Long time evolution of phase oscillator systems

Chaos

Volumn 19, Issue 2, 2009, Pages

  Ott, Edward ^a   Antonsen, Thomas M ^a  

^a UNIVERSITY OF MARYLAND   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67651092270     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3136851     Document Type: Article

References (21)

1. Y. Kuramoto, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics Vol. 139, edited by, H. Araki, (Springer-Verlag, Berlin, 1975);Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984); For reviews of the Kuramoto model, see J. A. Acebron, L. L. Bonilla, C. J. P. Vincente, F. Ritortr, and R. Spigler, Rev. Mod. Phys. 0034-6861 77, 137 (2005) 10.1103/RevModPhys.77.137; S. H. Strogatz, Physica D 0167-2789 143, 1 (2000) 10.1016/S0167-2789(00)00094-4; E. Ott, Chaos in Dynamical Systems, 2nd ed. (Cambridge University Press, Cambridge, 2002), Chap.
2. S. A. Marvel and S. H. Strogatz, Chaos 1054-1500 19, 013132 (2009). 10.1063/1.3087132
3. M. M. Abdulrehem and E. Ott, Chaos 1054-1500 19, 013129 (2009). 10.1063/1.3087434
4. B. Eckhardt, E. Ott, S. H. Strogatz, D. M. Abrams, and A. McRobie, Phys. Rev. E 1063-651X 75, 021110 (2007) 10.1103/PhysRevE.75.021110; S. H. Strogatz, D. M. Abrams, A. McRobie, B. Eckhardt, and E. Ott, Nature (London) 0028-0836 438, 43 (2005). 10.1038/43843a
5. P. So, B. C. Cotton, and E. Barreto, Chaos 1054-1500 18, 037114 (2008). 10.1063/1.2979693
6. T. M. Antonsen, R. Faghih, M. Girvan, and E. Ott, Chaos 1054-1500 18, 037112 (2008) 10.1063/1.2952447; H. Sakaguchi, Prog. Theor. Phys. 0033-068X 79, 39 (1988). 10.1143/PTP.79.39
7. L. M. Childs and S. H. Strogatz, Chaos 1054-1500 18, 043128 (2008). 10.1063/1.3049136
8. M. Choi, H. J. Kim, D. Kim, and H. Hong, Phys. Rev. E 1063-651X 61, 371 (2000) 10.1103/PhysRevE.61.371; M. K. S. Yeung and S. H. Strogatz, Phys. Rev. Lett. 0031-9007 82, 648 (1999). 10.1103/PhysRevLett.82.648
9. W. S. Lee, E. Ott, and T. M. Antonsen, e-print arXiv:0903.1372.
10. J. D. Crawford, J. Stat. Phys. 0022-4715 74, 1047 (1994). 10.1007/BF02188217
11. E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott, P. So, and T. M. Antonsen, Phys. Rev. E 1063-651X 79, 026204 (2009). 10.1103/PhysRevE.79.026204
12. E. Barreto, B. Hunt, E. Ott, and P. So, Phys. Rev. E 1063-651X 77, 036107 (2008). 10.1103/PhysRevE.77.036107
13. A. Pikovsky and M. Rosenblum, Phys. Rev. Lett. 0031-9007 101, 264103 (2008). 10.1103/PhysRevLett.101.264103
14. D. W. Abrams, R. Mirollo, S. H. Strogatz, and D. A. Wiley, Phys. Rev. Lett. 0031-9007 101, 084103 (2008). 10.1103/PhysRevLett.101.084103
15. E. Ott and T. M. Antonsen, Chaos 1054-1500 18, 037113 (2008). 10.1063/1.2930766
16. S. Watanabe and S. H. Strogatz, Physica D 0167-2789 74, 197 (1994). For Δ=0 this paper shows that resistively coupled Josephson junctions can desplay chaos, while Ref. shows that the dynamics of the order parameter on the reduced manifold of this system is two dimensional and hence cannot be chaotic. Thus, there is long time dynamics not on the reduced manifold in this system if Δ=0. 10.1016/0167-2789(94)90196-1
17. 0.
18. Equation, together with the final condition Z (z, t′ =t) =z, can be viewed as generating a conformal mapping from the complex z -plane to the complex Z -plane. Since Eq. with ṽ (Z, t′) given by Eq. is a Riccati equation, this mapping is a Möbius transformation, Z=A (z-B) / (z-C), where the coefficients A,B and C depend on t. [See, for example, Ref., Eq. (4.1.6).] Thus the unit disk in z (i.e., z 1) is mapped into a disk in Z, and since, by Eq., dρ /d t′ =Δ0 at ρ =1 for t′ t, this disk is contained within the region Z 1. Our result, Re [ (z,t)] →+∞ as t→+∞, implies that the radius of this Z -disk shrinks to zero as t→+∞.
19. E. Ott, J. H. Platig, T. M. Antonsen, and M. Girvan, Chaos 1054-1500 18, 037115 (2008) (in particular, see Appendix C). 10.1063/1.2973816
20. nθ- γ-t Δ) for Lorentzian g (ω), and that this term increases exponentially with t for t<γ but then decreases exponentially with t for t>γ. This general type of behavior is what is responsible for the "echo" phenomenon in Ref. We also note the result that f+ tends to zero even though F+ need not be similar to the behavior of the distribution function for linear, Landau damped, waves in collisionless plasmas. For example, see Ref. and S. H. Strogatz, R. E. Mirollo, and P. C. Matthews, Phys. Rev. Lett. 0031-9007 68, 2730 (1992). 10.1103/PhysRevLett.68.2730
21. E. Hille, Ordinary Differential Equations in the Complex Domain (Courier Dover, New York, 1997), p. 105.