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General Properties of Nonlinear Dynamic Systems in Phase Space

  • M. I. Rabinovich
  • D. I. Trubetskov
Part of the Mathematics and Its Applications (Soviet Series) book series (MASS, volume 50)

Abstract

We begin with a system with one degree of freedom. Such a system is described by a second-order equation and may be completely investigated qualitatively by analyzing the behavior of the trajectory on the phase plane [1–6].

Keywords

Phase Space Phase Plane Periodic Motion Nonlinear Dynamic System Phase Trajectory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A.A. Andronov, A.A. Vitt, and S.E. Khaikin, Oscillation Theory [in Russian], Fizmatgiz, Moscow (1959).Google Scholar
  2. 2.
    A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, Qualitative Theory of Second-Order Dynamic Systems [in Russian], Nauka, Moscow (1966).Google Scholar
  3. 3.
    H. Poincare, On the Curves Determinable by Differential Equations [Russian translation], Gostekhizdat, Moscow (1947).Google Scholar
  4. 4.
    A.V. Gaponov-Grekhov and M.I. Rabinovich, “L.I. Mandel’shtam and the modern theory of nonlinear oscillations,” UFN, 128, 579–624 (1979).MathSciNetCrossRefGoogle Scholar
  5. 5.
    N.V. Butenin, Yu.I. Neimark, N.A. Fufaev, Introduction to Nonlinear Oscillations [in Russian] Nauka, Moscow (1976), Ch.7, Sec.4.Google Scholar
  6. 6.
    Yu.I. Neimark, Method of Point Mappings in Nonlinear Oscillation Theory [in Russian], Nauka, Moscow (1972).Google Scholar
  7. 7.
    L.P. Shil’nikov, “On the birth of periodic motion from trajectories bisymmetric about a saddle type equilibrium,” Matem. Sbor., 77, No. 119, 461–472 (1968).Google Scholar
  8. 8.
    V.M. Alekseev and M.V. Yakobson, “Symbolic dynamics and hyperbolic dynamic systems” in: R. Bowen (ed), Methods of Symbolic Dynamics [Russian translation], Mir, Moscow (1979).Google Scholar
  9. 9.
    V.K. Mel’nikov, “On the stability of a center during temporally periodic disturbances,” Tr. Mosk. Matem. Ob., 12, 3–52 (1963)..Google Scholar
  10. 10.
    V.I. Arnol’d, Additional Chapters to the Theory of Ordinary Differential Equations [Russian translation], Nauka, Moscow (1978).Google Scholar
  11. 11.
    A.D. Morozov, “On a complete qualitative investigation of Duffing’s equation,” ZhVMMF, 12, 1134–1152 (1973).Google Scholar
  12. 12.
    J.E. Marsden and M. MacCracken, The Hopf Bifurcation and its Applications (1976) [Russian translation], Mir, Moscow (1980).Google Scholar
  13. 13.
    R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley Interscience) [Russian translation], Mir, Moscow (1978).Google Scholar
  14. 14.
    V.M. Alekseev, Symbolic Dynamics (XI Summer Mathematics School) [in Russian], Inst. Matem. Akad. Nauk Ukrain. SSR, Kiev (1976), p.212.Google Scholar
  15. 15.
    H. Poincare, Selected Works [Russian translation], Nauka, Moscow (1972), Vol.2, Ch.33.Google Scholar
  16. 16.
    N.V. Butenin, Yu.I. Heimark, N.A. Furaev, Introduction to the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1987), Ch.3, Sect.2.Google Scholar
  17. 17.
    V.C. Aframovich, “Short essay on the qualitative theory of dynamic systems” in: Lectures on Microwave Electronics and Radio Physics (Sixth Summer School-Seminar for Engineers [in Russian], Izd. Saratovskogo Univ., Saratov (1983), Vol.2, pp. 75–89.Google Scholar

Copyright information

© Kluwer Academic Publishers 1989

Authors and Affiliations

  • M. I. Rabinovich
    • 1
  • D. I. Trubetskov
    • 2
  1. 1.Institute of Applied PhysicsAcademy of Sciences of the USSRGorkyUSSR
  2. 2.Saratov State UniversityUSSR

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