×

Degenerated singular cycles of inclination-flip type. (English) Zbl 0903.34024

The authors consider two-parameter families of three-dimensional vector fields, unfolding a degenerate singular cycle. The singular cycle contains a hyperbolic singularity with a two-dimensional stable manifold; the linearized vector field at the singularity possesses three distinct real eigenvalues. It further contains a hyperbolic periodic orbit of saddle type, a heteroclinic connection from the singularity to the periodic orbit and a heteroclinic connection from the periodic orbit to the singularity. The degeneracy condition involves a quadratic contact between the stable manifold of the singularity and the unstable manifold of the periodic orbit, along the heteroclinic connection from the periodic orbit to the singularity.
Different cases occur, depending on eigenvalue conditions and geometric configurations. The authors state bifurcation theorems, in which the main thrust is on the prevalence of hyperbolic dynamics.
Bifurcations from singular cycles of codimension one had been studied by R. Bamón, R. Labarca, R. Mañe and M. J. Pacífico [Publ. Math., Inst. Hautes Étud. Sci. 78, 207-232 (1993; Zbl 0801.58010)] and by M. J. Pacifico and A. Rovella [Ann. Sci. Ec. Norm. Supér., IV. Sér. 26, No. 6, 691-700 (1994; Zbl 0802.58036)].

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] R. BAMÓN , R. LABARCA , R. MAÑ;É and M. J. PACÍFICO , The explosion of Singular Cycles (Publ. Math. IHES, Vol. 78, 1993 , pp. 207-232). Numdam | MR 94m:58152 | Zbl 0801.58010 · Zbl 0801.58010
[2] A. J. HOMBURG , H. KOKUBU and H. KRUPA , The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbits (Erg. Th. & Dyn. Sys., Vol. 14, 1994 , pp. 667-693). MR 96a:58134 | Zbl 0864.58044 · Zbl 0864.58044
[3] M. HIRSCH , C. C. PUGH and M. SHUB , Invariant Manifolds , LNM 583. MR 58 #18595 | Zbl 0355.58009 · Zbl 0355.58009
[4] C. A. MORALES , On inclination-flip homoclinic orbit associated to a saddle-node equilibria (Bol. Soc. Bras. Mat., Vol. 27, 1996 , pp. 145-160). MR 97h:58118 | Zbl 0895.58043 · Zbl 0895.58043
[5] V. NAUDOT , Hyperbolic Dynamic nearby an unfolded degenerate homoclinic orbit , To appear in Erg. Th. & Dyn. Sys.
[6] M. J. PACÍFICO and A. ROVELLA , Unfolding contracting singular cycles (Ann. Scient. É. Norm. Sup. 4e sér., Vol. 26, 1994 , pp. 691-700). Numdam | MR 94j:58133 | Zbl 0802.58036 · Zbl 0802.58036
[7] J. PALIS and F. TAKENS , Hyperbolicity and sensitive chaotic dynamic at homoclinic bifurcation , Cambridge University Press, Vol. 35, 1992 . MR 94h:58129 | Zbl 0790.58014 · Zbl 0790.58014
[8] M. R. RYCHLIC , Lorenz attractor through Silnikov-type bifurcations (Erg. Th. & Dyn. Sys., Vol. 10, 1990 , pp. 793-821). MR 92f:58103 | Zbl 0715.58027 · Zbl 0715.58027
[9] B. SAN MARTIN , Contracting Singular Cycles , Preprint IMPA to appear. · Zbl 0942.37039
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.