Comparative dynamics of chains of coupled van der Pol equations and coupled systems of van der Pol equations

S. A. Kaschenko

Teoretičeskaâ i matematičeskaâ fizika, Tome 207 (2021) no. 2, pp. 277-292

Résumé

We consider chains of van der Pol equations closed into a ring and chains of systems of two first-order van der Pol equations. We assume that the couplings are homogeneous and the number of chain elements is sufficiently large. We naturally realize a transition to functions depending continuously on the spatial variable. As $t\to\infty$, we study the behavior of all solutions of such chains with initial conditions sufficiently small in the norm. We identify critical cases in the stability problem and show that they all have an infinite dimension. We construct special nonlinear boundary value problems of parabolic type without small parameters, which play the role of normal forms. Their local dynamics determines the behavior of solutions of the original boundary value problems with two spatial variables. We formulate conditions under which the dynamical properties of both chains are close to each other. We establish that in several cases, the dynamics of chains of systems of van der Pol equations turns out to be essentially more complicated and diverse compared with the dynamics of chains of second-order van der Pol equations.

Détail

Keywords: nonlinear dynamics, stability, van der Pol system, asymptotic solution.

Bibliographie

[1] G. Heinrich, M. Ludwig, J. Qian, B. Kubala, F. Marquardt, “Collective dynamics in optomechanical arrays”, Phys. Rev. Lett., 107:4 (2011), 043603, 4 pp., arXiv: 1007.4819 | DOI

[2] M. Zhang, G. S. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, M. Lipson, “Synchronization of micromechanical oscillators using light”, Phys. Rev. Lett., 109:23 (2012), 233906, 5 pp., arXiv: 1112.3636 | DOI

[3] E. A. Martens, S. Thutupalli, A. Fourrière, O. Hallatschek, “Chimera states in mechanical oscillator networks”, Proc. Nat. Acad. Sci. USA, 110:26 (2013), 10563–10567, arXiv: 1301.7608 | DOI

[4] M. R. Tinsley, S. Nkomo, K. Showalter, “Chimera and phase-cluster states in populations of coupled chemical oscillators”, Nature Phys., 8:9 (2012), 662–665 | DOI

[5] V. Vlasov, A. Pikovsky, “Synchronization of a Josephson junction array in terms of global variables”, Phys. Rev. E, 88:2 (2013), 022908, 5 pp., arXiv: 1304.2181 | DOI

[6] T. E. Lee, H. R. Sadeghpour, “Quantum synchronization of quantum van der Pol oscillators with trapped ions”, Phys. Rev. Lett., 111:23 (2013), 234101, 5 pp., arXiv: 1306.6359 | DOI

[7] A. P. Kuznetsov, S. P. Kuznetsov, I. R. Sataev, L. V. Turukina, “About Landau–Hopf scenario in a system of coupled self-oscillators”, Phys. Lett. A, 377:45–48 (2013), 3291–3295, arXiv: 1303.5767 | DOI | MR

[8] D. Pazó, M. A. Matías, “Direct transition to high-dimensional chaos through a global bifurcation”, Europhys. Lett., 72:2 (2005), 176–182, arXiv: nlin/0407039 | DOI | MR

[9] G. V. Osipov, A. S. Pikovsky, M. G. Rosenblum, J. Kurths, “Phase synchronization effects in a lattice of nonidentical Rössler oscillators”, Phys. Rev. E, 55:3 (1997), 2353–2361 | DOI | MR

[10] J. M. T. Thompson, H. B. Stewart, Nonlinear Dynamics and Chaos, John Wiley and Sons, Chichester, 1986 | MR

[11] E. Simonotto, M. Riani, C. Seife, M. Roberts, J. Twitty, F. Moss, “Visual perception of stochastic resonance”, Phys. Rev. Lett., 78:6 (1997), 1186–1189 | DOI

[12] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer Series in Synergetics, 19, Springer, Berlin, 1984 | DOI | MR

[13] V. S. Afraimovich, V. I. Nekorkin, G. V. Osipov, V. D. Shalfeev, Stability, Structures and Chaos in Nonlinear Synchronization Networks, World Scientific Series on Nonlinear Science. Ser. A: Monographs and Treatises, 6, World Sci., Singapore, 1994 | MR

[14] A. S. Pikovsky, M. G. Rosenblum, J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, 12, Cambridge Univ. Press, Cambridge, 2001 | DOI | MR

[15] G. V. Osipov, J. Kurths, C. Zhou, Synchronization in Oscillatory Networks, Springer, Berlin, 2007 | DOI | MR

[16] M. M. Vainberg, “Integro-differentsialnye uravneniya”, Itogi nauki. Ser. Matem. anal. Teor. veroyatn. Regulir. 1962, VINITI, M., 1964, 5–37 | MR | Zbl

[17] Yu. M. Daletskii, M. G. Krein, Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970 | MR | Zbl

[18] S. S. Orlov, Obobschennye resheniya integro-differentsialnykh uravnenii vysokikh poryadkov v banakhovykh prostranstvakh, Izd-vo Irkut. gos. un-ta, Irkutsk, 2014

[19] I. S. Kashchenko, S. A. Kashchenko, “Dynamics of the Kuramoto equationwith spatially distributed control”, Commun. Nonlinear Sci. Numer. Simul., 34 (2016), 123–129 | DOI | MR

[20] S. A. Kaschenko, “O kvazinormalnykh formakh dlya parabolicheskikh uravnenii s maloi diffuziei”, Dokl. AN SSSR, 299:5 (1988), 1049–1052 | MR | Zbl

[21] S. A. Kaschenko, “Normalization in the systems with small diffusion”, Internat. J. Bifur. Chaos Appl. Sci. Eng., 6:6 (1996), 1093–1109 | DOI | MR

[22] S. A. Kaschenko, “Dinamika sistem s zapazdyvaniem i bystro ostsilliruyuschimi koeffitsientami”, Differents. uravneniya, 54:1 (2018), 15–29 | DOI | DOI

[23] S. A. Kaschenko, “Bifurkatsii v uravnenii Kuramoto–Sivashinskogo”, TMF, 192:1 (2017), 23–40 | DOI | DOI | MR

[24] S. A. Kaschenko, “Asimptotiki regulyarnykh reshenii v zadache Kamassa–Kholma”, Zhurn. vychisl. matem. i matem. fiz., 60:2 (2020), 253–266 | DOI

[25] S. A. Kaschenko, “Asimptotika periodicheskikh reshenii avtonomnykh parabolicheskikh uravnenii s maloi diffuziei”, Sib. matem. zhurn., 27:6 (1986), 116–127 | DOI | MR | Zbl

[26] T. S. Akhromeeva, S. P. Kurdyumov, G. G. Malinetskii, A. A. Samarskii, Nestatsionarnye struktury i diffuzionnyi khaos, Nauka, M., 1992 | MR

[27] I. S. Kashchenko, S. A. Kashchenko, “Infinite process of forward and backward bifurcations in the logistic equation with two delays”, Nonlinear Phenom. Complex Syst., 22:4 (2019), 407–412 | DOI