zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Bifurcation of homoclinic orbits with saddle-center equilibrium. (English) Zbl 1127.34022
The paper presents new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in general nondegenerate systems with action-angle variables: $$ \aligned \dot z&=f(z,I)+\varepsilon g^z(z,I,\theta ,\lambda ,\varepsilon ),\\ \dot I&=\varepsilon g^I(z,I,\theta ,\lambda ,\varepsilon ),\\ \dot \theta &=\omega, \endaligned \tag 1$$ where $(z,I,\theta )\in\Bbb R^n\times\Bbb R^m\times\Bbb T^l$, $\lambda\in\Bbb R^k$, $0\le\varepsilon\ll 1$, $\vert \lambda\vert \ll 1$, and $g^z$,$g^I$ are $2\pi$-periodic in $\theta$. The unperturbed system ($\varepsilon =0$) is assumed to have a saddle-center type equilibrium whose stable and unstable manifolds intersect in a one dimensional manifold, and is not completely integrable or near-integrable. By constructing local coordinate systems near the unperturbed homoclinic orbit, conditions for the existence of a transversal homoclinic orbit are obtained. Conditions for the existence of periodic orbits bifurcating from the homoclinic orbit are also given.

34C23Bifurcation (ODE)
34C37Homoclinic and heteroclinic solutions of ODE
37C29Homoclinic and heteroclinic orbits
Full Text: DOI
[1] Wiggins, S., Global Bifurcation and chaos, Springer-Verlag, New York, 1988. · Zbl 0661.58001
[2] Wiggins, S. and Holmes, P., Homoclinic orbits in slowly varying oscillators, SIAM J. Math. Anal., 18, 1987, 612--629. · Zbl 0622.34041 · doi:10.1137/0518047
[3] Yagasaki, K., The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems, Nonlinearity, 12, 1999, 799--822. · Zbl 0967.34042 · doi:10.1088/0951-7715/12/4/304
[4] Huang, D., Liu, Z. and Cheng, Z., Global dynamics near the resonance in the Sine-Gordon equation, J. Shanghai Univ., 2, 1998, 259--261. · Zbl 0927.34038 · doi:10.1007/s11741-998-0036-6
[5] Feckan, M., Bifurcation of multi-bump homoclinics in systems with normal and slow variables, J. Differential Equations, 41, 2000, 1--17.
[6] Kovacic, G., Singular perturbation theory for homoclinic orbits in a class of near-integrable dissipative system, SIAM J. Math. Anal., 26, 1995, 1611--1643. · Zbl 0835.34049 · doi:10.1137/S0036141093245422
[7] Zhu, D. M., Problems in homoclinic bifurcations with higher dimensions, Acta Math. Sinica, New Ser., 14, 1998, 341--352. · Zbl 0932.37032 · doi:10.1007/BF02580437
[8] Zhu, D. M. and Xia, Z. H., Bifurcations of heteroclinic loops, Sci. China Ser. A, 41, 1998, 837--848. · Zbl 0993.34040 · doi:10.1007/BF02871667
[9] Zhu, D. M. and Han, M. A., Bifurcation of homoclinic orbits in fast variable space (in chinese), Chin. Ann. Math, 23A(4), 2002, 438--449. · Zbl 1016.34040
[10] Deng, B., Homoclinic bifurcations with nonhyperbolic equilibria, SIAM J. Math. Anal., 3, 1990, 693--720. · Zbl 0698.34037 · doi:10.1137/0521037