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


For citation:

Leonov G. A. Effective criteria for the existence of homoclinic bifurcations in dissipative systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2005, vol. 13, iss. 3, pp. 20-26. DOI: 10.18500/0869-6632-2005-13-3-20-26

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):
(downloads: 138)
Language: 
Russian
Article type: 
Article
UDC: 
531.36

Effective criteria for the existence of homoclinic bifurcations in dissipative systems

Autors: 
Leonov Gennadij Alekseevich, Saint Petersburg State University
Abstract: 

The path bifurcation problem is formulated. The application of it for the classical result of F. Tricomi on the existence of homoclinic bifurcations in a dissipative pendulum system is discussed. The survey of results concerning to the solving of the path homoclinic bifurcation problems for Lorenz system is given.

Key words: 
Reference: 
  1. Tricomi F. Integrazione di un'equazione differenziale presentatasi in elettrotecnica. Ann. di Matem. Pura Ed Appl. 1933;2(1):1–20 (in Italian).
  2. Andronov AA, Vitt AA, Khaikin SE. Theory of Oscillators. Pergamon; 1966. 815 p. DOI: 10.1016/C2013-0-06631-5.
  3. Barbashin EA, Tabueva VA. Dynamical Systems with Cylindrical Phase Space. Moscow: Nauka; 1969. 300 p. (in Russian).
  4. Bautin NN. Qualitative study of a certain equation of the theory of phase automatic frequency control. Journal of Applied Mathematics and Mechanics. 1970;34(5):812–821. DOI: 10.1016/0021-8928(70)90063-8.
  5. Belykh VN, Nekorkin VI. Qualitative investigation of a system of three differential equations in the theory of phase synchronization. Journal of Applied Mathematics and Mechanics. 1975;39(4):615–622. DOI: 10.1016/0021-8928(75)90062-3.
  6. Belyustina LN. On one equation from the theory of electrical machines. In: Collection of Memory of A.A. Andronov. Moscow: Publishing House of the Academy of Sciences of the USSR; 1955. P. 173–186 (in Russian).
  7. Belyustina LN. Investigation of a nonlinear phase-locked loop system. Radiophys. Quantum Electron. 1959;2(2):277–291 (in Russian).
  8. Belyustina LN. On the capture band and the numerical study of point mappings in some synchronization problems. In: System Dynamics. No. 11. Gorky: GSU; 1976. P. 18 (in Russian).
  9. Belyustina LN, Brykov VV, Kiveleva KG, Shalfeev VD. On the magnitude of the locking band of a phase-shift automatic frequency control system with a proportionally integrating filter. Radiophys. Quantum Electron. 1970;13(4)437–440. DOI: 10.1007/BF01030651.
  10. Belyustina LN, Belykh VN. Qualitative study of a dynamic system on a cylinder. Differential Equations. 1973;9(3):403–415 (in Russian).
  11. Gubar’ NA. Investigation of a piecewise linear dynamical system with three parameters. Journal of Applied Mathematics and Mechanics. 1961;25(6):1519–1535. DOI: 10.1016/0021-8928(62)90132-6.
  12. Amerio L. Studio asimptotika del moto un punto su una chiusa per azione diforze independenti dal tempo. Ann. R. Scuola Norm. Sup. Piza. 1950;3(3):17–57 (in Italian).
  13. Amerio L. Determinazione della condizioni di stabilita per gli integrali di un’equazione interessante l’electrotecnica. Ann. di Matem. Pura Ed Appl. 1949;30(4):75–90 (in Italian).
  14. Hayes WD. On the equation for a damped pendulum under constant torque. Z. Ang. Math. Phys. 1953;4(5):398–401. DOI: 10.1007/BF02074983.
  15. Zeifert G. On the existence of certain solutions of nonlinear differential equations. Z. Ang. Math. Phys. 1952;3(6):468–471. DOI: 10.1007/BF02025575.
  16. Zeifert G. On stability questions for pendulum-like equations. Z. Ang. Math. Phys. 1956;7(3):238–247. DOI: 10.1007/BF02044469.
  17. Shakhgildyan VV, Belyustina LN. Phase Synchronization Systems. Moscow: Radio I Svyaz; 1982. 288 p. (in Russian).
  18. Shakhgildyan VV, Lyakhovkin AA. Phase Locking Systems. Moscow: Svyaz; 1972. 446 p. (in Russian).
  19. Yanko-Trinitsky AA. A New Method for Analyzing the Operation of Synchronous Motors Under Abruptly Varying Loads. Moscow–Leningrad: Gosenergoizdat; 1958. 103 p. (in Russian).
  20. Gupta S. Phase-locked frequency control. Proceedings of the Institute of Electrical Engineering and Radio Engineering. 1975;63(2):50 (in Russian).
  21. Stocker JJ. Nonlinear Vibrations in Mechanical and Electrical Systems. New York: Interscience; 1950. 273 p.
  22. Viterbi AJ. Principles of Coherent Communications. New York: McGraw-Hill; 1966. 321 p.
  23. Lindsey WC. Synchronization Systems in Communication and Control. New York: Prentice-Hall; 1972. 695 p.
  24. Belykh VN. On the bifurcation of the separatrices of the saddle of the Lorentz system. Differential Equations. 1984;20(10):1666–1674 (in Russian).
  25. Leonov GA, Reitman V. Attraktoreingrenzung für nichtlineare Systeme. Leipzig: Teubner; 1987. 196 p. DOI: 10.1007/978-3-322-91271-8.
  26. Leonov GA. On the estimation of the bifurcation parameters of the separatrix loop of the Lorentz system. Differential Equations. 1988;24(6):972–977 (in Russian).
  27. Leonov GA. On estimates of the bifurcation values of the parameters of a Lorentz system. Russian Math. Surveys. 1988;43(3):216–217. DOI: 10.1070/RM1988v043n03ABEH001766.
  28. Leonov GA. On the existence of homoclinic trajectories in the Lorentz system. Bulletin of St. Petersburg University. Mathematics, Mechanics, Astronomy. 1999;(1):13 (in Russian).
  29. Hastings SP, Troy WC. A shooting approach to chaos in the Lorenz equations. Journal of Differential Equations. 1996;127(1):41–53. DOI: 10.1006/jdeq.1996.0060.
  30. Chen X. Lorenz equations. Pt. 1. Existence and nonexistence of homoclinic orbits. SIAM J. Math. Analysis. 1966;27(4):1057–1069. DOI: 10.1137/S0036141094264414.
  31. Leonov GA. Bounds for attractors and the existence of homoclinic orbits in the lorenz system. Journal of Applied Mathematics and Mechanics. 2001;65(1):19–32. DOI: 10.1016/S0021-8928(01)00004-1.
  32. Leonov GA, Ponomarenko DV, Smirnova VB. Frequency-Domain Methods for Nonlinear Analysis. Theory and Applications. Singapore: World Scientific; 1996. 512 p. DOI: 10.1142/2638.
Received: 
02.06.2005
Accepted: 
02.06.2005
Published: 
31.10.2005
Short text (in English):
(downloads: 66)