# American Institute of Mathematical Sciences

May  2011, 10(3): 937-961. doi: 10.3934/cpaa.2011.10.937

## Nonautonomous continuation of bounded solutions

 1 Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, D-85758 Garching

Received  March 2009 Revised  September 2009 Published  December 2010

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.
Citation: Christian Pötzsche. Nonautonomous continuation of bounded solutions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 937-961. doi: 10.3934/cpaa.2011.10.937
##### References:
 [1] E. L. Allgower and K. Georg, "Numerical Continuation Methods. An Introduction,", Springer Series in Computational Mathematics 13, (1990).   Google Scholar [2] A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations,, Computers & Mathematics with Applications, 38 (1998), 41.   Google Scholar [3] H. Amann, "Ordinary Differential Equations: An Introduction to Nonlinear Analysis,", Studies in Mathematics 13, (1990).   Google Scholar [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.  doi: doi:10.1080/10236199608808060.  Google Scholar [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.  doi: doi:10.1080/10236190108808310.  Google Scholar [6] A. G. Baskakov, Invertibility and the Fredholm property of difference operators,, Mathematical Notes, 67 (2000), 690.  doi: doi:10.1007/BF02675622.  Google Scholar [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.  doi: doi:10.1007/BF01200554.  Google Scholar [8] A. Berger, Counting uniformly attracting solutions of nonautonomous differential equations,, Discrete and Continuous Dynamical Systems (Series S), 1 (2008), 15.   Google Scholar [9] A. Berger and S. Siegmund, Uniformly attracting solutions of nonautonomous differential equations,, Nonlinear Analysis (TMA), 68 (2008), 3789.  doi: doi:10.1016/j.na.2007.04.020.  Google Scholar [10] W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps,, {SIAM} Journal of Numerical Analysis, 34 (1997), 1209.  doi: doi:10.1137/S0036142995281693.  Google Scholar [11] Z. Bishnani and R. S. Mackay, Safety criteria for aperiodically forced systems,, Dynamical Systems, 18 (2003), 107.  doi: doi:10.1080/1468936031000080795.  Google Scholar [12] O. Boichuk, Solutions of linear and nonlinear difference equations bounded on the whole line,, Nonlinear Oscillations, 4 (2001), 16.   Google Scholar [13] A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-auton-omous attractors,, Cadernos de Matem{\'a}tica, 07 (2006), 277.   Google Scholar [14] C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition,, Texts in Applied Mathematics 34, (2006).   Google Scholar [15] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs 70, (1999).   Google Scholar [16] W. A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Math. 629, (1978).   Google Scholar [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.  doi: doi:10.1016/S0362-546X(01)00576-4.  Google Scholar [18] A. Hagen, Hyperbolic trajectories of time discretizations,, Nonlinear Analysis (TMA), 59 (2004), 121.   Google Scholar [19] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences 99, (1993).   Google Scholar [20] J. K. Hale and M. Weedermann, On perturbations of delay-differential equations with periodic orbits,, Journal of Differential Equations, 197 (2004), 219.  doi: doi:10.1016/S0022-0396(02)00063-3.  Google Scholar [21] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Math. 840, (1981).   Google Scholar [22] M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,", Academic Press, (1974).   Google Scholar [23] J. M. Holtzman, Explicit $\epsilon$ and $\delta$ for the implicit function theorem,, SIAM Review, 12 (1970), 284.  doi: doi:10.1137/1012051.  Google Scholar [24] T. Hüls, Homoclinic trajectories of non-autonomous maps,, Journal of Difference Equations and Applications, (2009).   Google Scholar [25] G. Iooss, "Bifurcation of Maps and Applications,", Mathematics Studies 36, (1979).   Google Scholar [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.  doi: doi:10.1142/S0218127403007321.  Google Scholar [27] J. Kalkbrenner, "Exponentielle Dichotomie und chaotische Dynamik nichtinvertierbarer Differenzengleichungen,", Ph.D. thesis, (1994).   Google Scholar [28] H. Kielhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs,", Applied Mathematical Sciences 156, (2004).   Google Scholar [29] B. Krauskopf, H. M. Osinga and J. Galán-Vioque, "Numerical Continuation Methods for Dynamical Systems. Path Following and Boundary Value Problems,", Springer, (2007).   Google Scholar [30] S. Lang, "Real and Functional Analysis,", Graduate Texts in Mathematics 142, (1993).   Google Scholar [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.  doi: doi:10.3934/dcds.1997.3.401.  Google Scholar [32] C. Pötzsche, A note on the dichotomy spectrum,, Journal of Difference Equations and Applications, 15 (2009), 1021.   Google Scholar [33] C. Pötzsche and M. Rasmussen, Taylor approximation of invariant fiber bundles,, Nonlinear Analysis (TMA), 60 (2005), 1303.   Google Scholar [34] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences 143, (2002).   Google Scholar [35] E. Zeidler, "Nonlinear Functional Analysis and its Applications I (Fixed-Points Theorems),", Springer, (1993).   Google Scholar

show all references

##### References:
 [1] E. L. Allgower and K. Georg, "Numerical Continuation Methods. An Introduction,", Springer Series in Computational Mathematics 13, (1990).   Google Scholar [2] A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations,, Computers & Mathematics with Applications, 38 (1998), 41.   Google Scholar [3] H. Amann, "Ordinary Differential Equations: An Introduction to Nonlinear Analysis,", Studies in Mathematics 13, (1990).   Google Scholar [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.  doi: doi:10.1080/10236199608808060.  Google Scholar [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.  doi: doi:10.1080/10236190108808310.  Google Scholar [6] A. G. Baskakov, Invertibility and the Fredholm property of difference operators,, Mathematical Notes, 67 (2000), 690.  doi: doi:10.1007/BF02675622.  Google Scholar [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.  doi: doi:10.1007/BF01200554.  Google Scholar [8] A. Berger, Counting uniformly attracting solutions of nonautonomous differential equations,, Discrete and Continuous Dynamical Systems (Series S), 1 (2008), 15.   Google Scholar [9] A. Berger and S. Siegmund, Uniformly attracting solutions of nonautonomous differential equations,, Nonlinear Analysis (TMA), 68 (2008), 3789.  doi: doi:10.1016/j.na.2007.04.020.  Google Scholar [10] W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps,, {SIAM} Journal of Numerical Analysis, 34 (1997), 1209.  doi: doi:10.1137/S0036142995281693.  Google Scholar [11] Z. Bishnani and R. S. Mackay, Safety criteria for aperiodically forced systems,, Dynamical Systems, 18 (2003), 107.  doi: doi:10.1080/1468936031000080795.  Google Scholar [12] O. Boichuk, Solutions of linear and nonlinear difference equations bounded on the whole line,, Nonlinear Oscillations, 4 (2001), 16.   Google Scholar [13] A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-auton-omous attractors,, Cadernos de Matem{\'a}tica, 07 (2006), 277.   Google Scholar [14] C. Chicone, "Ordinary Differential Equations with Applications," 2nd edition,, Texts in Applied Mathematics 34, (2006).   Google Scholar [15] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs 70, (1999).   Google Scholar [16] W. A. Coppel, "Dichotomies in Stability Theory,", Lecture Notes in Math. 629, (1978).   Google Scholar [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.  doi: doi:10.1016/S0362-546X(01)00576-4.  Google Scholar [18] A. Hagen, Hyperbolic trajectories of time discretizations,, Nonlinear Analysis (TMA), 59 (2004), 121.   Google Scholar [19] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences 99, (1993).   Google Scholar [20] J. K. Hale and M. Weedermann, On perturbations of delay-differential equations with periodic orbits,, Journal of Differential Equations, 197 (2004), 219.  doi: doi:10.1016/S0022-0396(02)00063-3.  Google Scholar [21] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Math. 840, (1981).   Google Scholar [22] M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,", Academic Press, (1974).   Google Scholar [23] J. M. Holtzman, Explicit $\epsilon$ and $\delta$ for the implicit function theorem,, SIAM Review, 12 (1970), 284.  doi: doi:10.1137/1012051.  Google Scholar [24] T. Hüls, Homoclinic trajectories of non-autonomous maps,, Journal of Difference Equations and Applications, (2009).   Google Scholar [25] G. Iooss, "Bifurcation of Maps and Applications,", Mathematics Studies 36, (1979).   Google Scholar [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.  doi: doi:10.1142/S0218127403007321.  Google Scholar [27] J. Kalkbrenner, "Exponentielle Dichotomie und chaotische Dynamik nichtinvertierbarer Differenzengleichungen,", Ph.D. thesis, (1994).   Google Scholar [28] H. Kielhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs,", Applied Mathematical Sciences 156, (2004).   Google Scholar [29] B. Krauskopf, H. M. Osinga and J. Galán-Vioque, "Numerical Continuation Methods for Dynamical Systems. Path Following and Boundary Value Problems,", Springer, (2007).   Google Scholar [30] S. Lang, "Real and Functional Analysis,", Graduate Texts in Mathematics 142, (1993).   Google Scholar [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.  doi: doi:10.3934/dcds.1997.3.401.  Google Scholar [32] C. Pötzsche, A note on the dichotomy spectrum,, Journal of Difference Equations and Applications, 15 (2009), 1021.   Google Scholar [33] C. Pötzsche and M. Rasmussen, Taylor approximation of invariant fiber bundles,, Nonlinear Analysis (TMA), 60 (2005), 1303.   Google Scholar [34] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences 143, (2002).   Google Scholar [35] E. Zeidler, "Nonlinear Functional Analysis and its Applications I (Fixed-Points Theorems),", Springer, (1993).   Google Scholar
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