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Local and global behavior near homoclinic orbits. (English) Zbl 0588.58041
We study the local behavior of systems near homoclinic orbits to stationary points of saddle-focus type. We explicitly describe how a periodic orbit approaches homoclinicity and, with the help of numerical examples, discuss how these results relate to global patterns of bifurcations.
MSC:
37D45Strange attractors, chaotic dynamics
37G99Local and nonlocal bifurcation theory
References:
[1]L. P. Sil’nikov, A Case of the Existence of a Denumerable Set of Periodic Motions,Sov. Math. Dokl. 6:163-166 (1965).
[2]L. P. Sil’nikov, 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 (1970). · Zbl 0216.11201 · doi:10.1070/SM1970v010n01ABEH001588
[3]A. Arneodo, P. Coullet, E. Speigel, and C. Tresser, Asymptotic Chaos, Preprint, Universite de Nice (1982).
[4]J. Guckenheimer, Multiple Bifurcation Problems of Codimension Two, Preprint, U.C. Santa Cruz (1980).
[5]J. Guckenheimer, On a Co-Dimension Two Bifurcation, inDynamical Systems and Turbulence: Warwick 1980, D. Rand and L.-S. Young, eds., Lecture Notes in Mathematics No. 898 (Springer-Verlag, Berlin, 1981).
[6]P. Gaspard, Memoire de License, Universite de Bruxelles (1982).
[7]A. Arneodo, P. Coullet, and C. Tresser, Possible New Strange Attractors with Spiral Structure,Commun. Math. Phys. 79:573-579 (1981). · Zbl 0485.58013 · doi:10.1007/BF01209312
[8]G. R. Belitskii, Equivalence and Normal Forms of Germs of Smooth Mappings,Russ. Math. Surv. 33:107-177 (1978). · Zbl 0398.58009 · doi:10.1070/RM1978v033n01ABEH002237
[9]C. Tresser, thesis, Universite de Nice (1981).
[10]H. W. Broer and G. Vegter, Subordinate Sil’nikov Bifurcations near Some Singularities of Vector Fields Having Low Codimension, Preprint ZW-8208, Rijksuniversitat, Groningen (1982).
[11]J. A. Yorke and K. T. Alligood, Cascades of Period-Doubling Bifurcations: A Prerequisite for Horseshoes, Preprint, University of Maryland (1982).
[12]P. Gaspard, Generation of a Countable Set of Homoclinic Flows Through Bifurcation,Phys. Lett. 97A:1-4 (1983).
[13]S. P. Hastings, Single and Multiple Pulse Waves for the Fitzhugh-Nagumo Equations,SIAM J. Appl. Math. 42(2):247-260 (1982). · Zbl 0503.92009 · doi:10.1137/0142018
[14]J. Evans, N. Fenichel, and J. A. Feroe, Double Impulse Solutions in Nerve Axon Equations,SIAM J. Appl. Math. 42(2):219-234 (1983). · Zbl 0512.92006 · doi:10.1137/0142016
[15]J. A. Feroe, Existence and Stability of Multiple Impulse Solutions of a Nerve Axon Equation,SIAM J. Appl. Math. 42(2):235-246 (1983). · Zbl 0502.92002 · doi:10.1137/0142017
[16]C. Sparrow,The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Appl. Math. Sci. No. 41 (Springer-Verlag, New York, 1982).
[17]E. N. Lorenz, Deterministic Non-Periodic Flows,J. Atmos. Sci. 20:130-141 (1963). · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[18]R. Rössler, F. Gotz, and O. E. Rössler, Chaos in Endocrinology,Biophys. J. 25:216A (1979).
[19]C. Sparrow, Chaos in a Three-Dimensional Single Loop Feedback System with a Piece-wise Linear Feedback Function,J. Math. Anal. Appl. 83:275-291 (1981). · Zbl 0518.34037 · doi:10.1016/0022-247X(81)90263-8
[20]O. E. Rössler, The Gluing Together Principle and Chaos, inNon-linear Problems of Analysis in Geometry and Mechanics, M. Atteia, D. Bancel, and I. Gumowski, eds. (Pitman, New York, 1981), pp. 50-56.
[21]B. Uehleke, Chaos in einem stuckweise linearen System: Analytische Resultate, Ph.D. thesis, Tübingen (1982).
[22]A. Arneodo, P. Coullet, and C. Tresser, Oscillators with Chaotic Behavior: An Illustration of a Theorem by Sil’nikov,J. Stat. Phys. 27:171-182 (1982). · Zbl 0522.58033 · doi:10.1007/BF01011745
[23]O. E. Rössler, Continuous Chaos: Four Prototype Equations, inBifurcation Theory and Applications in Scientific Disciplines, O. Gurel and O. E. Rössler, eds., Proc. N.Y. Acad. Sci. No. 316, pp. 376-394 (1978).
[24]E. Knobloch and N. O. Weiss, Bifurcations in a Model of Magnetoconvection,Physica 9D:379-407 (1983).
[25]A. Bernoff, Preprint, University of Cambridge (1984).
[26]E. Knobloch and N. O. Weiss, Bifurcations in a Model of Double-Diffusive Convection,Phys. Lett. 85A(3):127-130 (1981).
[27]D. R. Moore, J. Toomre, E. Knobloch, and N. O. Weiss, Chaos in Thermosolutal Convection: Period Doubling for Partial Differential Equations,Nature 303: (1983).
[28]P. Gaspard and G. Nicolis, What Can We Learn from Homoclinic Orbits in Chaotic Dynamics?J. Stat. Phys. 31:499-518 (1983). · Zbl 0587.58035 · doi:10.1007/BF01019496
[29]P. Gaspard, R. Kapral, and G. Nicolis, Bifurcation Phenomena near Homoclinic Systems: A Two-Parameter Analysis, This Issue,J. Stat. Phys. 35:697 (1984). · Zbl 0588.58055 · doi:10.1007/BF01010829