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

  • Vladimir Igorevich Arnold

Abstract

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.

Keywords

Convection Eter 

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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

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