Loss of Stability of Equilibrium and of Self-Oscillating Modes of Behaviour

  • Vladimir Igorevich Arnold


Loss of stability of an equilibrium state on change of parameter is not necessarily associated with the bifurcation of this equilibrium state: it can lose stability not only by colliding with another state, but also by itself.


Equilibrium State Strange Attractor Phase Curve Catastrophe Theory Hydrodynamic Stability 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical Comments

Poincares’ These (1879)

  1. H. Poincaré: Sur les propriétés des fonctions définies par les équations aux différences partielles, in: Oeuvres de Henri Poincaré, Tome I. Gauthier-Villars, Paris 1951, XLIX–CXXIX.Google Scholar
  2. contains among other things a versai deformation theorem for 0-dimensional complete intersections (Lemme IV, p. LXI) and the method of normal forms.Google Scholar

Andronov’s first presentation of the theory of structural stability and bifurcation theory:

  1. A. A. Andronov: Mathematical problems of the theory of self-oscillations (in Russian), in: I Vsesoyuznaya konferentsiya po kolebaniyam, M. L., GTTI, 1933, pp. 32–72 (reprinted in Andronov’s Collected Works, Moscow 1956, pp. 85-124).Google Scholar

The 1939 paper with E. A. Leontovich contains the theory for both kinds of bifurcations in which a cycle of codimension one is born : the local kind (birth out of an equilibrium point) and the nonlocal kind (birth out of a loop which is a saddle separatrix). See:

  1. A. A. Andronov, E. A. Leontovich: Some cases of the dependence of limit cycles on parameters (in Russian). Uchenye Zapiski Gor’kovskogo Gosudarstvennogo Universiteta, Issue 6, 1939, 3–24.Google Scholar

For applications of exponential instability to hydrodynamics, see :

  1. V. I. Arnol’d: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier, 16, 1 (1966), 319–361.MathSciNetCrossRefGoogle Scholar

Recent papers referred to in the chapter:

  1. Yu. S. Il’yashenko: Weakly contracting systems and attractors of the Galerkin approximations of the Navier-Stokes equation (in Russian). Usp. Mat. Nauk 36:3 (1981), 243–244.Google Scholar
  2. Yu. S. Il’yashenko, A. N. Chetaev: Weakly contracting systems and attractors of the Galerkin approximations of the Navier-Stokes equations on a two-dimensional torus (in Russian). Uspekhi Mekh. 5:1/2 (1982), 31–63.Google Scholar
  3. A. V. Babin, M. I. Vishik: Attractors of partial differential evolution equations and estimates of their dimension. Usp. Mat. Nauk 38:4 (1983), 133–187 (English translation: Russ. Math. Surv. 38:4 (1983), 151-213).MathSciNetGoogle Scholar

The Bogdanov theorem was announced in:

  1. V. I. Arnol’d: Lectures on bifurcations in versai families. Usp. Mat. Nauk 27:5 (1972), 119–184 (English translation: Russ. Math. Surv. 27:5(1972), 54-123).Google Scholar

The proofs are given in :

  1. R. I. Bogdanov: Bifurcation of the limit cycle of a family of plane vector fields. Tr. Semin. Im. I. G. Petrovskogo 2 (1976), 23–35 (English translation: Sel. Math. Sov. 1 (1981), 373-387).MathSciNetGoogle Scholar
  2. R. I. Bogdanov: Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues. Tr. Semin. Im. I. G. Petrovskogo 2 (1976), 37–65 (English translation: Sel. Math. Sov. 1 (1981), 389-421).MathSciNetGoogle Scholar

The cases of symmetries of orders 2, 3, or 2≥5:

  1. V. K. Melnikov: Qualitative description of resonance phenomena in nonlinear systems (in Russian). Dubna, O. I. Ya. F., P-1013 (1962), 1–17Google Scholar
  2. E. I. Khorozov: Versal deformations of equivariant vector fields for the cases of symmetry of order 2 and 3 (in Russian). Tr. Semin. Im. I. G. Petrovskogo 5 (1979), 163–192.MathSciNetMATHGoogle Scholar

Symmetry of order 4:

  1. V. I. Arnol’d: Loss of stability of self-oscillations close to resonance and versai deformations of equivariant vector fields. Funkts. Anal. Prilozh. 11:2 (1977), 1–10 (English translation: Funct. Anal. Appl. 11(1977), 85-92).Google Scholar
  2. A. I. Nejshtadt: Bifurcations of the phase pattern of an equation system arising in the problem of stability loss of selfoscillations close to 1:4 resonance. Prikl. Mat. Mekh. 42 (1978), 830–840 (English translation: J. Appl. Math. Mech. 42 (1978), 896-907).Google Scholar
  3. F. S. Berezovskaya, A. I. Khibnik: On the bifurcation of séparatrices in the problem of stability loss of auto-oscillations near 1:4 resonance. Prikl. Mat. Mekh. 44 (1980), 938–943 (English translation: J. Appl. Math. Mech. 44 (1980), 663-667).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Vladimir Igorevich Arnold
    • 1
  1. 1.Department of MathematicsUniversity of MoscowMoscowUSSR

Personalised recommendations