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Bifurcation from a saddle connection in functional differential equations: An approach with inclination lemmas

Seria
Rozprawy Matematyczne tom/nr w serii: 291 wydano: 1990
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Warianty tytułu
Abstrakty
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CONTENTS
Introduction...........................................................................................................................................5
  I.........................................................................................................................................................15
1. Preliminaries...................................................................................................................................15
2. Solutions of a family of differential delay equations with periodic nonlinearity.................................16
3. Linearization of the semiflow............................................................................................................21
4. The saddle point property...............................................................................................................24
  II........................................................................................................................................................30
5. The transformed semiflow...............................................................................................................30
6. A shift along the transformed semiflow............................................................................................36
7. The level sets $H⁺_{ak}$................................................................................................................44
8. Inclinations of tangent vectors.........................................................................................................46
9. Estimating D₁Σ(ψ,a) at BC-points ψ in level sets $H⁺_{ak}$............................................................48
10. End of the proof of Theorem 6.1...................................................................................................51
  III.......................................................................................................................................................53
11. Šilnikov continuation and return map............................................................................................53
12. Smoothness properties of f...........................................................................................................56
13. Bifurcation.....................................................................................................................................58
14. Proof of Theorem 13.2 (vi.2) and (vi.3), for a parameter interval $A_{19}$ instead of A...............63
15. Proof of Theorem 13.2 (vi.1).........................................................................................................70
References.........................................................................................................................................74
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 291
Liczba stron
74
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCXCI
Daty
wydano
1990
Twórcy
  • Mathematisches Institut, Universität München, München, BRD
Bibliografia
  • [1] R. Abraham, J. Robbin, Transversal Mappings and Flows, Benjamin, New York-Amsterdam 1967.
  • [2] U. an der Heiden, H. O. Walther, Existence of chaos in control systems with delayed feedback, J. Differential Equations 47 (1983), 273-295.
  • [3] A. A. Andronov, C. E. Chaikin, Theory of Oscillations, Princeton Univ. Press, Princeton, N. J., 1949.
  • [4] C. M. Blazquez, Bifurcation from a homoclinic orbit in parabolic differential equations, Nonlinear Analysis 10 (1986), 1277-1291.
  • [5] S. N. Chow, B. Deng, Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite dimensional systems. Trans. Amer. Math. Soc. 312 (1989), 539-587.
  • [6] S. N. Chow, J. K. Hale, Methods of Bifurcation Theory, Springer, New York 1982.
  • [7] J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York 1960.
  • [8] T. Furumochi, Existence of periodic solutions of one-dimensional differential-delay equations, Tôhoku Math. J. 30 (1978), 13-35.
  • [9] J. K. Hale, Functional Differential Equations, Springer, New York 1971.
  • [10] J. K. Hale, Theory of Functional Differential Equations, Springer, New York 1977.
  • [11] J. K. Hale, X. B. Lin, Symbolic dynamics and nonlinear semiflows, Ann. Mat. Pura Appl. 144 (1986), 229-259.
  • [12] J. K. Hale, X. B. Lin, Examples of transverse homoclinic orbits in delay equations, Nonlinear Analysis 10 (1986), 693-709.
  • [13] J. K. Hale, L. T. Magalhaes, W. M. Oliva, An Introduction to Infinite Dimensional Dynamical Systems - Geometric Theory, Springer, New York 1984.
  • [14] D. Henry, Invariant Manifolds, University of Sāo Paulo, IME, 1983.
  • [15] Z. J. Jelonek, C. I. Cowan, Synchronized systems with time delay in the loop, Proc. Inst. Electrical Engineers 104 C, 388-397 (1957) [IEEE Monograph No 229 R, March 1957].
  • [16] J. Palis, On Morse-Smale dynamical systems, Topology 8 (1969), 385-104.
  • [17] J. Palis, W. de Melo, Geometrical Theory of Dynamical Systems, Springer, New York 1982.
  • [18] L. P. Silnikov, On the generation of periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Math. USSR-Sbornik 6 (1968), 427-438 [Transl. of Mat. Sbornik 77 (119), 1968].
  • [19] Y. Ueda, Self-excited oscillations and their bifurcations in systems described by nonlinear differential difference equations, In: 24th Midwest symposium on circuits and systems, Ed. S. Karni, pp. 549-553, Univ. of New Mexico, Albuquerque, N. M., 1981.
  • [20] H. O. Walther, Homoclinic solution and chaos in ẋ(t) = f(x(t - 1)), Nonlinear Analysis 5 (1981), 775-788.
  • [21] H. O. Walther, Bifurcation from periodic solutions in functional differential equations, Math. Z. 182 (1983), 269-289.
  • [22] H. O. Walther, Bifurcation from a heteroclinic solution in differential delay equations, Trans. Amer. Math. Soc. 290 (1985), 213-233.
  • [23] H. O. Walther, Inclination lemmas with dominated convergence, J. Appl. Math. Phys. 32 (1987), 327-337.
Języki publikacji
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Uwagi
Identyfikator YADDA
bwmeta1.element.zamlynska-4390bd52-8c33-45d1-a2e3-692e6b316f53
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ISBN
83-01-09920-8
ISSN
0012-3862
Kolekcja
DML-PL
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