Nonlocal Bifurcations

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


In this chapter we describe the bifurcations of systems on the boundary of the set of Morse-Smale systems. We recall that a point P is a nonwandering point of a flow {f t } (or a diffeomorphism f) if, for any neighborhood U containing P, there exist sequences {t i} or {k i} with k i ∈ ℤ), diverging to ∞ as i → ∞, such that \( (f^{t_i } u) \cap u \ne 0((f^{k_i } u) \cap u \ne 0) \).


Vector Field Unstable Manifold Rotation Number Klein Bottle Saddle Connection 
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  1. 11.
    We recall (see Andronov and Pontryagin (1937) or Lefschetz (1957)) that the original definition of structural stability differed from that of roughness by the absence of the requirement of nearness to the identity of the homeomorphism realizing the topological equivalence between the original and the perturbed systems. The set of vector fields generating structurally stable systems is open. This follows immediately from their definition, in contrast to the case for rough systems. On the other hand, we do not know of any structurally stable systems that are also not rough. Therefore, at the present time “structural stability” is often used as a synonym for “roughness”. Translator’s Note: More explicitly, f ε C 1 is structurally stable (according to Lefschetz’s definition) if for any g that is sufficiently close to f there is a homeomorphism h such that g o h = h o f. According to Andronov and Pontryagin, an f ε C1 is rough if for any ε > 0 there is a δ (ε) > 0 such that, for any g, distc1(f, g) < δ5(δ), there exists a homeomorphism h, distco(hi. Id) < ε such that g o h = h o f. This translation uses “structurally stable” as a synonym for “rough”, corresponding to the standard English usage.Google Scholar
  2. 12.
    In the literature, a term “cycle” is also used, and it is not always assumed that k ⩾ 2.Google Scholar
  3. 13.
    We recall that a boundary point v 0 of an open set U is called accessible if there exists a path (a homeomorphic image of a closed interval) all of whose points, except for the boundary point that coincides with v 0, lie in U.Google Scholar
  4. 14.
    A set of second Baire category is an intersection of countably many open, everywhere dense sets (otherwise known as “residual sets” or “thick sets”).Google Scholar
  5. 15.
    A separatrix of a saddle-node equilibrium is understood here to be the part of a center manifold not belonging to a two-dimensional stable or unstable set, in other words, the common boundary of two hyperbolic sectors.Google Scholar
  6. 16.
    A condition on the topological type of the surface appears here because, for surfaces not included in the statement of the theorem, the closing lemma is unproved in the C r-topology for r≥2.Google Scholar
  7. 17.
    We recall that a vector field with a cycle corresponds to a rational rotation number.Google Scholar
  8. 18.
    The definitions of topological equivalence and weak equivalence of families and their structural stability are analogous to those presented in Sect. 2.2. It is only necessary to replace the interval I by a neighborhood of the bifurcation value.Google Scholar
  9. 19.
    A cycle is called completely unstable if it becomes stable upon reversing time.Google Scholar
  10. 20.
    A repeller is an invariant set of a dynamical system that turns into an attractor upon reversal of time.Google Scholar
  11. 21.
    The analog of this theorem for the case of a saddle in its hyperbolic variables (in which instead of a strange attractor a complicated invariant set is born) is announced in Afrajmovich and Shil’nikov (1982) We note that a complete proof of this theorem has not been published up to this time, and, probably one has not been obtained. Some progress has been made by F. Przytycki, “Chaos after bifurcation of a Morse-Smale diffeomorphism through a one-cycle saddle-node and iterations of an interval and a cycle,” Preprint 347, Inst. of Math Polish Acad. Sci., 1985, 62 pp.Google Scholar
  12. 22.
    See EMS, Dynamical Systems 2, pp. 115-118 for a description of a Smale horseshoe, or Guckenheimer and Holmes (1983, Sect. 51).Google Scholar
  13. 23.
    That is, periodic trajectories correspond to periodic trajectories, and asymptotic trajectories correspond to asymptotic trajectories, etc.Google Scholar
  14. 24.
    It is required here to be compact.Google Scholar
  15. 25.
    A nonleading submanifold intersects a transversal tangent to the invariant linear subspace corresponding to the multipliers \( \lambda _2 \ldots ,\lambda _m {\text{ if }}\lambda _1 \) is real, and to \( \lambda _3 \ldots ,\lambda _m {\text{ }} \) otherwise.Google Scholar
  16. 26.
    Suppose L is a hyperbolic cycle and Γis a homoclinic trajectory of it. Cycles in the sequence L n are called k-circuiting if for any two neighborhoods U of the cycle L, and V of the trajectory Γ, there exist a natural number N and a neighborhood W of L such that for all n \( n > N,L_n \subset U \cup V \) , and the set Ln/ W consists of k connected components.Google Scholar
  17. 27.
    The boundary points of a Cantor set are the ends of the deleted intervals; the rest are interior points.Google Scholar
  18. 28.
    We recall that a set is called a topological limit if it is both the upper and the lower topological limit of a family of sets. A set is called the upper (lower) topological limit of a family of sets Cε if it is the set of all points p such that for any neighborhood of each p there exists a value of ε arbitrarily close to ε such that the intersection of Cε, with this neighborhood is nonempty (beginning with some ε0 < ε the intersection of C t with this neighborhood is nonempty for ε0 < ε < ε).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia

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