Kuramoto oscillators with inertia: A fast-slow dynamical systems approach
Authors:
Young-Pil Choi, Seung-Yeal Ha, Sungeun Jung and Marshall Slemrod
Journal:
Quart. Appl. Math. 73 (2015), 467-482
MSC (2010):
Primary 91G80, 97M30
DOI:
https://doi.org/10.1090/qam/1380
Published electronically:
June 18, 2015
MathSciNet review:
3400753
Full-text PDF Free Access
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Abstract: We present a fast-slow dynamical systems theory for a Kuramoto type model with inertia. The fast part of the system consists of $N$-decoupled pendulum equations with constant friction and torque as the phase of individual oscillators, whereas the slow part governs the evolution of order parameters that represent the amplitude and phase of the centroid of the oscillators. In our new formulation, order parameters serve as orthogonal observables in the framework of Artstein-Kevrekidis-Slemrod-Titi’s unified theory of singular perturbation. We show that Kuramoto’s order parameters become stationary regardless of the coupling strength. This generalizes an earlier result (Ha and Slemrod (2011)) for Kuramoto oscillators without inertia.
References
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- V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR 0121520
- Robert E. O’Malley Jr., Singular perturbation methods for ordinary differential equations, Applied Mathematical Sciences, vol. 89, Springer-Verlag, New York, 1991. MR 1123483
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- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044
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- Steven H. Strogatz, Norbert Wiener’s brain waves, Frontiers in mathematical biology, Lecture Notes in Biomath., vol. 100, Springer, Berlin, 1994, pp. 122–138. MR 1348659, DOI https://doi.org/10.1007/978-3-642-50124-1_7
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- H. A. Tanaka, A. J. Lichtenberg, and S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses, Physica D 100 (1997), 279-300.
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
- A. N. Tihonov, On systems of differential equations containing parameters, Mat. Sbornik N.S. 27(69) (1950), 147–156 (Russian). MR 0036902
- Michel Valadier, A course on Young measures, Rend. Istit. Mat. Univ. Trieste 26 (1994), no. suppl., 349–394 (1995). Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1993). MR 1408956
- S. Watanabe and J. W. Swift, Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance, J. Nonlinear Sci. 7 (1997), no. 6, 503–536. MR 1474640, DOI https://doi.org/10.1007/s003329900038
- S. Watanabe, and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D 74 (1994), 197-253.
- K. Wiesenfeld, R. Colet, and S. H. Strogatz, Synchronization transitions in a disordered Josephson series arrays, Phys. Rev. Lett. 76 (1996), 404-407.
- K. Wiesenfeld, R. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: connection with the Kuramoto model, Phys. Rev. E 57 (1988), 1563-1569.
- K. Wiesenfeld and J. W. Swift, Averaged equations for Josephson junction series arrays, Phys. Review E 51 (1995), 1020-1025.
- L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. Foreword by Wendell H. Fleming. MR 0259704
- L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, Comtes Rendus de la Societe des Sciences et des Lettres de Varsovie, Classe III, 30 (1937), pp. 212-234.
References
- J. A. Acebrón, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77 (2005), 137-185.
- J. A. Acebrón, L. L. Bonilla, and R. Spigler, Synchronization in populations of globally coupled oscillators with inertial effects, Phys. Rev. E (3) 62 (2000), no. 3, 3437–3454. MR 1788951 (2002e:34059), DOI https://doi.org/10.1103/PhysRevE.62.3437
- J. A. Acebrón and R. Spigler, Adaptive frequency model for phase-frequency synchronization in large populations of globally coupled nonlinear oscillators. Phys. Rev. Lett. 81 (1998), 2229-2332.
- Zvi Artstein, Ioannis G. Kevrekidis, Marshall Slemrod, and Edriss S. Titi, Slow observables of singularly perturbed differential equations, Nonlinearity 20 (2007), no. 11, 2463–2481. MR 2361254 (2008j:34087), DOI https://doi.org/10.1088/0951-7715/20/11/001
- Zvi Artstein and Alexander Vigodner, Singularly perturbed ordinary differential equations with dynamic limits, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 3, 541–569. MR 1396278 (97g:34073), DOI https://doi.org/10.1017/S0308210500022903
- J. M. Ball, A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice, 1988) Lecture Notes in Phys., vol. 344, Springer, Berlin, 1989, pp. 207–215. MR 1036070 (91c:49021), DOI https://doi.org/10.1007/BFb0024945
- Neil J. Balmforth and Roberto Sassi, A shocking display of synchrony, Bifurcations, patterns and symmetry. Phys. D 143 (2000), no. 1-4, 21–55. MR 1783383 (2001g:82007), DOI https://doi.org/10.1016/S0167-2789%2800%2900095-6
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968. MR 0233396 (38 \#1718)
- Young-Pil Choi, Seung-Yeal Ha, and Seok-Bae Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Phys. D 240 (2011), no. 1, 32–44. MR 2740100 (2012b:34093), DOI https://doi.org/10.1016/j.physd.2010.08.004
- Young-Pil Choi, Seung-Yeal Ha, Sungeun Jung, and Yongduck Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D 241 (2012), no. 7, 735–754. MR 2897541, DOI https://doi.org/10.1016/j.physd.2011.11.011
- B. C. Daniels, S. T. Dissanayake and B. R. Trees, Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model. Phys. Rev. E. 67 (2003), 026216.
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2005. MR 2169977 (2006d:35159)
- Florian Dörfler and Francesco Bullo, On the critical coupling for Kuramoto oscillators, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 3, 1070–1099. MR 2837521 (2012i:34043), DOI https://doi.org/10.1137/10081530X
- G. Bard Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol. 22 (1985), no. 1, 1–9. MR 802731 (86m:92010), DOI https://doi.org/10.1007/BF00276542
- Seung-Yeal Ha, Sungeun Jung, and Marshall Slemrod, Fast-slow dynamics of planar particle models for flocking and swarming, J. Differential Equations 252 (2012), no. 3, 2563–2579. MR 2860630 (2012k:34119), DOI https://doi.org/10.1016/j.jde.2011.09.014
- Seung-Yeal Ha and Marshall Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations 251 (2011), no. 10, 2685–2695. MR 2831709 (2012h:37157), DOI https://doi.org/10.1016/j.jde.2011.04.004
- H. Hong, M. Y. Choi, J. Yi, and K.-S. Soh, Inertial effects on periodic synchronization in a system of coupled oscillators, Phys. Rev. E. 59 (1999), 353-363.
- H. Hong, Gun Sang Jeon, and M. Y. Choi, Spontaneous phase oscillation induced by inertia and time delay, Phys. Rev. E (3) 65 (2002), no. 2, 026208, 5. MR 1908304, DOI https://doi.org/10.1103/PhysRevE.65.026208
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432 (87e:92054)
- Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators (Kyoto Univ., Kyoto, 1975),, Lecture Notes in Phys., 39, Springer, Berlin, 1975, pp. 420–422. MR 0676492 (58 \#32705)
- Y. Kuramoto and I. Nishikawa, Onset of collective rhythms in large populations of coupled oscillators, in Cooperative dynamics in complex physical systems, edited by H. Takayama. 300-306 (1988).
- M. Levi, F. C. Hoppensteadt, and W. L. Miranker, Dynamics of the Josephson junction, Quart. Appl. Math. 36 (1978/79), no. 2, 167–198. MR 0484023 (58 \#3972)
- V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR 0121520 (22 \#12258)
- Robert E. O’Malley Jr., Singular perturbation methods for ordinary differential equations, Applied Mathematical Sciences, vol. 89, Springer-Verlag, New York, 1991. MR 1123483 (92i:34071)
- K. Park and M. Y. Choi, Synchronization in networks of superconducting wires, Phys. Rev. B 56 (1997), 387-394.
- Arkady Pikovsky, Michael Rosenblum, and Jürgen Kurths, Synchronization, A universal concept in nonlinear sciences. Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. MR 1869044 (2002m:37001)
- J. A. Sanders and F. Verhulst, Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences, vol. 59, Springer-Verlag, New York, 1985. MR 810620 (87d:34065)
- Marshall Slemrod, Averaging of fast-slow systems, Coping with complexity: model reduction and data analysis, Lect. Notes Comput. Sci. Eng., vol. 75, Springer, Berlin, 2011, pp. 1–7. MR 2757569 (2012c:34170), DOI https://doi.org/10.1007/978-3-642-14941-2_1
- J. J. Stoker, Nonlinear vibrations in mechanical and electrical systems, Interscience Publishers, Inc., New York, N.Y., 1950. MR 0034932 (11,666a)
- Steven H. Strogatz, Norbert Wiener’s brain waves, Frontiers in mathematical biology, Lecture Notes in Biomathematics, 100, Springer-Verlag, 1994, pp. 122–138. MR 1348659, DOI https://doi.org/10.1007/978-3-642-50124-1_7
- Steven H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382 (2001g:82008), DOI https://doi.org/10.1016/S0167-2789%2800%2900094-4
- H. A. Tanaka, A. J. Lichtenberg, and S. Oishi, First order phase transitions resulting from finite inertia in coupled oscillators systems, Phys. Rev. Lett. 78 (1997), 2104.
- H. A. Tanaka, A. J. Lichtenberg, and S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses, Physica D 100 (1997), 279-300.
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass., 1979, pp. 136–212. MR 584398 (81m:35014)
- A. N. Tihonov, On systems of differential equations containing parameters, Mat. Sbornik N.S. 27(69) (1950), 147–156 (Russian). MR 0036902 (12,181d)
- Michel Valadier, A course on Young measures, Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1993), Rend. Istit. Mat. Univ. Trieste 26 (1994), suppl., 349–394 (1995). MR 1408956 (97k:28009)
- S. Watanabe and J. W. Swift, Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance, J. Nonlinear Sci. 7 (1997), no. 6, 503–536. MR 1474640 (98f:34051), DOI https://doi.org/10.1007/s003329900038
- S. Watanabe, and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D 74 (1994), 197-253.
- K. Wiesenfeld, R. Colet, and S. H. Strogatz, Synchronization transitions in a disordered Josephson series arrays, Phys. Rev. Lett. 76 (1996), 404-407.
- K. Wiesenfeld, R. Colet, and S. H. Strogatz, Frequency locking in Josephson arrays: connection with the Kuramoto model, Phys. Rev. E 57 (1988), 1563-1569.
- K. Wiesenfeld and J. W. Swift, Averaged equations for Josephson junction series arrays, Phys. Review E 51 (1995), 1020-1025.
- L. C. Young, Lectures on the calculus of variations and optimal control theory, foreword by Wendell H. Fleming, W. B. Saunders Co., Philadelphia, 1969. MR 0259704 (41 \#4337)
- L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, Comtes Rendus de la Societe des Sciences et des Lettres de Varsovie, Classe III, 30 (1937), pp. 212-234.
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Additional Information
Young-Pil Choi
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
Email:
young-pil.choi@imperial.ac.uk
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Sungeun Jung
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
Email:
mirinsil@naver.com
Marshall Slemrod
Affiliation:
Department of Mathematics, University of Wisconsin-Madison – and – Department of Mathematics, Weizmann Institute of Science, Rehobot 76100, Israel
MR Author ID:
163635
Email:
slemrod@math.wisc.edu
Keywords:
Fast-slow dynamical system,
Kuramoto oscillators,
mean-field limit,
order parameter,
synchronization
Received by editor(s):
August 19, 2013
Published electronically:
June 18, 2015
Additional Notes:
The work of the second author was partially supported by KRF-2011-0015388, and the work of the fourth author was supported by KMRS-KAIST
Article copyright:
© Copyright 2015
Brown University