theory Statist. Phys. 25 733-51 MR 1420187 (97k:82034) (Pubitemid 126328764)
3. Bates P W, Lu K and Zeng C 1997 Normally hyperbolic invariant manifolds for semiflow in a Banach space Differential Equations and Applications (Cambridge, MA: Hangzhou International Press) pp 22-9
4. Bertini L, Giacomin G and Pakdaman K 2010 Dynamical aspects of mean field plane rotators and the Kuramoto model J. Stat. Phys. 138 270-90
5. Cazenave T and Haraux A 1998 An Introduction to Semilinear Evolution Equations (Oxford Lecture Series in Mathematics and its Applications vol 13) (New York: Clarendon/Oxford University Press) (Translated from the 1990 French original by Y Martel and revised by the authors)
6. Constantin P, Kevrekidis I G and Titi E S 2004 Asymptotic states of a Smoluchowski equation Arch. Ration. Mech. Anal. 174 365-84
7. Constantin P, Titi E S and Vukadinovic J 2005 Dissipativity and Gevrey regularity of a Smoluchowski equation Indiana Univ. Math. J. 54 949-69 (Pubitemid 41421372)
8. Constantin P and Vukadinovic J 2005 Note on the number of steady states for a two-dimensional Smoluchowski equation Nonlinearity 18 441-3 (Pubitemid 40097537)
9. Eden A, Foias C, Nicolaenko B and Temam R 1994 Exponential Attractors for Dissipative Evolution Equations (RAM: Research in Applied Mathematics vol 37) (Paris: Masson)
10. Eden A, Foias C, Nicolaenko B and Temam R 1990 Ensembles inertiels pour des équations d'évolution dissipatives C. R. Acad. Sci. Paris Sér. I Math. 310 559-62
11. Foias C and Temam R 1989 Gevrey class regularity for the solutions of the Navier-Stokes equations J. Funct. Anal. 87 359-69
12. Foias C, Sell G R and Temam R 1988 Inertial manifolds for nonlinear evolutionary equations J. Diff. Eqns 73 309-53
13. Frank T D 2002 Stability analysis of mean field models described by Fokker-Planck equations Ann. Phys. 11 707-16
14. Giacomin G, Pakdaman K, Pellegrin X and Poquet C 2011 Transitions in generalized active rotators systems: invariant hyperbolic manifold approach (arXiv:1106.0758v2)
15. Hale J K 1988 Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs vol 25) (Providence, RI: American Mathematical Society)
16. Haragus M and Iooss G 2011 Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems (London: Springer) (Engl. transl)
17. Henry D 1981 Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics vol 840) (Berlin: Springer)
18. Luo C, Zhang H and Zhang P 2005 The structure of equilibrium solutions of the one-dimensional DOI equation Nonlinearity 18 379
19. Mallet-Paret J and Sell G R 1988 Inertial manifolds for reaction diffusion equations in higher space dimensions J. Am. Math. Soc. 1 805-66
20. Marvel S A, Mirollo R E and Strogatz S H 2009 Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action Chaos 19 043104
21. Ott E and Antonsen T M 2008 Low dimensional behavior of large systems of globally coupled oscillators Chaos 18 037113
22. Rosa R and Temam R 1996 Inertial manifolds and normal hyperbolicity Acta Appl. Math. 45 1-50 (Pubitemid 126706562)
23. Sell G R and You Y 2002 Dynamics of Evolutionary Equations (Applied Mathematical Sciences vol 143) (New York: Springer)
24. Silver H, Frankel N E and Ninham B W 1972 A class of mean field models J. Math. Phys. 13 468-74
25. Strogatz S H 2000 From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators Physica D 143 1-20
26. Vukadinovic J 2008 Finite-dimensional description of the long-term dynamics for the 2D Doi-Hess model for liquid crystalline polymers in a shear flow Commun. Math. Sci. 6 975-93
27. Vukadinovic J 2008 Inertial manifolds for a Smoluchowski equation on a circle Nonlinearity 21 1533-45
28. Vukadinovic J 2009 Inertial manifolds for a Smoluchowski equation on the unit sphere Commun. Math. Phys. 285 975-90
29. Vukadinovic J 2011 Global dissipativity and inertial manifolds for diffusive Burgers equations with low-wavenumber instability Discrete Contin. Dyn. Syst. 29 327-41