# 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 & Continuous Dynamical Systems, 2011, 31 (3) : 941-973. doi: 10.3934/dcds.2011.31.941
##### References:
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Notes Math., 840, Springer, Berlin-New York, 1981.  Google Scholar [18] J. M. Holtzman, Explicit $\epsilon$ and $\delta$ for the implicit function theorem, SIAM Review, 12 (1970), 284-286. doi: 10.1137/1012051.  Google Scholar [19] T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064. doi: 10.1137/090754509.  Google Scholar [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.  Google Scholar [21] T. Kato, "Perturbation Theory for Linear Operators," reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar [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.  Google Scholar [23] _____, Exponential dichotomies for almost periodic equations, Proc. Am. Math. Soc., 101 (1987), 293-298. doi: 10.1090/S0002-9939-1987-0902544-6.  Google Scholar [24] _____, Exponential dichotomies and Fredholm operators, Proc. Am. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1.  Google Scholar [25] G. Papaschinopoulos, Exponential dichotomy for almost periodic linear difference equations, Ann. Soc. Sci. Bruxelles, Sér. I, 102 (1988), 19-28.  Google Scholar [26] _____, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109. doi: 10.1080/00036819108839996.  Google Scholar [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.  Google Scholar [28] _____, A note on the dichotomy spectrum, J. Difference Equ. Appl., 15 (2009), 1021-1025.  Google Scholar [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.  Google Scholar [30] _____, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] _____, Taylor approximation of integral manifolds, J. Dyn. Differ. Equations, 18 (2006), 427-460.  Google Scholar [34] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dyn. Differ. Equations, 14 (2002), 243-258.  Google Scholar [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.  Google Scholar [36] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.  Google Scholar [37] E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. (Fixed-Point Theorems)," Springer-Verlag, New York, 1986.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [3] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.  Google Scholar [4] B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Difference Equ. Appl., 7 (2001), 895-913.  Google Scholar [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.  Google Scholar [6] B. Aulbach, A reduction principle for nonautonomous differential equations, Archiv der Mathematik, 39 (1982), 217-232. doi: 10.1007/BF01899528.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] A. G. Baskakov, Invertibility and the Fredholm property of difference operators, Mathematical Notes, 67 (2000), 690-698.  Google Scholar [11] W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations," D. C. Heath and Co., Boston, Mass., 1965.  Google Scholar [12] _____, "Dichotomies in Stability Theory," Lect. Notes Math., 629, Springer-Verlag, Berlin-New York, 1978.  Google Scholar [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.  Google Scholar [14] L. Dieci and E. S. van Vleck, Lyapunov and Sacker-Sell spectral intervals, J. Dyn. Differ. Equations, 19 (2007), 265-293.  Google Scholar [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.  Google Scholar [16] S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Difference Equ. Appl., 3 (1998), 417-448.  Google Scholar [17] D. Henry, Geometric theory of semilinear parabolic equations, Lect. Notes Math., 840, Springer, Berlin-New York, 1981.  Google Scholar [18] J. M. Holtzman, Explicit $\epsilon$ and $\delta$ for the implicit function theorem, SIAM Review, 12 (1970), 284-286. doi: 10.1137/1012051.  Google Scholar [19] T. Hüls, Computing Sacker-Sell spectra in discrete time dynamical systems, SIAM J. Numer. Anal., 48 (2010), 2043-2064. doi: 10.1137/090754509.  Google Scholar [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.  Google Scholar [21] T. Kato, "Perturbation Theory for Linear Operators," reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar [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.  Google Scholar [23] _____, Exponential dichotomies for almost periodic equations, Proc. Am. Math. Soc., 101 (1987), 293-298. doi: 10.1090/S0002-9939-1987-0902544-6.  Google Scholar [24] _____, Exponential dichotomies and Fredholm operators, Proc. Am. Math. Soc., 104 (1988), 149-156. doi: 10.1090/S0002-9939-1988-0958058-1.  Google Scholar [25] G. Papaschinopoulos, Exponential dichotomy for almost periodic linear difference equations, Ann. Soc. Sci. Bruxelles, Sér. I, 102 (1988), 19-28.  Google Scholar [26] _____, On exponential trichotomy of linear difference equations, Appl. Anal., 40 (1991), 89-109. doi: 10.1080/00036819108839996.  Google Scholar [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.  Google Scholar [28] _____, A note on the dichotomy spectrum, J. Difference Equ. Appl., 15 (2009), 1021-1025.  Google Scholar [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.  Google Scholar [30] _____, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] _____, Taylor approximation of integral manifolds, J. Dyn. Differ. Equations, 18 (2006), 427-460.  Google Scholar [34] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dyn. Differ. Equations, 14 (2002), 243-258.  Google Scholar [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.  Google Scholar [36] G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.  Google Scholar [37] E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. (Fixed-Point Theorems)," Springer-Verlag, New York, 1986.  Google Scholar
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