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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:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G99 Local and nonlocal bifurcation theory for dynamical systems
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[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
[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). · Zbl 0498.58014
[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
[8] G. R. Belitskii, Equivalence and Normal Forms of Germs of Smooth Mappings,Russ. Math. Surv. 33:107-177 (1978). · Zbl 0398.58009
[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). · Zbl 0553.58024
[11] J. A. Yorke and K. T. Alligood, Cascades of Period-Doubling Bifurcations: A Prerequisite for Horseshoes, Preprint, University of Maryland (1982). · Zbl 0541.58039
[12] P. Gaspard, Generation of a Countable Set of Homoclinic Flows Through Bifurcation,Phys. Lett. 97A:1-4 (1983). · Zbl 0555.58021
[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
[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
[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
[16] C. Sparrow,The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Appl. Math. Sci. No. 41 (Springer-Verlag, New York, 1982). · Zbl 0504.58001
[17] E. N. Lorenz, Deterministic Non-Periodic Flows,J. Atmos. Sci. 20:130-141 (1963). · Zbl 1417.37129
[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
[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
[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). · Zbl 0583.76057
[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
[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
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