The mathematics of Kuramoto models which describe synchronization phenomena
Matematica, cultura e società, Série 1, Tome 1 (2016) no. 2, pp. 123-132.

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The so-called "Kuramoto models" and similar ones represent a paradigmatic way to describe a number of synchronization phenomena. These are states into which incoherent systems may go, often as it occurs in phase transition, and concern a variety of cases, ranging form Physics to Neuroscience, from Biology to Engineering and even Social Sciences. They explain, at least qualitatively, a large variety of complex processes. In this paper, we review such models and the underlying mathematics, showing some of their peculiarities.
I cosiddetti "modelli di Kuramoto", e altri simili ad essi, rappresentano un modo paradigmatico per descrivere una serie di fenomeni di sincronizzazione, cioè stati a cui possono passare sistemi incoerenti, come capita spesso nelle transizioni di fase e in una moltitudine di casi, che vanno dalla Fisica alle Neuroscienze, dalla Biologia all'Ingegneria e persino alle Scienze Sociali. Questi fenomeni spiegano, almeno qualitativamente, una grande varietà di processi complessi. In questo articolo, passiamo in rassegna tali modelli e la matematica sottostante, mostrando alcune delle loro peculiarità
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Spigler, Renato. The mathematics of Kuramoto models which describe synchronization phenomena. Matematica, cultura e società, Série 1, Tome 1 (2016) no. 2, pp. 123-132. http://geodesic.mathdoc.fr/item/RUMI_2016_1_1_2_a3/

[1] Acebrón, J.A., Bonilla, L.L., De Leo, S., and Spigler, R., "Breaking the symmetry in bimodal frequency distributions of globally coupled oscillators", Phys. Rev. E 57, May 1998, 5287-5290. | DOI | MR

[2] Acebrón, J.A. and Spigler, R., "Adaptive frequency model for phase-frequency synchronization in large populations of globally coupled nonlinear oscillators", Phys. Rev. Lett. 81, 14 September 1998, 229-2232.

[3] Acebrón, J.A. and Spigler, R., "Uncertainty in phasefrequency synchronization of large populations of globally coupled nonlinear oscillators", Phys. D 141, Nos. 1-2, July 2000, 65-79. | DOI | MR

[4] Acebrón, J.A., Bonilla, L.L., and Spigler, R., "Synchronization in populations of globally coupled oscillators with inertial effects", Phys. Rev. E 62, September 2000, 3437-3454. | DOI | MR

[5] Acebrón, J.A., Lavrentiev, M.M., Jr., and Spigler, R., "Spectral analysis and computation for the Kuramoto-Sakaguchi integroparabolic equation", IMA J. Numer. Anal. 21, no. 1 (2001), 239-263. | DOI | MR | Zbl

[6] Acebrón, J.A., Bonilla, L.L., Pérez Vicente, C.J., Ritort, F., and Spigler, R., "The Kuramoto model: a simple paradigm for synchronization phenomena", Rev. Modern Phys., 77 (2005), 137-185.

[7] Acebrón, J.A. and Spigler, R., "The TV remote control and beyond: The legacy of Robert Adler", SIAM News, Vol. 40, N. 5, June 2007, pp. 2-3.

[8] Akhmetov, D.R., Lavrentiev, M.M., Jr., and Spigler, R., "Nonlinear integroparabolic equations on unbounded domain: Existence of classical solutions with special properties", Siberian Math. J. 42 (2001), 495-516. | fulltext EuDML | DOI | MR

[9] Akhmetov, D.R., Lavrentiev, M.M., Jr., and Spigler, R., "Regularizing a nonlinear integroparabolic Fokker-Planck equation with space-periodic solutions: Existence of strong solutions", Siberian Math. J. 42 (2001), 693-714. | fulltext EuDML | DOI | MR

[10] Akhmetov, D.R., Lavrentiev, M.M., Jr., and Spigler, R., "Existence and uniqueness of classical solutions to certain nonlinear integrodifferential Fokker-Planck-type equations", Electron. J. Differential Equations, Vol. 2002 (2002), No. 24, pp. 1-17. | fulltext EuDML | MR | Zbl

[11] Akhmetov, D.R. and Spigler, R., "Uniform and optimal estimates for solutions to singularly perturbed parabolic equations", J. Evol. Equ. 7 (2007), 347-372. | DOI | MR | Zbl

[12] Bonilla, L.L., Spigler, R., and Neu, J.C., "Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators", J. Statist. Phys. 67 (1992), 313-330. | DOI | MR | Zbl

[13] Bonilla, L.L., Pérez Vicente, C.J., and Spigler, R., "Time-periodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions", Phys. D, 113 (1998), 79-97. | DOI | MR | Zbl

[14] Ermentrout, B., "An adaptive model for synchrony in the firefly Pteroptyx malaccae", J. Math. Biol. 29, no. 6 (1991), 571-585. | DOI | MR | Zbl

[15] Kuramoto, Y., "Self-entrainment of a population of coupled non-linear oscillators", International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975), pp. 420-422. Lecture Notes in Phys., 39, Springer, Berlin, 1975. | MR | Zbl

[16] Kuramoto, Y., "Chemical Oscillations, Waves and Turbulence", Springer Series in Synergetics, Springer-Verlag, Berlin, 1984. | DOI | MR | Zbl

[17] Lavrentiev, M.M., Jr., and Spigler, R., "Existence and uniqueness of solutions to the Kuramoto-Sakaguchi parabolic integrodifferential equation", Differential Integral Equations 13 (2000), 649-667. | MR | Zbl

[18] Lavrentiev, M.M., Jr., and Spigler, R., "Time-independent estimates and a comparison theorem for a nonlinear integroparabolic equation of the Fokker-Planck type", Differential Integral Equations 17 (2004), no. 5-6, 549-570. | MR | Zbl

[19] Lavrentiev, M.M., Jr., Spigler, R., and Tani, A., "Existence, uniqueness, and regularity for the Kuramoto-Sakaguchi equation with unboundedly supported frequency distribution", Differential Integral Equations, 27, No. 9-10 (2014), 879-892. | MR | Zbl

[20] Sakaguchi H., "Cooperative phenomena in coupled oscillator systems under external fields", Progr. Theor. Phys. 79, 1 (1988), 39-46. | DOI | MR

[21] Sakaguchi H. and Kuramoto, Y., "A soluble active rotator model showing phase transitions via mutual entreinment", Progr. Theor. Phys. 76, 3 (1986), 576-581. | DOI | MR

[22] Sakaguchi H., Shinimoto, S., and Kuramoto, Y., "Local and global self-entrainment in oscillator lattices", Progr. Theor. Phys. 77, 5 (1987), 1005-1010.

[23] Sartoretto, F., Spigler, R., and Pérez Vicente, C.J., "Numerical solution of the Kuramoto-Sakaguchi equation governing populations of coupled oscillators", Math. Models Methods Appl. Sci. 8 (1998), 1023-1038. | DOI | MR | Zbl

[24] Strogatz, S.H., and Mirollo, R., "Stability of incoherence in a population of coupled oscillators", J. Statist. Phys. 63, Nos. 3/4 (1991), 613-635. | DOI | MR

[25] Tanaka, H., Lichtenberg, A.J., and Oishi, S., "Selfsynchronization of coupled oscillators with hysteretic responses", Phys. D 100 (1997), 279-300. | Zbl

[26] Tanaka, H., Lichtenberg, A.J., and Oishi, S., "First order phase transition resulting from finite inertia in coupled oscillator systems", Phys. Rev. Lett. 78, N. 11 (1997), 2104-2107.

[27] Turing, A.M., "The chemical basis of morphogenesis", Philos. Trans. R. Soc. Lond., Series B, Biologcal Sciences, 237, No. 641 (1952), 37-72. | MR