zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Homoclinic bifurcation at resonant eigenvalues. (English) Zbl 0703.34050
From the summary: The authors consider a bifurcation of homoclinic orbits, which is an analogue of periodic doubling in the limit of infinite period. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under some conditions, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. This paper contains two new theorems and a wide discussion of the problem under consideration.
Reviewer: D.Bobrowski

MSC:
34C15Nonlinear oscillations, coupled oscillators (ODE)
37-99Dynamic systems and ergodic theory (MSC2000)
37G99Local and nonlocal bifurcation theory
References:
[1]Abraham, R., and Marsden, J. (1978).Foundation of Mechanics, Benjamin/Cummings, Reading, Mass.
[2]Abraham, R., and Robbin, J. (1967).Transversal Mappings and Flows, Benjamin, Amsterdam.
[3]Afraimovich, V. S., and Shilnikov, L. P. (1974). On attainable transitions from Morse-Smale systems to systems with many periodic points.Math. USSR Izvest. 8, 1235-1270. · Zbl 0322.58007 · doi:10.1070/IM1974v008n06ABEH002146
[4]Alexander, J. C., and Antman, S. S. (1981). Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems.Arch. Rat. Mech. Anal. 76, 339-354. · Zbl 0479.58005 · doi:10.1007/BF00249970
[5]Alligood, K. T., Mallet-Paret, J., and Yorke, J. A. (1983). An index for the global continuation of relatively isolated sets of periodic orbits. In J. Palis (ed.),Geometric Dynamics, Springer-Verlag, New York, 1-21.
[6]Amick, C. J., and Kirchg?ssner, K. (1988). A theory of solitary water-waves in the presence of surface tension. Preprint.
[7]Angenent, S., and Fiedler, B. (1988). The dynamics of rotating waves in scalar reaction diffusion equations.Trans. AMS 307, 545-568. · doi:10.1090/S0002-9947-1988-0940217-X
[8]Armbruster, D., Guckenheimer, J., and Holmes, P. (1988). Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry.Physica 29D, 257-282.
[9]Arneodo, A., Coullet, P., and Tresser, C. (1981). A possible new mechanism for the onset of turbulence.Phys. Lett. 81A, 197-201.
[10]Arnol’d, V. I. (1972). Lectures on bifurcations and versal systems.Russ. Math. Surv. 27, 54-123. · doi:10.1070/RM1972v027n05ABEH001385
[11]Belitskii, G. R. (1973). Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class.Fund. Anal. Appl. 7, 268-277. · Zbl 0293.39005 · doi:10.1007/BF01075731
[12]Belyakov, L. A. (1980). Bifurcation in a system with homoclinic saddle curve.Mat. Zam. 28, 911-922.
[13]Belyakov, L. A. (1984). Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero.Mat. Zam. 36, 838-843.
[14]Berger, M. S. (1977).Nonlinearity and Functional Analysis, Academic Press, New York.
[15]Bogdanov, R. I. (1976). Bifurcation of the limit cycle of a family of plane vector fields (Russ.).Trudy Sem. I. G. Petrovskogo 2, 23-36; (Engl.) (1981).Sel. Mal. Sov. 1, 373-387.
[16]Bogdanov, R. I. (1981). Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues (Russ.).Trudy Sem. I. G. Petrovskogo 2, 37-65; (Engl.)Sel. Mat. Sov.1, 389-421.
[17]Bogdanov, R. I. (1985). Invariants of singular points in the plane (Russ.).Uspekhi Mat. Nauk 40, 199-200.
[18]Brunovsk?, P., and Fiedler, B. (1986). Numbers of zeros on invariant manifolds in reactiondiffusion equations.Nonlin. Anal. TMA 10, 179-193. · Zbl 0594.35056 · doi:10.1016/0362-546X(86)90045-3
[19]Brunovsk?, P., and Fiedler, B. (1988). Connecting orbits in scalar reaction diffusion equations. In U. Kirchgraber and H.-O. Walther (eds.),Dynamics Reported 1, 57/2-89.
[20]Brunovsk?, P., and Fiedler, B. (1989). Connecting orbits in scalar reaction diffusion equations II: The complete solution.J. Diff. Eg. 81, 106-135. · Zbl 0699.35144 · doi:10.1016/0022-0396(89)90180-0
[21]Bykov, V. V. (1980). Bifurcations of dynamical systems close to systems with a separatrix contour containing a saddle-focus (Russ.). In E. A. Leontovich-Andronova (ed.),Methods of the Qualitative Theory of Differential Equations, Gor’kov. Gos. Univ., Gorki, 44-72.
[22]Chow, S.-N., and Deng, B. (1989). Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems.Trans. AMS 312, 539-587. · doi:10.1090/S0002-9947-1989-0988882-6
[23]Chow, S.-N., Deng, B., and Terman, D. (1986). The bifurcation of a homoclinic orbit from two heteroclinic orbit-a topological approach. Preprint.
[24]Chow, S.-N., Deng, B., and Terman, D. (1990). The bifurcation of homoclinic and periodic orbits from two heteroclinic orbits.SIAM J. Math. Anal. (in press).
[25]Chow, S.-N., and Hale, J. K. (1982).Methods of Bifurcation Theory, Grundl. math. Wiss. 251, Springer-Verlag, New York.
[26]Chow, S.-N., and Lin, X.-B. (1988). Bifurcation of a homoclinic orbit with a saddle-node equilibrium. Preprint.
[27]Chow, S.-N., Mallet-Paret, J., and Yorke, J. A. (1983). A periodic orbit index which is a bifurcation invariant. In J. Palis (ed.),Geometric Dynamics, Springer-Verlag, New York, pp. 109-131.
[28]Coullet, P., Gambaudo, J.-M., and Tresser, C. (1984). Une nouvelle bifurcation de codimension 2: le collage de cycles.C.R. Acad. Sci. Paris 299, 253-256.
[29]de Hoog, P. and Weiss, R. (1979). The numerical solution of boundary value problems with an essential singularity.SIAM J. Numer. Anal. 16, 637-669. · Zbl 0417.65044 · doi:10.1137/0716049
[30]Deng, B. (1989a). The ?il’nikov problem, exponential expansion, strong ?-lemma, C1-linearization, and homoclinic bifurcation.J. Diff. Eq. 79, 189-231. · Zbl 0674.34040 · doi:10.1016/0022-0396(89)90100-9
[31]Deng, B. (1989b). Exponential expansion with ?il’nikov’s saddle-focus.J. Diff. Eq. 82, 156-173. · Zbl 0703.34041 · doi:10.1016/0022-0396(89)90171-X
[32]Deng, B. (1990a). The bifurcation of countable connections from a twisted heteroclinic loop.SIAM J. Math. Anal. (in press).
[33]Deng, B. (1990b). Homoclinic bifurcations with nonhyperbolic equilibria.SIAM J. Math. Anal. (in press).
[34]Deuflhard, P., Fiedler, B., and Kunkel, P. (1987). Efficient numerical pathfollowing beyond critical points.SIAM J. Numer. Anal. 24, 912-927. · Zbl 0632.65058 · doi:10.1137/0724059
[35]Doedel, E. J., and Kernevez, J. P. (1985). Software for continuation problems in ordinary differential equations with applications. CALTECH.
[36]Evans, J., Fenichel, N., and Feroe, J. A. (1982). Double impulse solutions in nerve axon equations.SIAM J. Appl. Math. 42, 219-234. · Zbl 0512.92006 · doi:10.1137/0142016
[37]Feroe, J. A. (1982). Existence and stability of multiple impulse solutions of a nerve axon equation.SIAM J. Appl. Math. 42, 235-246. · Zbl 0502.92002 · doi:10.1137/0142017
[38]Fiedler, B. (1985). An index for global Hopf bifurcation in parabolic systems.J. reine angew. Math. 359, 1-36. · Zbl 0554.35010 · doi:10.1515/crll.1985.359.1
[39]Fiedler, B. (1986). Global Hopf bifurcation of two-parameter flows.Arch. Rat. Mech. Anal. 94, 59-81. · Zbl 0603.58015 · doi:10.1007/BF00278243
[40]Fiedler, B., and Kunkel, P. (1987a). A quick multiparameter test for periodic solutions. In T. K?pper, R. Seydel, and H. Troger (eds.),Bifurcation: Analysis, Algorithms, Applications, ISNM 79, Birkh?user, Basel, pp. 61-70.
[41]Fiedler, B., and Kunkel, P. (1987b). Multistability, scaling, and oscillations. In J. Warnatz and W. J?ger (eds.),Complex Chemical Reaction Systems, Springer Ser. Chem. Phys. 47, Berlin, pp. 169-180.
[42]Fiedler, B., and Mallet-Paret, J. (1989). Connections between Morse sets for delay-differential equations.J. reine angew. Math. 397, 23-41.
[43]Fischer, G. (1984). Zentrumsmannigfaltigkeiten bei elliptischen Differentialfgleichungen.Math. Nachr. 115, 137-157. · Zbl 0565.35040 · doi:10.1002/mana.19841150111
[44]FitzHugh, R. (1969). Mathematical models of excitation and propagation in nerves. In H. P. Schwan (ed.),Biological Engineering, McGraw-Hill, New York, pp. 1-85.
[45]Gambaudo, J.-M., Glendinning, P., and Tresser, C. (1984). Collage de cycles et suites de Farey.C.R. Acad. Sci. Paris 299, 711-714.
[46]Glendinning, P. (1984). Bifurcation near homoclinic orbits with symmetry.Phys. Lett. 103A, 163-166.
[47]Glendinning, P. (1987). Travelling wave solutions near isolated double-pulse solitary waves of nerve axon equations.Phys. Lett. 121A, 411-413.
[48]Glendinning, P., and Sparrow, C. (1986). T-points: A codimension two heteroclinic bifurcation.J. Stat. Phys. 43, 479-488. · Zbl 0635.58031 · doi:10.1007/BF01020649
[49]Guckenheimer, J., and Holmes, P. (1983).Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci.42, Springer-Verlag, New York.
[50]Guckenheimer, J., and Holmes, P. (1988). Structurally stable heteroclinic cycles.Math. Proc. Camb. Phil. Soc. 103, 189-192. · Zbl 0645.58022 · doi:10.1017/S0305004100064732
[51]Hale, J. K., and Lin, X.-B. (1986). Heteroclinic orbits for retarded functional differential equations.J. Diff. Eq. 65, 175-202. · Zbl 0611.34074 · doi:10.1016/0022-0396(86)90032-X
[52]Hastings, S. P. (1982). Single and multiple pulse waves for the FitzHugh-Nagumo equations.SIAM J. Appl. Math. 42, 247-260. · Zbl 0503.92009 · doi:10.1137/0142018
[53]Hirsch, M. W., Pugh, C. C., and Shub, M. (1977).Invariant Manifolds, Lect. Notes Math. 583, Springer-Verlag, Berlin.
[54]Hofer, H., and Toland, J. (1984). Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems.Math. Ann. 268, 387-403. · Zbl 0569.70017 · doi:10.1007/BF01457066
[55]Holmes, P. (1980). A strange family of three-dimensional vector fields near a degenerate singularity.J. Diff. Eq. 37, 382-403. · Zbl 0435.58015 · doi:10.1016/0022-0396(80)90106-0
[56]Hurewicz, W., and Wallman, H. (1948).Dimension Theory, Princeton University Press.
[57]Kielh?fer, H., and Lauterbach, R. (1983). On the principle of reduced stability.J. Funct. Anal. 53, 99-111. · Zbl 0522.58013 · doi:10.1016/0022-1236(83)90048-4
[58]Kirchg?ssner, K. (1982). Wave-solutions of reversible systems and applications.J. Diff. Eq. 45, 113-127. · Zbl 0507.35033 · doi:10.1016/0022-0396(82)90058-4
[59]Kirchg?ssner, K. (1983). Homoclinic bifurcation of perturbed reversible systems. InEquadiff 82, H. W. Knobloch and K. Schmitt (eds.), Lect. Notes Math. 1017, Springer-Verlag, Heidelberg, pp. 328-363.
[60]Kirchg?ssner, K. (1988). Nonlinearly resonant surface waves and homoclinic bifurcation. Preprint.
[61]Kokubu, H. (1987). On a codimension 2 bifurcation of homoclinic orbits.Proc. Jap. Acad. 63A, 298-301.
[62]Kokubu, H. (1988). Homoclinic and heteroclinic bifurcations of vector fields.Jap. J. Appl. Math. 5, 455-501. · Zbl 0668.34039 · doi:10.1007/BF03167912
[63]Kubi?ek, M., and Marek, M. (1983).Computational Methods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, New York.
[64]Kupka, I. (1963). Contribution ? la th?orie des champs g?n?riques.Contrib. Diff. Eq. 2, 457-484; (1964)3, 411-420.
[65]Kuramoto, Y., and Koga, S. (1982). Anomalous period-doubling bifurcations leading to chemical turbulence.Phys. Lett. 92A, 1-4.
[66]Lentini, M., and Keller, H. B. (1980). Boundary value problems on semi-infinite intervals and their numerical solution.SIAM J. Numer. Anal. 17, 577-604. · Zbl 0465.65044 · doi:10.1137/0717049
[67]Leontovich, E. (1951). On the generation of limit cycles from separatrices (Russ.).Dokl. Akad. Nauk 78, 641-644.
[68]Lin, X.-B. (1986). Exponential dichotomies and homoclinic orbits in functional differential equations.J. Diff. Eq. 63, 227-254. · Zbl 0589.34055 · doi:10.1016/0022-0396(86)90048-3
[69]Lukyanov, V. I. (1982). Bifurcations of dynamical systems with a saddle-point separatrix loop.Diff. Eq. 18, 1049-1059.
[70]Lyubimov, D. V., and Zaks, M. A. (1983). Two mechanisms of the transition to chaos in finite-dimensional models of convection.Physica 9D, 52-64.
[71]Mallet-Paret, J., and Yorke, J. A. (1980). Two types of Hopf bifurcation points: Sources and sinks of families of periodic orbits. In R. H. G. Helleman (ed.),Nonlinear Dynamics, Proc. N.Y. Acad. Sci.357, pp. 300-304.
[72]Mallet-Paret, J., and Yorke, J. A. (1982). Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation.J. Diff. Eq. 43, 419-450. · Zbl 0487.34038 · doi:10.1016/0022-0396(82)90085-7
[73]A. Mielke. A reduction principle for nonautonomous systems in infinite-dimensional spaces.J. Diff. Eq. 65, 68-88.
[74]Mielke, A. (1986). Steady flows of inviscid fluids under localized perturbations.J. Diff. Eq. 65, 89-116. · Zbl 0672.76017 · doi:10.1016/0022-0396(86)90043-4
[75]Moser, J. (1973).Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, N.J.
[76]Nozdracheva, V. P. (1982). Bifurcation of a noncoarse separatrix loop.Diff. Eq. 18, 1098-1104.
[77]Ovsyannikov, I. M., and Shil’nikov, L. P. (1987). On systems with a saddle-focus homoclinic curve.Math. USSR Sbornik 58, 557-574. · Zbl 0628.58044 · doi:10.1070/SM1987v058n02ABEH003120
[78]Palmer, K. J. (1984). Exponential dichotomies and transversal homoclinic points.J. Diff. Eq. 55, 225-256. · Zbl 0539.58028 · doi:10.1016/0022-0396(84)90082-2
[79]Rabinowitz, P. H. (1971). Some global results for nonlinear eigenvalue problems.J. Funct. Anal. 7, 487-513. · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[80]Reyn, J. W. (1980). Generation of limit cycles from separatrix polygons in the phase plane. In R. Martini (ed.),Geometrical Approaches to Differential Equations, Lect. Notes Math. 810, Springer-Verlag, Berlin, pp. 264-289.
[81]Rinzel, J., and Terman. D. (1982). Propagation phenomena in a bistable reaction-diffusion system.SIAM J. Appl. Math. 42, 1111-1137. · Zbl 0522.92004 · doi:10.1137/0142077
[82]Robinson, C. (1988). Differentiability of the stable foliation for the model Lorenz equations. Preprint.
[83]Rodriguez, J. A. (1986). Bifurcations to homoclinic connections of the focus-saddle type.Arch. Rat. Mech. Anal. 93, 81-90. · Zbl 0594.34068 · doi:10.1007/BF00250846
[84]Sanders, J. A., and Cushman, R. (1986). Limit cycles in the Josephson equation.SIAM J. Math. Anal. 17, 495-511. · Zbl 0604.58041 · doi:10.1137/0517039
[85]Schecter, S. (1987a). The saddle-node separatrix-loop bifurcation.SIAM J. Math. Anal. 18, 1142-1156. · Zbl 0651.58025 · doi:10.1137/0518083
[86]Schecter, S. (1987b). Melnikov’s method at a saddle-node and the dynamics of the forced Josephson junction.SIAM J. Math. Anal. 18, 1699-1715. · Zbl 0637.58019 · doi:10.1137/0518122
[87]Sell, G. R. (1984). Obstacles to linearization.Diff. Eq. 20, 341-345.
[88]Sell, G. R. (1985). Smooth linearization near a fixed point.Am. J. Math. 107, 1035-1091. · Zbl 0574.34025 · doi:10.2307/2374346
[89]Seydel, R. (1988).From Equilibrium to Chaos, Elsevier, New York.
[90]Shil’nikov, L. P. (1962). Some cases of generation of periodic motions in an n-dimensional space.Soviet Math. Dokl. 3, 394-397.
[91]Shil’nikov, L. P. (1965). A case of the existence of a countable number of periodic motions.Soviet Math. Dokl. 6, 163-166.
[92]Shil’nikov, L. P. (1966). On the generation of a periodic motion from a trajectory which leaves and re-enters a saddle state of equilibrium.Soviet Math. Dokl. 7, 1155-1158.
[93]Shil’nikov, L. P. (1967). The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus.Soviet Math. Dokl. 8, 54-57.
[94]Shil’nikov, L. P. (1968). On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type.Math. USSR Sbornik 6, 427-437. · Zbl 0188.15303 · doi:10.1070/SM1968v006n03ABEH001069
[95]Shil’nikov, L. P. (1969). On a new type of bifurcation of multidimensional dynamical systems.Soviet Math. Dokl. 10, 1368-1371.
[96]Shil’nikov, L. P. (1970). A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type.Math. USSR Sbornik 10, 91-102. · Zbl 0216.11201 · doi:10.1070/SM1970v010n01ABEH001588
[97]Shub, M. (1987).Global Stability of Dynamical Systems, Springer-Verlag, New York.
[98]Smale, S. (1963). Stable manifolds for differential equations and diffeomorphisms.Ann. Sc. Norm. Sup. Pisa 17, 97-116.
[99]Sparrow, C. (1982).The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Appl. Math. Sci. 41, Springer-Verlag, New York.
[100]Sternberg, S. (1957). Local contractions and a theorem of Poincar?.Am. J. Math. 79, 809-824. · Zbl 0080.29902 · doi:10.2307/2372437
[101]Sternberg, S. (1958). On the structure of local homeomorphisms of Euclidean n-space.Am. J. Math. 80, 623-631. · Zbl 0083.31406 · doi:10.2307/2372774
[102]Takens, F. (1974). Singularities of vector fields.Publ. IHES 43, 47-100.
[103]Tresser, C. (1984). About some theorems by L. P. Shil’nikov.Ann. Inst. H.Poincar? 40, 441-461.
[104]Vanderbauwhede, A. (1989). Center manifolds, normal forms and elementary bifurcations. U. Kirchgraber and H.-O. Walther (eds.),Dynamics Reported 2, 89-169.
[105]Walther, H.-O. (1981). Homoclinic solution and chaos in x(t)=f(x(t?1)).Nonlin. Anal. TMA 5, 775-788. · Zbl 0459.34040 · doi:10.1016/0362-546X(81)90052-3
[106]Walther. H.-O. (1985). Bifurcation from a heteroclinic solution in differential delay equations.Trans. AMS 290, 213-233. · doi:10.1090/S0002-9947-1985-0787962-4
[107]Walther, H.-O. (1986). Bifurcation from a saddle connection in functional differential equations: An approach with inclination lemmas.Dissertationes Mathematicae, to appear.
[108]Walther, H.-O. (1987a). Inclination lemmas with dominated convergence.J. Appl. Math. Phys. (ZAMP) 32, 327-337.
[109]Walther, H.-O. (1987b). Homoclinic and periodic solutions of scalar differential delay equations. Preprint.
[110]Walther, H.-O. (1989). Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations.Memoirs of the AMS 402, Providence, R.I.
[111]Yanagida, E. (1987). Branching of double pulse solutions from single pulse solutions in nerve axon equations.J. Diff. Eq. 66, 243-262. · Zbl 0661.35003 · doi:10.1016/0022-0396(87)90034-9