ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Kuznetsov A. P., Roman J. P., Stankevich N. V., Turukina L. V. Pulsed synchronization and synchronization in coupled systems: new aspects of classical problem. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 3, pp. 88-111. DOI: 10.18500/0869-6632-2008-16-3-88-111

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Pulsed synchronization and synchronization in coupled systems: new aspects of classical problem

Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Roman Julija Pavlovna, Saratov State University
Stankevich Natalija Vladimirovna, National Research University "Higher School of Economics"
Turukina L. V., Saratov State University

Different features of the pulsed synchronization of self-oscillatory systems are considered. Namely nonisochronous, stabilization of the unstable systems, synchronization of the coupled oscillators in the region of the «oscillatory death» and etc. Illustrations for the coupled nonisochronously oscillators and nonidentical (controlling parameter and nonlinear dissipation) oscillators are presented.

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  1. Pikovsky A, Rosenblum M and Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge, UK: Cambridge University Press; 2001. 411 p. DOI: 10.1017/CBO9780511755743.
  2. Rabinovich MI, Trubetskov DI. Oscillations and Waves in Linear and Nonlinear Systems. Berlin: Springer; 1989. 578 p. DOI: 10.1007/978-94-009-1033-1.
  3. Glass L, Mackey M. From Clocks to Chaos. The Rhythms of Life. Princeton University Press; 1988. 248 p.
  4. Landa PS. Nonlinear Oscillations and Waves in Dynamical Systems. Dordrecht: Springer; 1996. 544 p. DOI: 10.1007/978-94-015-8763-1.
  5. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear Oscillations. 2 edition. Moscow: Fizmatlit; 2006. 292 p. (in Russian).
  6. Arnold VI. Cardiac arrhythmias and circle mappings. Chaos. 1991;1(1):13—19. DOI: 10.1063/1.165810.
  7. Glass L, Sun J. Periodic forcing of a limit-cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations. Phys. Rev. E. 1994;50(6):5077—5084. DOI: 10.1103/physreve.50.5077.
  8. Glass L et al. Global bifurcations of a periodically forced biological oscillator. Phys. Rev. A. 1983;29(3):1348—1357. DOI: 10.1103/PhysRevA.29.1348.
  9. Keener JP, Glass L. Global bifurcation of a periodically forced nonlinear oscillator. J. Math. Biology. 1984;21(2):175—190. DOI: 10.1007/bf00277669.
  10. Ding EJ. Analytic treatment of periodic orbit systematics for a nonlinear driven oscillator. Phys. Rev. A. 1986;34(4):3547—3550. DOI: 10.1103/PhysRevA.34.3547.
  11. Ding EJ. Analytic treatment of a driven oscillator with a limit cycle. Phys. Rev. A. 1987;35(6):2669—2683. DOI: 10.1103/physreva.35.2669.
  12. Ding EJ and Hemmer PC. Exact treatment of mode locking for a piecewise linear map. Journal of Statistical Physics. 1987;46(1–2):99—110. DOI: 10.1007/BF01010333.
  13. Ding EJ. Structure of parameter space for a prototype nonlinear oscillator. Phys. Rev. A. 1987;36(3):1488—1491. DOI: 10.1103/physreva.36.1488.
  14. Ding EJ. Structure of the parameter space for the van der Pol oscillator. Physica Scripta. 1988;38(1):9—17. DOI: 10.1088/0031-8949/38/1/001.
  15. Cecchi C, Keener JP, Glass L. Periodically kicked hard oscillators. Chaos. 1993;3(1):51—62. DOI: 10.1063/1.165978.
  16. Viana RL and Batista AM. Synchronization of coupled kicked limit cycle systems. Chaos, Solitons & Fractals. 1998;9(12):1931—1944. DOI: 10.1016/S0960-0779(98)00008-3.
  17. Ullmann K and Caldas IL. Transitions in the parameter space of a periodically forced dissipative system. Chaos, Solitons & Fractals. 1996;7(11):1913—1921. DOI: 10.1016/S0960-0779(96)00019-7.
  18. Campbell A et al. Isochrones and the dynamics of kicked oscillators. Physica A. 1989;155(3):565—584. DOI: 10.1016/0378-4371(89)90006-X.
  19. Kuznetsov AP, Tyuryukina LV. Impulsive Van der Pol oscillator: From differential equation to display. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(6):69 (in Russian).
  20. Kuznetsov AP, Tyuryukina LV. Synchronization of a self-oscillating van der Pol – Duffing system by short pulses. Izvestiya VUZ. Applied Nonlinear Dynamics. 2004;12(5):16 (in Russian).
  21. Kuznetsov AP, Tyuryukina LV. Forced synchronization in a system with unstable cycle. Tech. Phys. Lett. 2003;29(4):332–333. DOI: 10.1134/1.1573307.
  22. Kuznetsov AP, Turukina LV. Stable quasi-periodic and periodic regimes initiated by the short pulses in system with unstable limit cycle. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(1):72—81 (in Russian). DOI: 10.18500/0869-6632-2006-14-1-72-81.
  23. Ajdarova JS, Kuznetsov AP, Turukina LV. The comparative analysis of synchronization by a harmonious and pulse force by the example of lorentz system. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(4):55—67 (in Russian). DOI: 10.18500/0869-6632-2007-15-4-55-67.
  24. Kuznetsov AP, Stankevich NV, Tyuryukina LV. Features of pulsed synchronization of an autooscillatory system with a three-dimensional phase space. Tech. Phys. Lett. 2006;32(4):343–346. DOI: 10.1134/S1063785006040213.
  25. Kuznetsov AP, Stankevich NV, Turukina LV. Features of the synchronization picture by the pulses in the system with 3-dimensional phase space by the example of the Ressler system. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(6):43—53 (in Russian). DOI: 10.18500/0869-6632-2006-14-6-43-53.
  26. Kuznetsov AP, Stankevich NV, and Turukina LV. Picture of pulsed synchronization in the Dmitriev–Kislov generator. Nonlinear Phenomena in Complex Systems. 2007;10(4):407—412.
  27. Wang D, Li C, Chow SN. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press; 1994. 484 p.
  28. Kuznetsov SP. Dynamic Chaos. Moscow: Fizmatlit; 2006. 356 p. (in Russian).
  29. Schuster G. Deterministic Chaos. Wiley; 1995. 320 p.
  30. Mandelstam LI, Papaleksi ND. To the theory of asynchronous excitation. Tech. Phys. 1934;4(1):98 (in Russian).
  31. Anischenko VS. Complex Vibrations in Simple Systems. Moscow: Nauka; 1990. 312 p. (in Russian).
  32. Aronson DG, Ermentrout GB, Kopell N. Amplitude response of coupled oscillators. Physica D. 1990;41(3):403—449. DOI: 10.1016/0167-2789(90)90007-C.
  33. Rand R, Holmes P. Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mechanics. 1980;15(4—5):387—399. DOI: 10.1016/0020-7462(80)90024-4.
  34. Storti DW, Rand RH. Dynamics of two strongly coupled van der Pol oscillators. Int. J. Non-Linear Mechanics. 1982;17(3):143—152. DOI: 10.1016/0020-7462(82)90014-2.
  35. Chakraborty T, Rand RH. The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mechanics. 1988;23(5—6):369—376. DOI: 10.1016/0020-7462(88)90034-0.
  36. Poliashenko M, McKay SR, Smith CW. Chaos and nonisochronism in weakly coupled nonlinear oscillators. Phys. Rev. А. 1991;44(6):3452—3456. DOI: 10.1103/PhysRevA.44.3452.
  37. Poliashenko M, McKay SR, Smith CW. Hysteresis of synchronous – asynchronous regimes in a system of two coupled oscillators. Phys. Rev. A. 1991;43(10):5638—5641. DOI: 10.1103/PhysRevA.43.5638.
  38. Pastor I, Perez-Garcia VM, Encinas-Sanz F, Guerra JM. Ordered and chaotic behavior of two coupled van der Pol oscillators. Phys. Rev. E. 1993;48(1):171—182. DOI: 10.1103/PhysRevE.48.171.
  39. Camacho E, Rand RH, Howland H. Dynamics of two van der Pol oscillators coupled via a bath. Int. J. Solids Structures. 2004;41(8):2133—2143. DOI: 10.1016/j.ijsolstr.2003.11.035.
  40. Ivanchenko MV, Osipov GV, Shalfeev VD, Kurths J. Synchronization of two non-scalar-coupled limit-cycle oscillators. Physica D. 2004;189(1–2):8—30. DOI: 10.1016/j.physd.2003.09.035.
  41. Kuznetsov AP, Paksyutov VI. On the dynamics of two van der Pol – Duffing oscillators with dissipative coupling. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003;11(6):48—64 (in Russian).
  42. Kuznetsov AP, Paksjutov VI. Features of the parameter plane of two nonidentical coupled Van der Pol – Duffing oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(4):3—19 (in Russian. DOI: 10.18500/0869-6632-2005-13-4-3-19.
  43. Kuznetsov AP, Paksyutov VI, Roman YP. Features of the synchronization of coupled van der Pol oscillators with nonidentical control parameters. Tech. Phys. Lett. 2007;33(8):636–638. DOI: 10.1134/S1063785007080032.
  44. Kuznetsov AP, Paksjutov VI, Roman JP. Properties of synchronization in the system of nonidentical coupled van der pol and van der Pol – Duffing oscillators. Broadband synchronization. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007;15(4):3—15 (in Russian). DOI: 10.18500/0869-6632-2007-15-4-3-15.
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