Bifurcations of Limit Cycles

Part of the Encyclopaedia of Mathematical Sciences book series (volume 5)


Bifurcations of phase portraits in the neighborhood of a cycle are completely described by bifurcations of the corresponding monodromy transformations. Therefore the basic objects of study in this chapter are the bifurcations of germs of diffeomorphisms at a fixed point. Local families of germs of diffeomorphisms, their equivalence and weak equivalence, and induced and versal deformations of such germs are defined just as for germs of vector fields (see Sect. 1.5 of Chap. 1). Analogs of the Reduction Theorem are also true for germs of diffeomorphisms at a fixed point; see Sect. 1.6 of Chapter 1 and Arnol’d and Il’yashenko (1985, Sect. 2.4 of Chap.6). The restriction of a germ of a diffeomorphism to its center manifold is called the reduced germ of the diffeomorphism. We note that a reduced germ can change orientation, even if the original germ did not: for example, \( x \mapsto {\text{diag(1;2; - }}\frac{1}{2}{\text{)}}x,{\text{ }}x \in \mathbb{R}^4 . \). The bifurcations of germs of diffeomorphisms are described below, and then the results obtained are translated into the language of differential equations.


Vector Field Singular Point Center Manifold Bifurcation Theory Invariant Curve 
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  1. 6.
    We recall that the multipliers of a limit cycle are the eigenvalues of the Poincaré mapping on a disc into itself transversal to the cycle.Google Scholar
  2. 7.
    This is easy to accomplish with the help of the considerations in Marsden and McCracken (1976, ). However, it seems that an explicit formulation of the result and its proof is absent from the literature.Google Scholar
  3. 8.
    An s-critical saddle-node is defined and its bifurcations are studied in Sect. 4 of.Google Scholar
  4. 9.
    An elliptic periodic trajectory of a Hamiltonian system is a cycle with nonreal multipliers whose moduli equal one; a hyperbolic trajectory of a Hamiltonian system is a cycle with multipliers whose moduli are not equal to one.Google Scholar
  5. 10.
    For a detailed proof see: K.I. Babenko, V.Yu. Petrovich, “On proofs by computation on a computer”. Preprint, the M.V. Keldysh Institute of Appl. Math., Moscow, 1983, 183 pp.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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