Article Contents
Article Contents

# Nonautonomous continuation of bounded solutions

• We show the persistence of hyperbolic bounded solutions to nonautonomous difference and retarded functional differential equations under parameter perturbation, where hyperbolicity is given in terms of an exponential dichotomy in variation. Our functional-analytical approach is based on a formulation of dynamical systems as operator equations in ambient sequence or function spaces. This yields short proofs, in particular of the stable manifold theorem.
As an ad hoc application, the behavior of hyperbolic solutions and stable manifolds for ODEs under numerical discretization with varying step-sizes is studied.
Mathematics Subject Classification: 39A11, 37B55, 37C60, 37G10, 37G15.

 Citation:

•  [1] E. L. Allgower and K. Georg, "Numerical Continuation Methods. An Introduction," Springer Series in Computational Mathematics 13, Springer, Berlin etc., 1990. [2] A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations, Computers & Mathematics with Applications, 38 (1998), 41-49. [3] H. Amann, "Ordinary Differential Equations: An Introduction to Nonlinear Analysis," Studies in Mathematics 13, Walter De Gruyter, Berlin-New York, 1990. [4] B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations II, Journal of Difference Equations and Applications, 2 (1996), 251-262.doi: doi:10.1080/10236199608808060. [5] B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, Journal of Difference Equations and Applications, 7 (2001), 895-913.doi: doi:10.1080/10236190108808310. [6] A. G. Baskakov, Invertibility and the Fredholm property of difference operators, Mathematical Notes, 67 (2000), 690-698.doi: doi:10.1007/BF02675622. [7] A. Ben-Artzi and I. Gohberg, Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts, Integral Equations and Operator Theory, 14 (1991), 613-677.doi: doi:10.1007/BF01200554. [8] A. Berger, Counting uniformly attracting solutions of nonautonomous differential equations, Discrete and Continuous Dynamical Systems (Series S), 1 (2008), 15-25. [9] A. Berger and S. Siegmund, Uniformly attracting solutions of nonautonomous differential equations, Nonlinear Analysis (TMA), 68 (2008), 3789-3811.doi: doi:10.1016/j.na.2007.04.020. [10] W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps, SIAM Journal of Numerical Analysis, 34 (1997), 1209-1236.doi: doi:10.1137/S0036142995281693. [11] Z. Bishnani and R. S. Mackay, Safety criteria for aperiodically forced systems, Dynamical Systems, 18 (2003), 107-129.doi: doi:10.1080/1468936031000080795. [12] O. Boichuk, Solutions of linear and nonlinear difference equations bounded on the whole line, Nonlinear Oscillations, 4 (2001), 16-27. [13] A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-auton-omous attractors, Cadernos de Matemática, 07 (2006), 277-302. [14] C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition, Texts in Applied Mathematics 34, Springer, Berlin etc., 2006. [15] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs 70, AMS, Providence RI, 1999. [16] W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Math. 629, Springer, Berlin etc., 1978. [17] I. Győri and M. Pituk, The converse of the theorem on stability by the first approximation for difference equations, Nonlinear Analysis (TMA), 47 (2001), 4635-4640.doi: doi:10.1016/S0362-546X(01)00576-4. [18] A. Hagen, Hyperbolic trajectories of time discretizations, Nonlinear Analysis (TMA), 59 (2004), 121-132. [19] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences 99, Springer, Berlin etc., 1993. [20] J. K. Hale and M. Weedermann, On perturbations of delay-differential equations with periodic orbits, Journal of Differential Equations, 197 (2004), 219-246.doi: doi:10.1016/S0022-0396(02)00063-3. [21] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Math. 840, Springer, Berlin etc., 1981. [22] M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, Boston, 1974. [23] J. M. Holtzman, Explicit $\epsilon$ and $\delta$ for the implicit function theorem, SIAM Review, 12 (1970), 284-286.doi: doi:10.1137/1012051. [24] T. Hüls, Homoclinic trajectories of non-autonomous maps, Journal of Difference Equations and Applications, to appear, 2009. [25] G. Iooss, "Bifurcation of Maps and Applications," Mathematics Studies 36, North-Holland, Amsterdam, 1979. [26] N. Ju, D. Small and S. Wiggins, Existence and computation of hyperbolic trajectories of aperiodically time dependent vector fields and their approximations, International Journal of Bifurcation and Chaos, 13 (2003), 1449-1457.doi: doi:10.1142/S0218127403007321. [27] J. Kalkbrenner, "Exponentielle Dichotomie und chaotische Dynamik nichtinvertierbarer Differenzengleichungen," Ph.D. thesis, Universität Augsburg, Germany, 1994. [28] H. Kielhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs," Applied Mathematical Sciences 156, Springer, New York, 2004. [29] B. Krauskopf, H. M. Osinga and J. Galán-Vioque, "Numerical Continuation Methods for Dynamical Systems. Path Following and Boundary Value Problems," Springer, Berlin etc., 2007. [30] S. Lang, "Real and Functional Analysis," Graduate Texts in Mathematics 142, Springer, Berlin etc., 1993. [31] P. Perfetti, An infinite-dimensional extension of a Poincaré result concerning the continuation of periodic orbits, Discrete and Continuous Dynamical Systems, 3 (1997), 401-418.doi: doi:10.3934/dcds.1997.3.401. [32] C. Pötzsche, A note on the dichotomy spectrum, Journal of Difference Equations and Applications, 15 (2009), 1021-1025. [33] C. Pötzsche and M. Rasmussen, Taylor approximation of invariant fiber bundles, Nonlinear Analysis (TMA), 60 (2005), 1303-1330. [34] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences 143, Springer, Berlin etc., 2002. [35] E. Zeidler, "Nonlinear Functional Analysis and its Applications I (Fixed-Points Theorems)," Springer, Berlin etc., 1993.