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)


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].


Phase Space Phase Plane Periodic Motion Nonlinear Dynamic System Phase Trajectory 
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  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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