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

For citation:

Kuznetsov A. P., Seliverstova E. S., Trubetskov D. I., Turukina L. V. Phenomenon of the van der pol equation. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 4, pp. 3-42. DOI: 10.18500/0869-6632-2014-22-4-3-42

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon trubeckov_di_pnd_4_2014.pdf
(downloads: 977)
Language: 
Russian
Heading: 
Review of Actual Problems of Nonlinear Dynamics
Article type: 
Review
UDC: 
517.91, 517.938, 51.73
DOI: 
10.18500/0869-6632-2014-22-4-3-42

Phenomenon of the van der pol equation

Autors: 
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Seliverstova Ekaterina Sergeevna, Saratov State University
Trubetskov Dmitriy Ivanovich, Saratov State University
Turukina L. V., Saratov State University
Abstract: 

This review is devoted to the famous Dutch scientist Balthasar van der Pol, who made a significant contribution to the development of radio­engineering, physics and mathematics. The review outlines only one essential point of his work, associated with the equation that bears his  name, and has a surprisingly wide range of applications in natural sciences. In this review we discuss the following matters.
• The biography of van der Pol, history of his equation and supposed precursors.
• The contribution of A.A. Andronov in the theory of self­oscillations.
• Van der Pol equation and modeling of processes in the human body (the mode of the heart beat and of the «heart–vessels» system; modeling of processes in the large intestine; models of excitatory and inhibitory neural interactions; modeling synchronization in processing and transfer of information in neural networks; various problems related to human musculoskeletal apparatus; modeling the vocal cords).
• Development and modifications of the van der Pol equation.

Key words: 
Van der Pol equation
oscillations
biophysics
vacuum­tube oscillator
Neuron
synchronization
Reference: 
  1. Ginoux JM, Letellier C. Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept. Chaos. 2012;22(2):023120. DOI: 10.1063/1.3670008.
  2. Cartwright ML. Balthazar Van der Pol. J. London Math. Soc. 1960;35:367—376.
  3. Van der Pol B. Nonlinear Theory of Electrical Oscillations. Moscow: State Communications Technology Publishing House; 1935. 42 p. (in Russian).
  4. Rabinovich MI, Trubetskov DI. Oscillations and Waves in Linear and Nonlinear Systems. Netherlands: Springer; 1986. 578 p. DOI: 10.1007/978-94-009-1033-1.
  5. Landa PS. Self-Oscillations in Systems with a Finite Number of Degrees of Freedom. Moscow: Nauka; 1980. 360 p. (in Russian).
  6. Goryachenko VD. Andronov Alexander Alexandrovich. Nizhni Novgorod: NNUP; 2001. 80 p. (in Russian).
  7. Trubetskov DI. Synchronization: Scientist and Time. Lectures at Schools «Nonlinear Days in Saratov for the Young». Vol. 2. Saratov: «College»; 2006. 112 p. (in Russian).
  8. Andronov AA, Witt AA, Khaikin SE. Theory of Oscillators. Oxford: Pergamon Press; 1966. 961 p.
  9. Feinberg EL. The forefather (about Leonid Isaakovich Mandelstam). Physics-Uspekhi. 2002;45(1):81—100. DOI: 1070/PU2002v045n01ABEH001126.
  10. Fabrikant VA. About L. I. Mandelstam. Academician Mandelstam. To the 100th Anniversary of Birth. Мoscow: Nauka; 1979. 234 p. (in Russian).
  11. Filippov AT. Multi-Facial Soliton. Moscow: Nauka; 1986. 230 p. (in Russian).
  12. Cveticanin L. On the Van der Pol oscillator: An overview. Applied Mechanics and Materials. 2013;430:3—13. DOI: 10.4028/www.scientific.net/AMM.430.3.
  13. Kuang YC, Biernacki PD, Lahrichi A, Mickelson A. Analysis of an experimental technique for determining Van der Pol parameters of a transistor oscillator. IEEE Transactions on Microwave Theory and Techniques. 1998;46(7):914—922. DOI: 10.1109/22.701443.
  14. Van der Pol B. On relaxation-oscillations. Philosophical Magazine & Journal of Science. 1926;2(11):978—992. DOI: 10.1080/14786442608564127.
  15. Van der Pol B, van der Mark J. The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Philosophical Magazine & Journal of Science. 1928;6(38):763—775. DOI: 10.1080/14786441108564652.
  16. Privezencev AP, Sablin NI, Filippenko NM, Fomenko GP. Nonlinear vibrations of the virtual cathode in the triode system. Journal of Communications Technology and Electronics. 1992;37(7):1242 (in Russian).
  17. Magda II, Pashchenko AV, Romanov SS. To the theory of beam feedback in generators with a virtual cathode. Problems of atomic science and technology. Plasma Electronics and New Methods of Acceleration . 2003;(4):167—170 (in Russian).
  18. Sze H, Price D, Harteneck B. Phase locking of two strongly coupled vircators. J. Appl. Phys. 1990;67(5):2278—2282. DOI: 10.1063/1.345521.
  19. Repin BG, Dubinov AE. Phasing of three vircators simulated in terms of coupled van der Pol oscillators. Tech. Phys. 2006;51(4):489–494. DOI: 10.1134/S1063784206040153.
  20. Liao P, York RA. A new phase-shifterless beam-scanning technique using arrays of coupled oscillators. IEEE transactions on microwave theory and techniques. 1993;41(10):1810—1815. DOI: 10.1109/22.247927.
  21. Yabuno H, Kaneko H, Kuroda M, Kobayashi T. Van der Pol type self-excited micro-cantilever probe of atomic force microscopy. Nonlinear Dyn. 2008;54(1):137—149. DOI: 10.1007/s11071-008-9339-1.
  22. Menzel KO, Bockwoldt T, Arp O, Piel A. Modeling dust-density wave fields as a system of coupled van der Pol oscillators. IEEE Transactions on Plasma Science. 2013;41(4):735—739. DOI: 10.1109/TPS.2012.2227497.
  23. Miwadinou CH, Hinviy LA, Monwanou AV, Chabi Orou JB. Nonlinear dynamics of plasma oscillations modeled by a forced modified Van der Pol–Duffing oscillator. arXiv: 1308.6132. arXiv Preprint; 2013.
  24. Klinger T, Greiner F, Rohde A, Piel A. Van der Pol behavior of relaxation oscillations in a periodically driven thermionic discharge. Phys. Rev. E. 1995;52(4):4316—4327. DOI: 10.1103/PhysRevE.52.4316.
  25. Klinger T, Piel A, Seddighi F, Wilke C. Van der Pol Dynamics of ionization waves. Phys. Lett. A. 1993;182(23):312—318.
  26. Gembarzhevskii GV. Electric-discharge effect in a plasma wake flow: Pulsation energy redistribution toward lower frequencies. Tech. Phys. Lett. 2009;35(3):241–244. DOI: 10.1134/S1063785009030146
  27. Lashinsky H, Rosenberagn J, Detrick L. Power line radiation: Possible evidence of van der Pol oscillations in the magnetosphere. Geophysical Research Letters. 1980;7(10):837—840. DOI: 10.1029/GL007i010p00837.
  28. Sun ZZ, Sun Y, Wang XR, Cao JP, Wang YP, Wang YQ. Self-sustained current oscillations in superlattices and the van der Pol equation. Applied Physics Letters. 2005;87(18):182110. DOI: 10.1063/1.2126149.
  29. Tony EL, Sadeghpour HR. Quantum synchronization of quantum van der Pol oscillators with trapped ions. Phys. Rev. Lett. 2013;111(23):234101. DOI: 10.1103/PhysRevLett.111.234101.
  30. Lamb W. Theory of Optical Nasers. In: Quantum Optics and Quantum Radiophysics. Moscow: Mir; 1966. P. 281 (in Russian).
  31. Pampalon E, Lapucci A. Locking-range analysis for three coupled lasers. Optics letters. 1993;18(22):1881—1883. DOI: 10.1364/OL.18.001881.
  32. Khibnik AI, Braiman Y, Kennedy TAB, Wiesenfeld K. Phase model analysis of two lasers with injected field. Physica D. 1998;111(1—4):295—310. DOI: 10.1016/S0167-2789(97)80017-6.
  33. Braiman Y, Kennedy TAB, Wiesenfeld K, Khibnik AI. Entrainment of solid-state laser arrays. Phys. Rev. A. 1995;52(2):1500—1506. DOI: 10.1103/physreva.52.1500 .
  34. Khibnik AI, Braiman Y, Protopopescu V, Kennedy TAB, Wiesenfeld K. Amplitude dropout in coupled lasers. Phys. Rev. A. 2000;62(6):063815. DOI: 10.1103/PhysRevA.62.063815.
  35. Glova AF. Phase locking of optically coupled lasers. Quantum Electronics. 2003;33(4):28—306. DOI: 10.1070/QE2003v033n04ABEH002415
  36. Kaganov VI. Using wind power to prevent tropical cyclone development. Tech. Phys. Lett. 2006; 32(3):252–254. DOI: 10.1134/S1063785006030230.
  37. Skop RA, Griffin OM. A model for the vortex-excited resonant response of bluff cylinders. Journal of Sound and Vibration. 1973;27(2):225—233. DOI: 10.1016/0022-460X(73)90063-1.
  38. Facchinettia ML, Langre E, Biolley F. Vortex shedding modeling using diffusive van der Pol oscillators. Comptes Rendus Mecanique. 2002;330(7):451—456. DOI: 10.1016/S1631-0721(02)01492-4.
  39. Veskos P, Demiris Y. Developmental acquisition of entrainment skills in robot swinging using van der Pol oscillators. In: Proceedings of the Fifth International Workshop on Epigenetic Robotics: Modeling Cognitive Development in Robotic Systems Lund University Cognitive Studies. Lund University Cognitive Studies; 2005. P. 87—93.
  40. Paschenko RE, Paschenko EI, Maksyuta DV. Forming KFM fractal signals on the basis van der Pol equations. Systems of Control, Bavigation and Communication. 2009;11(3):225 (in Russian).
  41. Zaitsev VV, Zaitsev OV. Method of information protection using the algorithm for generating chaotic self-oscillations. Vestnik of Samara University. Natural Science Series. 2006;49(9):66—71 (in Russian).
  42. Lucero J, Schoentgen J. Modeling vocal fold asymmetries with coupled van der Pol oscillators. Proceedings of Meetings on Acoustics. 2013;19:060165. DOI: 10.1121/1.4798467.
  43. Long GR, Tubis A, Jones KL. Modeling synchronization and suppression of spontaneous otoacoustic emissions using Van der Pol oscillators: Effects of aspirin administration. J. Acoust. Soc. Am. 1991;89(3):1201—1212. DOI: 10.1121/1.400651.
  44. Dutra MS, de Pina Filho AC, Romano VF. Modeling of a bipedal locomotor using coupled nonlinear oscillators of Van der Pol. Biol. Cybern. 2003;88(4):286—292. DOI: 10.1007/s00422-002-0380-8.
  45. Buldakov NS, Samochetova NS, Sitnikov AV, Suyatinov SI. Modeling of connections in the «heart-vessel» system. Scientific Edition of Bauman MSTU. Science and Education. 2013;(1):123—134 (in Russain).
  46. Beek PJ, Schmidt RC, Morris AW, Sim MY, Turvey MT. Linear and nonlinear stiffness and friction in biological rhythmic movements. Biol. Cybern. 1995;73(6):499—507. DOI: 10.1007/bf00199542.
  47. Kawahara T. Coupled van der Pol oscillators – A model of excitatory and inhibitory interactions. Biol. Cybern. 1980;39(1):37—43. DOI: 10.1007/bf00336943.
  48. Linkens DA, Taylor I, Duthie HL. Mathematical modeling of the colorectal myo-electrical activity in humans. IEEE Transactions on Biomedical Engineering. 1976;BME-23(2):101—110. DOI: 10.1109/TBME.1976.324569.
  49. Wilson HR, Cowan JD. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 1972;12(1):1—24. DOI: 10.1016/S0006-3495(72)86068-5.
  50. Osipov GV. Synchronization Processing and Transmitting Information in Neural Networks. Teaching materials. Nizhny Novgorod; 2007. 99 p. (in Russian).
  51. Kuznetsov AP, Kuznetsov SP, Ryskin NM. Nonlinear Oscillations. Moscow: Fizmatlit; 2002. 292 p. (in Russian).
  52. Seliverstova ES. On the question of two self-exciting oscillation models in non-physical systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2013;21(3):112—118 (in Russian). DOI: 10.18500/0869-6632-2013-21-3-112-118.
  53. Khokhlov RV. To the theory of capture with a small amplitude of external force. Sov. Phys. Dokl. 1954;97(3):411—414 (in Russian).
  54. Pikovsky A, Rosenblum M, Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge: Cambridge University Press; 2001. 432 p.
  55. Balanov AG, Janson NB, Postnov DE, Sosnovtseva O. Synchronization: From Simple to Complex. Berlin: Springer; 2009. 426 p. DOI: 10.1007/978-3-540-72128-4.
  56. Rand R, Holmes PJ. 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.
  57. Ivanchenko M, Osipov G, Shalfeev V, 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.
  58. Adler RA. A study of locking phenomena in oscillators. Proc. IRE. 1946;34(34):351—357. DOI: 10.1109/JRPROC.1946.229930.
  59. Kuznetsov AP, Stankevich NV, Turukina LV. Coupled van der pol and van der Pol–Duffing oscillators: dynamics of phase and computer simulation. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(4):101—136 (in Russian). DOI: 10.18500/0869-6632-2008-16-4-101-136.
  60. Kuznetsov AP, Stankevich NV, Turukina LV. Coupled van der Pol–Duffing oscillators: Phase dynamics and structure of synchronization tongues. Physica D. 2009;238(14):1203—1215. DOI: 10.1016/j.physd.2009.04.001.
  61. 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.
  62. Kuznetsov AP, Roman JP. Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol–Duffing oscillators. Broadband synchronization. Physica D. 2009;238(16):1499—1506. DOI: 10.1016/j.physd.2009.04.016.
  63. Kuznetsov AP, Roman JP, Seleznev EP. Synchronization in coupled self­sustained oscillators with non­-identical parameters. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(2):62—78 (in Russian). DOI: 10.18500/0869-6632-2010-18-2-62-78.
  64. Astakhov V, Koblyanskii S, Shabunin A, Kapitaniak T. Peculiarities of the transitions to synchronization in coupled systems with amplitude death. Chaos. 2011;21(2):023127. DOI: 10.1063/1.3597643.
  65. Baesens С, Guckenheimer J, Kim S, MacKay RS. Three coupled oscillators: mode locking, global bifurcations and toroidal chaos. Physica D. 1991;49(3):387—475. DOI: 10.1016/0167-2789(91)90155-3.
  66. Emelianova YP, Kuznetsov AP, Sataev IR, Turukina LV. Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators. Physica D. 2013;244(1):36—49. DOI: 10.1016/j.physd.2012.10.012.
  67. Roman JP, Kuznetsov AP, Turukina LV. Dynamics of three coupled van der Pol oscillators with non-identical controlling parameters. Izvestiya VUZ. Applied Nonlinear Dynamics. 2011;19(5):76—90 (in Russian). DOI: 10.18500/0869-6632-2011-19-5-76-90.
  68. Kuznetsov AP, Kuznetsov SP, Sataev IR, Turukina LV. About Landau–Hopf scenario in a system of coupled self-oscillators. Phys. Lett. A. 2013;377(45—48):3291—3295. DOI: 10.1016/j.physleta.2013.10.013.
  69. Emelianova YP, Kuznetsov AP, Turukina LV, Sataev IR, Chernyshov NY. A structure of the oscillation frequencies parameter space for the system of dissipatively coupled oscillators. Communications in Nonlinear Science and Numerical Simulation. 2014;19(4):1203—1212. DOI: 10.1016/j.cnsns.2013.08.004.
  70. Mendelowitz L, Verdugo A, Rand R. Dynamics of three coupled limit cycle oscillators with application to artificial intelligence. Communications in Nonlinear Science and Numerical Simulation. 2009;14(1):270—283. DOI: 10.1016/j.cnsns.2007.08.009.
  71. Rand R, Wong J. Dynamics of four coupled phase-only oscillators. Communications in Nonlinear Science and Numerical Simulation. 2008;13(3):501—507. DOI: 10.1016/j.cnsns.2006.06.013.
  72. Hong H, Strogatz SH. Kuramoto model of coupled oscillators with positive and negative coupling parameters: An example of conformist and contrarian oscillators. Phys. Rev. Lett. 2011;106(5):054102. DOI: 10.1103/PhysRevLett.106.054102.
  73. Hong H, Strogatz SH. Mean-field behavior in coupled oscillators with attractive and repulsive interactions. Phys. Rev. E. 2012;85(5):056210. DOI: 10.1103/PhysRevE.85.056210.
  74. Borgers C, Kopell N. Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity. Neural Computation. 2003;15(3):509—538. DOI: 10.1162/089976603321192059.
  75. Rompala K, Rand R, Howland H. Dynamics of three coupled van der Pol oscillators with application to circadian rhythms. Communications in Nonlinear Science and Numerical Simulation. 2007;12(5):794—803. DOI: 10.1016/j.cnsns.2005.08.002.
  76. Kuznetsov AP, Turukina LV, Chernyschov NY. Dynamics and synchronization of the three coupled oscillators with reactive type of coupling. Russian Journal of Nonlinear Dynamics. 2013;9(1):11—25 (in Russian). DOI: 10.20537/nd1301002.
  77. Pikovsky A, Rosenau P. Phase compactons. Physica D. 2006;218(1):56—69. DOI: 10.1016/j.physd.2006.04.015.
  78. Topaj D, Pikovsky A. Reversibility vs. synchronization in oscillator lattices. Physica D. 2002;170(2):118—130. DOI: 10.1016/S0167-2789(02)00536-5.
  79. Bridge J, Rand R, Sah SM. Dynamics of a ring network of phase-only oscillators. Communications in Nonlinear Science and Numerical Simulation. 2009;14(11):3901—3913. DOI: 10.1016/j.cnsns.2008.08.011.
Received: 
11.07.2014
Accepted: 
11.07.2014
Published: 
31.12.2014
Short text (in English):
PDF icon a2014no4p003.doc_.pdf
(downloads: 131)
  • 2778 reads