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Qualitative study of a multidimensional phase system

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Literature Cited

  1. E. A. Barbashin and V. A. Tabueva, Dynamic Systems with Cylindrical Phase Space [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  2. Yu. N. Bakaev, “Some topics in the nonlinear theory of phase systems,” Tr. VVIA, 800 (1959).

  3. G. A. Leonov, “On asymptotically stable systems,” Sib. Mat. Zh.,15, No. 1, 49–60 (1974).

    Google Scholar 

  4. V. V. Shakhgil'dyan and A. A. Lyakhovkin, Automatic Frequency Control Phase Systems [in Russian], Svyaz', Moscow (1972).

    Google Scholar 

  5. E. I. Pavlyuk, “Study of the stability of phase synchronization systems by Lyapunov's direct method,” in: Phase Synchronization [in Russian], Svyaz', Moscow (1975), Chap. 9, pp. 123–134.

    Google Scholar 

  6. H. Poincaré, On Curves Defined by Differential Equations [Russian translation], Gostekhizdat (Classics of Science), Moscow-Leningrad (1947).

    Google Scholar 

  7. A. M. Lyapunov, General Problem on Stability of Motion [in Russian], Gostekhizdat, Moscow (1950).

    Google Scholar 

  8. V. N. Belykh and V. I. Nekorkin, “Qualitative study of a system of three differential equations in the theory of phase synchronization,” Prikl. Mat. Mekh.,39, 642–649 (1975).

    Google Scholar 

  9. L. N. Belyustina and V. N. Belykh, “Qualitative study of a dynamic system on a cylinder,” Differents. Uravn.,9, No. 3, 403–415 (1973).

    Google Scholar 

  10. V. S. Afraimovich and L. P. Shil'nikov, “On singular sets of Morse-Smale systems,” Tr. Mosk. Mat. Ob-va,28, 181–214 (1973).

    Google Scholar 

  11. E. A. Leontovich-Andronova and L. P. Shil'nikov, “Current state of bifurcation theory of dynamic systems,” in: Proceedings of the Fifth International Conference on Nonlinear Oscilations [in Russian], Vol. 2, Izd. Akad. Nauk SSSR, Kiev (1970), pp. 282–290.

    Google Scholar 

  12. Yu. N. Bakaev, “Construction of working zones and phase control systems,” Izv. Akad. Nauk SSSR, Energet. Avtom., No. 2, 132–136 (1960).

    Google Scholar 

  13. Yu. N. Bakaev, “Synchronization properties of a third-order automatic phase control system,” Radiotekh. Elektron., No. 6, 1083–1087 (1965).

    Google Scholar 

  14. G. A. Leonov, “On boundedness of trajectories of phase systems,” Sib. Mat. Zh.,15, No. 3, 687–692 (1974).

    Google Scholar 

  15. L. N. Belyustina and V. N. Belykh, “On the global structure of the decomposition of a cylindrical phase space of a certain nonautonomous system,” Differents. Uravn.,9, No. 4, 595–608 (1973).

    Google Scholar 

  16. V. N. Belykh, “On the qualitative study of a nonautonomous nonlinear second-order equation,” Differents. Uravn.,9, No. 10, 1738–1753 (1975).

    Google Scholar 

  17. E. A. Leontovich-Andronova and L. N. Belyustina, “Bifurcation theory of second-order dynamic systems and its application to the study of nonlinear problems in the theory of vibration,” in: Proceedings of the International Symposium on Nonlinear Vibrations [in Russian], Vol. 2, Izd. Akad. Nauk SSSR, Kiev (1963), pp. 7–28.

    Google Scholar 

  18. V. M. Safonov, “Automatic frequency phase control with second-order filters,” Radiotekh. Elektron., No. 4, 114–128 (1958).

    Google Scholar 

  19. L. N. Belyustina and V. V. Bykov, “Bifurcation and certain qualitative characteristics of an automatic frequency control system with second-order filter,” in: Proceedings of the Symposium on Applied Mathematics and Cybernetics [in Russian], Nauka, Moscow (1973), pp. 28–32.

    Google Scholar 

  20. V. M. Popov, “On the absolute stability of nonlinear control systems,” Avtomat. Telemekh., No. 8, 961–979 (1961).

    Google Scholar 

  21. V. A. Yakubovich, “Absolute stability of nonlinear control systems,” Avtomat. Telemekh., No. 12, 5–14 (1970).

    Google Scholar 

  22. Yu. I. Neimark, Point Transformation Method in the Theory of Nonlinear Vibrations [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  23. L. P. Shil'nikov, “On some cases of generation of periodic motions from singular trajectories,” Mat. Sb.,61 (104), No. 4, 443–466 (1963).

    Google Scholar 

  24. L. P. Shil'nikov, “On generation of a periodic motion from a trajectory doubly asymptotic to a saddletype equilibrium state,” Mat. Sb.,77(119), No. 3, 461–472 (1968).

    Google Scholar 

  25. L. P. Shil'nikov, “On the structure of the extended neighborhood of a rough equilibrium state of saddlefocus type,” Mat. Sb.,81(123), No. 1, 92–103 (1970).

    Google Scholar 

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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 18, No. 4, pp. 723–735, July–August, 1977.

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Belykh, V.N., Nekorkin, V.I. Qualitative study of a multidimensional phase system. Sib Math J 18, 511–520 (1977). https://doi.org/10.1007/BF00967190

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  • DOI: https://doi.org/10.1007/BF00967190

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