Robustness of hyperbolic solutions under parametric perturbations

Journal of Difference Equations and Applications

Volumn 15, Issue 8-9, 2009, Pages 803-819

  Pötzsche, Christian ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Almost periodic; Hyperbolic solution; Nonautonomous difference equation; Parametric perturbation; Persistence; Structural stability

Indexed keywords

EID: 70549103598     PISSN: 10236198     EISSN: 15635120     Source Type: Journal    
DOI: 10.1080/10236190802357644     Document Type: Article

References (40)

1. A. Alonso, J. Hong, and R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl. 38 (1998), pp. 41-49.
2. B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations II, J. Difference Equ. Appl. 2 (1996), pp. 251-262.
3. C. Corduneano, Almost periodic functions, Tracts in pure and applied mathematics, 22, Interscience, New York, 1968.
4. S. Elaydi and R. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Differential Equ. 208 (2005), pp. 258-273.
5. S. Elaydi and R. Sacker, Global stability of periodic orbits of non-autonomous difference equations and population biology and the Cushing-Henson conjectures, in Proceedings of the Eighth International Conference of Difference Equations and Application (Brno, Czech Republic, 2003), S. Elaydi, G. Ladas, B. Aulbach, and O. Dosly, eds., Chapman & Hall/CRC, London/Boca Raton, FL, 2005, pp. 113-126.
6. S. Elaydi and R. Sacker, Nonautonomous Beverton-Hold equations and the Cushing-Henson conjectures, J. Difference Equ. Appl. 11 (2005), pp. 337-346.
7. S. Elaydi and R. Sacker, Periodic difference equations, population biology and the Cushing-Henson conjecture, Math. Biosci. 2001 (2006), pp. 195-207.
8. K. Ey and C. Potzsche, Asymptotic behavior of recursions via fixed point theory, J. Math. Anal. Appl. 337 (2008), pp. 1125-1141.
9. P. Giesl and M. Rasmussen, A note on almost periodic variational equations, Submitted (2008). publication.
10. I. Gyo{double acute}ri and M. Pituk, The converse of the theorem on stability by the first approximation for difference equations, Nonlinear Anal. (TMA) 47 (2001), pp. 4635-4640.
11. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer, Berlin, 1981.
12. M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, Boston/New York/San Diego, 1974.
13. J. Holtzmann, Explicit e and d for the implicit function theorem, SIAM Rev. 12 (1970), pp. 284-286.
14. J. Hong and C. Núñez, The almost periodic type difference equation, Math. Comput. Model. 28 (1998), pp. 21-31.
15. T. Huls, Homoclinic trajectories of non-autonomous maps, (2008), preprint, Universitat Bielefeld.
16. R. Johnson, Remarks on linear differential systems with measurable coefficients, Proc. Am. Math. Soc. 100 (1987), pp. 491-504.
17. T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften 132, corrected 2nd ed., Springer, Berlin, 1980.
18. S. Lang, Real and Functional Analysis, Graduate Texts in Mathematics, 142, Springer, Berlin, 1993.
19. K. Palmer, The structurally stable linear systems on the half-line are those with an exponential dichotomy, J. Differ. Equ. 33 (1979), pp. 16-25.
20. K. Palmer, A perturbation theorem for exponential dichotomies, Proc. R. Soc. Edinburgh, Sect. A 106 (1987), pp. 25-37.
21. V. Pliss and G. Sell, Robustness of exponential dichotomies in infinite dimensional dynamical systems, J. Dyn. Differ. Equ. 11 (1999), pp. 471-513.
22. C. Potzsche, Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients, J. Math. Anal. Appl. 289 (2004), pp. 317-335.
23. C. Potzsche, A functional-analytical approach to the asymptotics of recursions, Proc. Am. Math. Soc. to appear, (2007).
24. C. Potzsche and M. Rasmussen, Local approximation of invariant fiber bundles: An algorithmic approach, in Difference Equations and Discrete Dynamical Systems, L. Allen, B. Aulbach, S. Elaydi, and R. Sacker, eds., World Scientific, New Jersey, London, 2005, pp. 155-170.
25. C. Potzsche and M. Rasmussen, Taylor approximation of invariant fiber bundles, Nonlinear Anal. (TMA) 60 (2005), pp. 1303-1330.
26. Y. Rong and J. Hong, The existence of almost periodic solutions for a class of differential equations with piecewise constant argument, Nonlinear Anal. (TMA) 28 (1997), pp. 1439-1450.
27. R. Sacker, Existence of dichotomies and invariant splittings for linear differential equations IV, J. Differ. Equ. 27 (1978), pp. 106-137.
28. R. Sacker, Dedication to George R. Sell, J. Differ. Equ. Appl. 9 (2003), pp. 437-440.
29. R. Sacker, A note on periodic Ricker maps, J. Differ.Equ. Appl. 13 (2007), pp. 89-92.
30. R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential equations I, J. Differ. Equ. 15 (1974), pp. 429-458.
31. R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential equations II, J. Differ. Equ. 22 (1976), pp. 478-496.
32. R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential equations III, J. Differ. Equ. 22 (1976), pp. 497-522.
33. R. Sacker and G. Sell, A spectral theory for linear differential systems, J. Differ. Equ. 27 (1978), pp. 320-358.
34. R. Sacker and G. Sell, Dichotomies for linear evolutionary equations in Banach spaces, J. Differ. Equ. 113 (1994), pp. 17-67.
35. R. Sacker and H. von Bremen, A new approach to cycling in a 2-locus 2-allele genetic model, J. Differ. Equ. Appl. 9 (2003), pp. 441-448.
36. R. Sacker and H. von Bremen, Global asymptotic stability in the Jia Li model for genetically altered mosquitoes, in Difference Equations and Discrete Dynamical Systems, L. Allen, B. Aulbach, S. Elaydi, and R. Sacker, eds., World Scientific, New Jersey, London, 2005, pp. 87-100.
37. R. Sacker and H. von Bremen, Dynamic reduction with applications to mathematical biology and other areas, J. Biol. Dyn. 1 (2007), pp. 1-17.
38. G. Sell, Almost periodic and periodic solutions of difference equations, Bull. AMS 72 (1966), pp. 261-265.
39. G. Sell and Y. You, Dynamics of Evolutionary Equations, Appllied Mathematical Science, 143, Springer, Berlin, 2002.
40. T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, Applied Mathematical Science, 14, Springer, Berlin, 1975.