Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory

Robert Skiba; Nils Waterstraat

doi:10.3934/dcds.2022183

Article Contents

2023, Volume 43, Issue 5: 1878-1904. Doi: 10.3934/dcds.2022183

This issue Previous Article Non-autonomous stochastic lattice systems with Markovian switching Next Article Vacuum and singularity formation for compressible Euler equations with time-dependent damping

Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory

Robert Skiba^1, , and
Nils Waterstraat^2,

1.
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń, ul. Chopina 12/18, 87-100 Toruń, Poland
2.
Institut für Mathematik, Naturwissenschaftliche Fakultät Ⅱ, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany

^*Corresponding author: Robert Skiba
^*Corresponding author: Robert Skiba

Received: May 2022

Revised: November 2022

Early access: December 2022

Published: May 2023

The first author is supported by the Polish National Science Center (NCN) under grant DEC-2017/01/X/ST1/00889 and by the Polish National Agency for Academic Exchange (NAWA)
The second author is supported by the German Academic Exchange Service (DAAD)

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Abstract

In a previous work, we proved an index theorem for families of asymptotically hyperbolic discrete dynamical systems and obtained applications to bifurcation theory. A weaker and far more common assumption than asymptotic hyperbolicity is the existence of an exponential dichotomy. In this paper we generalize all our previous results to the latter setting, which requires substantial modifications of our arguments. In addition, we generalize previous results on continuity and differentiability of Nemitski operators for discrete dynamical systems to further weaken the assumptions of our bifurcation theorems.

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Mathematics Subject Classification: Primary: 58E07, 37C29; Secondary: 19L20, 47A53.

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References

[1]	H. Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis, Walter de Gruyter, Berlin, 1990. doi: 10.1515/9783110853698.
[2]	D. Arlt, Zusammenziehbarkeit der allgemeinen linearen Gruppe des Raumes $c_0$ der Nullfolgen, Invent. Math., 1 (1966), 36-44. doi: 10.1007/BF01389697.
[3]	M. F. Atiyah, Thom complexes, Proc. Lond. Math. Soc., 11 (1961), 291-310. doi: 10.1112/plms/s3-11.1.291.
[4]	M. F. Atiyah, K-Theory, Addison-Wesley, 1989.
[5]	B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations Ⅱ, J. Difference Equ. Appl., 2 (1996), 251-262. doi: 10.1080/10236199608808060.
[6]	B. Aulbach, N. Van Minh and P. P. Zabreiko, The concept of spectral dichotomy for linear difference equations, J. Math. Anal. Appl., 185 (1994), 275-287. doi: 10.1006/jmaa.1994.1248.
[7]	L. Barreira and C. Valls, Smooth robustness of exponential dichotomies, Proc. Am. Math. Soc., 139 (2011), 999-1012. doi: 10.1090/S0002-9939-2010-10531-2.
[8]	T. Bartsch, The global structure of the zero set of a family of semilinear Fredholm maps, Nonlinear Analysis: Theory, Methods and Applications, 17 (1991), 313-331. doi: 10.1016/0362-546X(91)90074-B.
[9]	P. M. Fitzpatrick and J. Pejsachowicz, Nonorientability of the index bundle and several-parameter bifurcation, J. Funct. Anal., 98 (1991), 42-58. doi: 10.1016/0022-1236(91)90090-R.
[10]	P. Fitzpatrick and J. Pejsachowicz, Orientation and the Leary-Schauder Theory for Fully Nonlinear Elliptic Boundary Value Problems, Memoris of the American Mathematical Society, 483, 1993. doi: 10.1090/memo/0483.
[11]	I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators Vol. I, Operator Theory: Advances and Applications, 49, Birkhäuser, 1990. doi: 10.1007/978-3-0348-7509-7.
[12]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981.
[13]	S. Lang, Differential and Riemannian Manifolds, Third edition, Graduate Texts in Mathematics, 160, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4182-9.
[14]	K. J. Palmer, Exponential dichotomies and transversal homoclinic points, Journal of Differential Equations, 55 (1984), 225-256. doi: 10.1016/0022-0396(84)90082-2.
[15]	K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported, 1 (1988), 265-306.
[16]	J. Pejsachowicz, Bifurcation of homoclinics, Proc. Amer. Math. Soc., 136 (2008), 111-118. doi: 10.1090/S0002-9939-07-09088-0.
[17]	J. Pejsachowicz, Bifurcation of Fredholm maps Ⅱ. The dimension of the set of bifurcation points, Topol. Methods Nonlinear Anal., 38 (2011), 291-305.
[18]	J. Pejsachowicz and R. Skiba, Global bifurcation of homoclinic trajectories of discrete dynamical systems, Central European Journal of Mathematics, 10 (2012), 2088-2109. doi: 10.2478/s11533-012-0121-8.
[19]	J. Pejsachowicz and R. Skiba, Topology and homoclinic trajectories of discrete dynamical systems, Discrete and Continuous Dynamical Systems, Ser. S, 6 (2013), 1077-1094. doi: 10.3934/dcdss.2013.6.1077.
[20]	M. Pituk, The local spectral radius of a nonnegative orbit of compact linear operators, Mathematica Slovaca, 66 (2016), 707-714. doi: 10.1515/ms-2015-0172.
[21]	C. Pötzsche, A Note on the dichotomy spectrum, J. Difference Equ. Appl., 15 (2009), 1021-1025. doi: 10.1080/10236190802320147.
[22]	C. Pötzsche, Nonautonomous bifurcation of bounded solutions Ⅰ: A Lyapunov-Schmidt approach, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 739-776. doi: 10.3934/dcdsb.2010.14.739.
[23]	C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, vol. 2002, Lect. Notes Math., Springer, Berlin, 2010. doi: 10.1007/978-3-642-14258-1.
[24]	C. Pötzsche, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961. doi: 10.3934/cpaa.2011.10.937.
[25]	C. Pötzsche, Bifurcations in Nonautonomous Dynamical Systems: Results and tools in discrete time, Proceedings of the International Workshop Future Directions in Difference Equations, 2011,163-212.
[26]	C. Pötzsche, Smooth roughness of exponential dichotomies, revisited, Discrete Contin. Dyn. Syst., Ser. B, 20 (2015), 853-859. doi: 10.3934/dcdsb.2015.20.853.
[27]	C. Pötzsche and R. Skiba, Global continuation of homoclinic solutions, Zeitschrift für Analysis und ihre Anwendungen, 37 (2018), 159-187. doi: 10.4171/ZAA/1608.
[28]	E. Russ, Dichotomy spectrum for difference equations in Banach spaces, Journal of Difference Equations and Applications, 23 (2016), 574-617. doi: 10.1080/10236198.2016.1263622.
[29]	R. Skiba and N. Waterstraat, The index bundle and multiparameter bifurcation for discrete dynamical systems, Discrete Contin. Dyn. Syst., Ser. A, 37 (2017), 5603-5629. doi: 10.3934/dcds.2017243.
[30]	N. Waterstraat, The index bundle for Fredholm morphisms, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 299-315.
[31]	N. Waterstraat, A remark on bifurcation of Fredholm maps, Adv. Nonlinear Anal., 7 (2018), 285-292. doi: 10.1515/anona-2016-0067.