# American Institute of Mathematical Sciences

September  2011, 31(3): 941-973. doi: 10.3934/dcds.2011.31.941

## Nonautonomous bifurcation of bounded solutions II: A Shovel-Bifurcation pattern

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

Received  May 2010 Revised  June 2011 Published  August 2011

This paper continues our work on local bifurcations for nonautonomous difference and ordinary differential equations. Here, it is our premise that constant or periodic solutions are replaced by bounded entire solutions as bifurcating objects in order to encounter right-hand sides with an arbitrary time dependence.
We introduce a bifurcation pattern caused by a dominant spectral interval (of the dichotomy spectrum) crossing the stability boundary. As a result, differing from the classical autonomous (or periodic) situation, the change of stability appears in two steps from uniformly asymptotically stable to asymptotically stable and finally to unstable. During the asymptotically stable regime, a whole family of bounded entire solutions occurs (a so-called "shovel"). Our basic tools are exponential trichotomies and a quantitative version of the surjective implicit function theorem yielding the existence of strongly center manifolds.
Citation: Christian Pötzsche. Nonautonomous bifurcation of bounded solutions II: A Shovel-Bifurcation pattern. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 941-973. doi: 10.3934/dcds.2011.31.941
##### References:
 [1] R. P. Agarwal, "Difference Equations and Inequalities," 2nd edition, Pure and Applied Mathematics, 228, Marcel Dekker, Inc., New York, 2000. [2] A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl., 38 (1999), 41-49. doi: 10.1016/S0898-1221(99)00167-4. [3] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [4] B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Difference Equ. Appl., 7 (2001), 895-913. [5] _____, A spectral theory for nonautonomous difference equations, New Trends in Difference Equations (Temuco, Chile, 2000) (eds. J. López-Fenner, et al.), Taylor & Francis, London, (2002), 45-55. [6] B. Aulbach, A reduction principle for nonautonomous differential equations, Archiv der Mathematik, 39 (1982), 217-232. doi: 10.1007/BF01899528. [7] B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations, II, J. Difference Equ. Appl., 2 (1996), 251-262. [8] A. Ben-Artzi and I. Gohberg, Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts, Integral Equations Oper. Theory, 14 (1991), 613-677. doi: 10.1007/BF01200554. [9] _____, Dichotomies of perturbed time varying systems and the power method, Indiana Univ. Math. J., 42 (1993), 699-720. doi: 10.1512/iumj.1993.42.42031. [10] A. G. Baskakov, Invertibility and the Fredholm property of difference operators, Mathematical Notes, 67 (2000), 690-698. [11] W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations," D. C. Heath and Co., Boston, Mass., 1965. [12] _____, "Dichotomies in Stability Theory," Lect. Notes Math., 629, Springer-Verlag, Berlin-New York, 1978. [13] J. L. Dalec'kiĭ and M. G. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space," Translations of Mathematical Monographs, 43, American Mathematical Society, Providence, RI, 1974. [14] L. Dieci and E. S. van Vleck, Lyapunov and Sacker-Sell spectral intervals, J. Dyn. Differ. Equations, 19 (2007), 265-293. [15] S. Elaydi and O. Hajek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362-374. doi: 10.1016/0022-247X(88)90255-7. [16] S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Difference Equ. Appl., 3 (1998), 417-448. [17] D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math., 840, Springer, Berlin-New York, 1981. [18] J. M. Holtzman, Explicit $\epsilon$ and $\delta$ for the implicit function theorem, SIAM Review, 12 (1970), 284-286. doi: 10.1137/1012051. [19] T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064. doi: 10.1137/090754509. [20] R. A. Johnson, P. E. Kloeden and R. Pavani, Two-step transitions in nonautonomous bifurcations: An explanation, Stoch. Dyn., 2 (2002), 67-92. doi: 10.1142/S0219493702000297. [21] T. Kato, "Perturbation Theory for Linear Operators," reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. [22] K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differ. Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. [23] _____, Exponential dichotomies for almost periodic equations, Proc. Am. Math. Soc., 101 (1987), 293-298. doi: 10.1090/S0002-9939-1987-0902544-6. [24] _____, Exponential dichotomies and Fredholm operators, Proc. Am. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. [25] G. Papaschinopoulos, Exponential dichotomy for almost periodic linear difference equations, Ann. Soc. Sci. Bruxelles, Sér. I, 102 (1988), 19-28. [26] _____, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109. doi: 10.1080/00036819108839996. [27] C. Pötzsche, Stability of center fiber bundles for nonautonomous difference equations, in "Difference and Differential Equations" (eds. S. Elaydi, et al), Fields Institute Communications, 42, American Mathematical Society, Providence, RI, (2004), 295-304. [28] _____, A note on the dichotomy spectrum, J. Difference Equ. Appl., 15 (2009), 1021-1025. [29] _____, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach, Discrete and Continuous Dynamical Systems (Series B), 14 (2010), 739-776. doi: 10.3934/dcdsb.2010.14.739. [30] _____, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961. [31] C. Pötzsche and M. Rasmussen, Local approximation of invariant fiber bundles: An algorithmic approach, in "Difference Equations and Discrete Dynamical Systems" (eds. R. J. Sacker, et al), World Scientific, Hackensack, NJ, 2005, 155-170. [32] _____, Taylor approximation of invariant fiber bundles for nonautonomous difference equations, Nonlin. Analysis, 60 (2005), 1303-1330. doi: 10.1016/j.na.2004.10.019. [33] _____, Taylor approximation of integral manifolds, J. Dyn. Differ. Equations, 18 (2006), 427-460. [34] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dyn. Differ. Equations, 14 (2002), 243-258. [35] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differ. Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. [36] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [37] E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. (Fixed-Point Theorems)," Springer-Verlag, New York, 1986.

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##### References:
 [1] R. P. Agarwal, "Difference Equations and Inequalities," 2nd edition, Pure and Applied Mathematics, 228, Marcel Dekker, Inc., New York, 2000. [2] A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl., 38 (1999), 41-49. doi: 10.1016/S0898-1221(99)00167-4. [3] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [4] B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Difference Equ. Appl., 7 (2001), 895-913. [5] _____, A spectral theory for nonautonomous difference equations, New Trends in Difference Equations (Temuco, Chile, 2000) (eds. J. López-Fenner, et al.), Taylor & Francis, London, (2002), 45-55. [6] B. Aulbach, A reduction principle for nonautonomous differential equations, Archiv der Mathematik, 39 (1982), 217-232. doi: 10.1007/BF01899528. [7] B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations, II, J. Difference Equ. Appl., 2 (1996), 251-262. [8] A. Ben-Artzi and I. Gohberg, Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts, Integral Equations Oper. Theory, 14 (1991), 613-677. doi: 10.1007/BF01200554. [9] _____, Dichotomies of perturbed time varying systems and the power method, Indiana Univ. Math. J., 42 (1993), 699-720. doi: 10.1512/iumj.1993.42.42031. [10] A. G. Baskakov, Invertibility and the Fredholm property of difference operators, Mathematical Notes, 67 (2000), 690-698. [11] W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations," D. C. Heath and Co., Boston, Mass., 1965. [12] _____, "Dichotomies in Stability Theory," Lect. Notes Math., 629, Springer-Verlag, Berlin-New York, 1978. [13] J. L. Dalec'kiĭ and M. G. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space," Translations of Mathematical Monographs, 43, American Mathematical Society, Providence, RI, 1974. [14] L. Dieci and E. S. van Vleck, Lyapunov and Sacker-Sell spectral intervals, J. Dyn. Differ. Equations, 19 (2007), 265-293. [15] S. Elaydi and O. Hajek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362-374. doi: 10.1016/0022-247X(88)90255-7. [16] S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Difference Equ. Appl., 3 (1998), 417-448. [17] D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math., 840, Springer, Berlin-New York, 1981. [18] J. M. Holtzman, Explicit $\epsilon$ and $\delta$ for the implicit function theorem, SIAM Review, 12 (1970), 284-286. doi: 10.1137/1012051. [19] T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064. doi: 10.1137/090754509. [20] R. A. Johnson, P. E. Kloeden and R. Pavani, Two-step transitions in nonautonomous bifurcations: An explanation, Stoch. Dyn., 2 (2002), 67-92. doi: 10.1142/S0219493702000297. [21] T. Kato, "Perturbation Theory for Linear Operators," reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995. [22] K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differ. Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2. [23] _____, Exponential dichotomies for almost periodic equations, Proc. Am. Math. Soc., 101 (1987), 293-298. doi: 10.1090/S0002-9939-1987-0902544-6. [24] _____, Exponential dichotomies and Fredholm operators, Proc. Am. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1. [25] G. Papaschinopoulos, Exponential dichotomy for almost periodic linear difference equations, Ann. Soc. Sci. Bruxelles, Sér. I, 102 (1988), 19-28. [26] _____, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109. doi: 10.1080/00036819108839996. [27] C. Pötzsche, Stability of center fiber bundles for nonautonomous difference equations, in "Difference and Differential Equations" (eds. S. Elaydi, et al), Fields Institute Communications, 42, American Mathematical Society, Providence, RI, (2004), 295-304. [28] _____, A note on the dichotomy spectrum, J. Difference Equ. Appl., 15 (2009), 1021-1025. [29] _____, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach, Discrete and Continuous Dynamical Systems (Series B), 14 (2010), 739-776. doi: 10.3934/dcdsb.2010.14.739. [30] _____, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961. [31] C. Pötzsche and M. Rasmussen, Local approximation of invariant fiber bundles: An algorithmic approach, in "Difference Equations and Discrete Dynamical Systems" (eds. R. J. Sacker, et al), World Scientific, Hackensack, NJ, 2005, 155-170. [32] _____, Taylor approximation of invariant fiber bundles for nonautonomous difference equations, Nonlin. Analysis, 60 (2005), 1303-1330. doi: 10.1016/j.na.2004.10.019. [33] _____, Taylor approximation of integral manifolds, J. Dyn. Differ. Equations, 18 (2006), 427-460. [34] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dyn. Differ. Equations, 14 (2002), 243-258. [35] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differ. Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. [36] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. [37] E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. (Fixed-Point Theorems)," Springer-Verlag, New York, 1986.
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