×

zbMATH — the first resource for mathematics

Bifurcation and stability of families of hyperbolic vector fields in dimension three. (English) Zbl 0879.58056
Let \(M\) be a \({\mathcal C}^\infty\) compact boundaryless 3-dimensional manifold and \(\chi^\infty (M)\) denote the space of \({\mathcal C}^\infty\) vector fields on \(M\). The object of the study is the space \(\chi_1^\infty (M)\) of all \({\mathcal C}^\infty\) arcs \(\xi: I=[-1,1] \to \chi^\infty (M)\). For \(\xi\in \chi_1^\infty(M)\) we let \(\xi= \{X_\mu\}\) where \(X_\mu =\xi (\mu)\) for each \(\mu \in I\). Hence, \(\xi\) is a one-parameter family of vector fields on \(M\).
We say that \(\{X_\mu\}\) is stable at \(\overline \mu\in I\), if there exists a neighborhood \({\mathfrak U}\) of \(\{X_\mu\}\) in \(\chi_1^\infty (M)\) such that for each \(\{Y_\mu\} \in {\mathfrak U}\), there is a parameter value \(\widetilde \mu \in I\) near \(\overline\mu\) and a homeomorphism \(H:M \times \overline I\to M \times \widetilde I\) where \(\overline I\), respectively \(\widetilde I\), is a neighborhood of \(\overline\mu\), respectively of \(\widetilde\mu\), in \(I\) and \(H(x,\mu) =(h_\mu (x), \rho (\mu))\), with \(\rho: (\overline I, \overline\mu) \to (\widetilde I, \widetilde \mu)\) a reparametrization and \(h_\mu: M\to M\) is a topological equivalence between \(X_\mu\) and \(Y_{\rho (\mu)}\), and the map \(\mu \mapsto h_\mu\) is continuous.
In the paper the authors investigate the conditions under which the family \(\{X_\mu\}\) is stable at \(\overline \mu\in I\). Roughly speaking, \(\{X_\mu\}\) is stable at \(\overline \mu\), if \(\overline \mu\) is its first bifurcation value in \(I\) and for \(\mu= \overline \mu\) the vector field \(X_{\overline \mu}\) has one and only one orbit \(\overline \gamma\) along which it is not locally stable. The authors consider only the cases in which \(\overline \gamma\) is an orbit of the following type: (1) an isolated saddle-node singularity; (2) an isolated Hopf singularity; (3) an isolated flip periodic orbit; (4) an isolated saddle-node periodic orbit; (5) a flip periodic orbit arising from two hyperbolic periodic orbits inside a basic set.
This interesting paper is well written and organized.
Reviewer: A.Klíč (Praha)
MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
37D99 Dynamical systems with hyperbolic behavior
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Carneiro, M.J.; Palis, J., Bifurcations and global stability of two-parameter families of gradient vector fields, Publications math. I.H.E.S., Vol. 10, 103-168, (1990) · Zbl 0706.58042
[2] Labarca, R., Stability of parametrized families of vector fields, Pitman research notes in mathematics series, Vol. 160, 121-213, (1987)
[3] Labarca, R.; Plaza, S., Global stability of vector fields, Erg. theory and dyn. systems, Vol. 13, 737-766, (1993) · Zbl 0798.58044
[4] Malta, I.; Palis, J., Families of vector fields with finite modulus of stability, Lectures notes in math., Vol. 898, 212-229, (1980)
[5] Newhouse, S., On simple arcs between structurally stable flows, Lectures notes in math., Vol. 468, 209-233, (1975) · Zbl 0307.58011
[6] Newhouse, S.; Palis, J., Cycles and bifurcations theory, Asterisque, Vol. 31, 44-140, (1976) · Zbl 0322.58009
[7] Newhouse, S.; Palis, J.; Takens, F., Bifurcation and stability of families of diffeomorphisms, Publ. math. I.H.E.S., Vol. 57, 5-71, (1983) · Zbl 0518.58031
[8] Palis, J.; Takens, F., Stability of parametrized families of gradient vector fields, Ann. of math., Vol. 118, 383-421, (1983) · Zbl 0533.58018
[9] Plaza, S., Global stability of saddle-node bifurcation of periodic orbit for vector fields, Annales de la faculté de sciences de Toulouse, Vol. III, n^o 3, 411-448, (1994) · Zbl 0831.58036
[10] Robin, J., Structural stability theorem, Annals of math., Vol. 94, 447-493, (1971) · Zbl 0224.58005
[11] Robinson, C., Structurally stability for vector fields, Annals of math., Vol. 99, 154-175, (1974) · Zbl 0275.58012
[12] Smale, S., Differentiable dynamical systems, Bull. A.M.S., Vol. 73, 747-817, (1967) · Zbl 0202.55202
[13] Takens, F., Normal form for certain singularities of vector fields, Ann. inst. Fourier, Vol. 23, 2, 163-195, (1973) · Zbl 0266.34046
[14] Vera, J., Stability of quasi-transversal bifurcation of vector fields on 3-dimensional manifolds, (1994), Preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.