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)
Constructing dynamical systems having homoclinic bifurcation points of codimension two. (English) Zbl 0879.34051
Summary: A procedure is derived which allows for a systematic construction of three-dimensional ordinary differential equations having homoclinic solutions. The equations are proved to exhibit codimension-two homoclinic bifurcation points. Examples include the non-orientable resonant bifurcation, the inclination-flip, and the orbit-flip. In addition, an equation is constructed which has a homoclinic orbit converging to a saddle-focus satisfying Shilnikov’s condition. The vector fields are polynomial and non-stiff in that the eigenvalues are of moderate size.

34C37Homoclinic and heteroclinic solutions of ODE
37G99Local and nonlocal bifurcation theory
Full Text: DOI
[1] L. A. Belyakov. The bifurcation set in a system with a homoclinic saddle curve.Mat. Zam. 28 (1980), 910--916. · Zbl 0471.34032
[2] W.-J. Beyn. The numerical computation of connecting orbits in dynamical systems.IMA J. Numer. Anal. 9 (1990), 379--405. · Zbl 0706.65080 · doi:10.1093/imanum/10.3.379
[3] S.-N. Chow, B. Deng, and B. Fiedler. Homoclinic bifurcation at resonant eigenvalues.J. Dyn. Diff. Eq. 2 (1990), 177--244. · Zbl 0703.34050 · doi:10.1007/BF01057418
[4] A. R. Champneys, J. HÄrterich, and B. Sandstede. A non-transverse homoclinic orbit to a saddle-node equilibrium.Ergod. Theory Dyn. Syst. 16 (1996), 431--450. · Zbl 0853.58079 · doi:10.1017/S0143385700008919
[5] A. R. Champneys and Yu. A. Kuznetsov. Numerical detection and continuation of codimension-two homoclinic bifurcations.Int. J. Bifurc. Chaos 4 (1994), 795--822. · Zbl 0873.34030
[6] A. R. Champneys, Yu. A. Kuznetsov, and B. Sandstede.HomCont: An AUTO86 Driver for Homoclinic Bifurcation Analysis, Version 2.0, Technical report, CWI, Amsterdam, 1995.
[7] A. R. Champneys, Yu. A. Kuznetsov, and B. Sandstede. A numerical toolbox for homoclinic bifurcation analysis.Int. J. Bifurc. Chaos 6 (1996), 867--887. · Zbl 0877.65058 · doi:10.1142/S0218127496000485
[8] B. Deng. Constructing homoclinic orbits and chaotic attractors.Int. J. Bifurc. Chaos 4 (1994), 823--841. · Zbl 0873.34036 · doi:10.1142/S0218127494000599
[9] F. Dumortier, H. Kokubu, and H. Oka. A degenerate singularity generating geometric Lorenz attractors.Ergod. Theory Dyn. Syst. 15 (1995), 833--856. · Zbl 0836.58030 · doi:10.1017/S0143385700009664
[10] M. J. Friedman and E. J. Doedel. Numerical computation and continuation of invariant manifolds connecting fixed points.SIAM J. Numer. Anal. 28 (1991), 789--808. · Zbl 0735.65054 · doi:10.1137/0728042
[11] A. J. Homburg, H. Kokubu, and M. Krupa. The cusp horseshoe and its bifurcations from inclination-flip homoclinic orbits.Ergod. Theory Dyn. Syst. 14 (1994), 667--693. · Zbl 0864.58044 · doi:10.1017/S0143385700008117
[12] M. Kisaka, H. Kokubu, and H. Oka. Bifurcation toN-homoclinic orbits andN-periodic orbits in vector fields.J. Dyn. Diff. Eq. 5 (1993), 305--358. · Zbl 0784.34038 · doi:10.1007/BF01053164
[13] K. J. Palmer. Exponential dichotomies and transversal homoclinic points.J. Diff. Eq. 55 (1984), 225--256. · Zbl 0539.58028 · doi:10.1016/0022-0396(84)90082-2
[14] B. Sandstede.Verzweigungstheorie homokliner Verdopplungen, Doctoral thesis, University of Stuttgart, Stuttgart, 1993.
[15] B. Sandstede. Convergence estimates for the numerical approximation of homoclinic solutions.IMA J. Numer. Anal. (1997), to appear. · Zbl 0899.65044
[16] B. Sandstede. A unified approach to homoclinic bifurcations with codimension two. II. Applications, in preparation (1996).
[17] L. P. Shilnikov. A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type.Mat. USSR Sb. 10 (1970), 91--102. · Zbl 0216.11201 · doi:10.1070/SM1970v010n01ABEH001588
[18] B. Sandstede and A. Scheel. Forced symmetry breaking of heteroclinic cycles.Nonlinearity 8 (1995), 333--365. · Zbl 0841.58048 · doi:10.1088/0951-7715/8/3/003
[19] D. Terman. The transition from bursting to continuous spiking in excitable membrane models.J. Nonl. Sci. 2 (1992), 135--182. · Zbl 0900.92059 · doi:10.1007/BF02429854
[20] E. Yanagida. Branching of double pulse solutions from single solutions in nerve axon equations.J. Diff. Eq. 66 (1987), 243--262. · Zbl 0661.35003 · doi:10.1016/0022-0396(87)90034-9