A local unstable manifold for differential equations with state-dependent delay

Discrete and Continuous Dynamical Systems

Volumn 9, Issue 4, 2003, Pages 993-1028

  Krisztin, Tibor ^a  

^a UNIVERSITY OF SZEGED   (Hungary)

Author keywords

Functional differential equation; Lyapunov Perron method; State dependent delay; Substitution operator; Unstable manifold; Variation of constans formula

Indexed keywords

EID: 0038062904     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2003.9.993     Document Type: Article

References (25)

1. W. Alt, Some periodicity criteria for functional differential equations, Manuscripta Mathematica 23 (1978), 295-318.
2. O. Arino, K.P. Hadeler and M.L. Hbid, Existence of periodic solutions for delay differential equations with state dependent delay, J. Differential Equations 144 (1998), 263-301.
3. M. Bartha, Convergence of solutions for an equation with state-dependent delay, J. Math. Anal. Appl. 254 (2001), 410-432.
4. J. Bélair and M.C. Mackey, Consumer memory and price fluctuations on commodity markets: an integrodifferential model, J. Dynamics and Differential Equations 1 (1989), 299-325.
5. M. Büger and M.R.W. Martin, The escaping disaster: A problem related to state-dependent delays, preprint.
6. K.L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc. 124 (1996), 1417-1426.
7. O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel and H.-O. Walther, Delay Equations, Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
8. R.D. Driver, A two-body problem of classical electrodynamics, Annals of Physics 21 (1963), 122-142.
9. T. Faria, Normal forms for semilinear functional differential equations in Banach spaces and applications, Discrete and Continuous Dynamical Systems 7 (2001), 155-176.
10. J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
11. F. Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, preprint.
12. F. Hartung and J. Turi, On differentiability of solutions with respect to parameters in state-dependent delay equations, J. Differential Equations 135 (1997), 192-237.
13. D. Henry, Variedades invariantes perto dum ponto fixo, Lecture Notes, Univ. Sao Paulo, Brasil, 1983.
14. H.P. Krishnan, An analysis of singularly perturbed delay-differential equations and equations with state-dependent delays, Ph.D. Thesis, Brown University, Providence, RI, 1998.
15. T. Krisztin and O. Arino, The 2-dimensional attractor of a differential equation with state-dependent delay, J. Dynamics and Differential Equations 13 (2001), 453-522.
16. Y. Kuang and H.L. Smith, Periodic solutions of differential delay equations with threshold-type delays, in "Oscillation and Dynamics in Delay Equations", J.R. Graef and J.K. Hale eds., pp. 153-176, Contemporary Mathematics vol. 120, AMS, Providence, RI, 1992.
17. Y. Kuang and H.L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay differential equations, Nonlinear Anal. 19 (1992), 855-872.
18. P. Magal and O. Arino, Existence of periodic solutions for a state dependent delay differential equation, J. Differential Equations 165 (2000), 61-95.
19. J. Mallet-Paret and R.D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags: I, Arch. Rational Mech. Anal. 120 (1992), 99-146.
20. J. Mallet-Paret and R.D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags: II, J. reine angew. Math. 477 (1996), 129-197.
21. J. Mallet-Paret and R.D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discrete and Continuous Dynamical Systems 8 (2002), 519-562.
22. J. Mallet-Paret, R.D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal. 3 (1994), 101-162.
23. H.-O. Walther, Contracting maps for monotone delayed feedback, Discrete and Continuous Dynamical Systems 7 (2001), 155-176.
24. H.-O. Walther, Stable periodic motion of a system with state dependent delay, Differential and Integral Equations 15 (2002), 923-944.
25. 1 -smoothness for differential equations with state dependent delay, preprint, 2002.