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November  2017, 37(11): 5603-5629. doi: 10.3934/dcds.2017243

The index bundle and multiparameter bifurcation for discrete dynamical systems

Robert Skiba 1, and  Nils Waterstraat 2,

1. 

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Torun, Poland
2. 

School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, Kent CT2 7NF, United Kingdom

Received  March 2017 Revised  May 2017 Published  July 2017

We develop a K-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author.

Keywords: Homoclinic solutions, index bundle, bifurcation points, Stiefel-Whitney classes, Fredholm maps.
Mathematics Subject Classification: Primary: 58E07; Secondary: 37C29, 19L20, 47A53.
Citation: Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243
References:
[1]

A. Abbondandolo and P. Majer, On the global stable manifold, Studia Math., 177 (2006), 113-131.  doi: 10.4064/sm177-2-2.

[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. London Math. Soc.(3), 11 (1961), 291-310.  doi: 10.1112/plms/s3-11.1.291.

[4]

M. F. Atiyah and I. M. Singer, The index of elliptic operators. Ⅳ, Ann. of Math.(2), 93 (1971), 119-138.  doi: 10.2307/1970756.

[5]

M. F. Atiyah, K-Theory, Addison-Wesley, 1989.

[6]

T. Bartsch, The global structure of the zero set of a family of semilinear Fredholm maps, Nonlinear Anal, 71 (1991), 313-331.  doi: 10.1016/0362-546X(91)90074-B.

[7]

G. E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, 139 Springer, 1993. doi: 10.1007/978-1-4757-6848-0.

[8]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., vol. 629, Springer-Verlag, New York, 1978.

[9]

M. Crabb and I. James, Fibrewise Homotopy Theory, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1998. doi: 10.1007/978-1-4471-1265-5.

[10]

A. Dold, ÜUber fasernweise Homotopieäquivalenz von Faserräumen, Math. Z., 62 (1955), 111-136.  doi: 10.1007/BF01180627.

[11]

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.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981.

[13]

W. Hurewicz and H. Wallmann, Dimension Theory, Princeton Mathematical Series, 4 Princeton University Press, 1941.

[14]

T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31.  doi: 10.1080/10236190902932742.

[15]

K. Jänich, Vektorraumbündel und der Raum der Fredholmoperatoren, Math. Ann., 161 (1965), 129-142.  doi: 10.1007/BF01360851.

[16]

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.

[17]

H. B. Lawson and M. -L. Michelsohn, Spin Geometry, Princeton Mathematical Series, 38 Princeton University Press, Princeton, NJ, 1989.

[18]

J. P. May, A Concise Course in Algebraic Topology, Chicago University Press, 2nd edition, 1999.

[19]

J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton University Press, 1974.

[20]

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.

[21]

K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported, 1 (1988), 265-306. 

[22]

E. Park, Complex Topological K-theory, Cambridge Studies in Advanced Mathematics, 111 Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511611476.

[23]

J. Pejsachowicz, K-theoretic methods in bifurcation theory, Fixed point theory and its applications (Berkeley, CA, 1986), Contemp. Math, 72 (1988), 193-206.  doi: 10.1090/conm/072/956492.

[24]

J. Pejsachowicz, Index bundle, Leray-Schauder reduction and bifurcation of solutions of nonlinear elliptic boundary value problems, Topol. Methods Nonlinear Anal., 18 (2001), 243-267.  doi: 10.12775/TMNA.2001.033.

[25]

J. Pejsachowicz, Bifurcation of homoclinics, Proc. Amer. Math. Soc., 136 (2008), 111-118.  doi: 10.1090/S0002-9939-07-09088-0.

[26]

J. Pejsachowicz, Bifurcation of homoclinics of Hamiltonian systems, Proc. Amer. Math. Soc., 136 (2008).

[27]

J. Pejsachowicz, Bifurcation of Fredholm maps Ⅰ. The index bundle and bifurcation, Topol. Methods Nonlinear Anal, 38 (2011), 115-168. 

[28]

J. Pejsachowicz, Bifurcation of Fredholm maps Ⅱ. The dimension of the set of bifurcation points, Topol. Methods Nonlinear Anal., 38 (2011), 291-305. 

[29]

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.

[30]

J. Pejsachowicz and R. Skiba, Topology and homoclinic trajectories of discrete dynamical systems, Discrete and Continuous Dynamical Systems, Series S, 6 (2013), 1077-1094.  doi: 10.3934/dcdss.2013.6.1077.

[31]

J. Pejsachowicz, The index bundle and bifurcation from infinity of solutions of nonlinear elliptic boundary value problems, J. Fixed Point Theory Appl., 17 (2015), 43-64.  doi: 10.1007/s11784-015-0237-0.

[32]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.

[33]

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.

[34]

C. Pötzsche, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961.  doi: 10.3934/cpaa.2011.10.937.

[35]

C. Pötzsche, Bifurcations in nonautonomous dynamical systems: Results and tools in discrete time, Proceedings of the International Workshop Future Directions in Difference Equations, 69 (2011), 163-212. 

[36]

S. Secchi and C. A. Stuart, Global Bifurcation of homoclinic solutions of Hamiltonian systems, Discrete Contin. Dyn. Syst., 9 (2003), 1493-1518.  doi: 10.3934/dcds.2003.9.1493.

[37]

M. Starostka and N. Waterstraat, A remark on singular sets of vector bundle morphisms, Eur. J. Math., 1 (2015), 154-159.  doi: 10.1007/s40879-014-0010-8.

[38]

N. Waterstraat, The index bundle for Fredholm morphisms, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 299-315. 

[39]

N. Waterstraat, A remark on bifurcation of Fredholm maps accepted for publication in Adv. Nonlinear Anal., arXiv: 1602.02320 [math. FA] doi: 10.1515/anona-2016-0067.

[40]

M. G. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov, Banach bundles and linear operators, Russian Math. Surveys, 30 (1975), 101-157. 

show all references

References:
[1]

A. Abbondandolo and P. Majer, On the global stable manifold, Studia Math., 177 (2006), 113-131.  doi: 10.4064/sm177-2-2.

[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. London Math. Soc.(3), 11 (1961), 291-310.  doi: 10.1112/plms/s3-11.1.291.

[4]

M. F. Atiyah and I. M. Singer, The index of elliptic operators. Ⅳ, Ann. of Math.(2), 93 (1971), 119-138.  doi: 10.2307/1970756.

[5]

M. F. Atiyah, K-Theory, Addison-Wesley, 1989.

[6]

T. Bartsch, The global structure of the zero set of a family of semilinear Fredholm maps, Nonlinear Anal, 71 (1991), 313-331.  doi: 10.1016/0362-546X(91)90074-B.

[7]

G. E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, 139 Springer, 1993. doi: 10.1007/978-1-4757-6848-0.

[8]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., vol. 629, Springer-Verlag, New York, 1978.

[9]

M. Crabb and I. James, Fibrewise Homotopy Theory, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1998. doi: 10.1007/978-1-4471-1265-5.

[10]

A. Dold, ÜUber fasernweise Homotopieäquivalenz von Faserräumen, Math. Z., 62 (1955), 111-136.  doi: 10.1007/BF01180627.

[11]

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.

[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1981.

[13]

W. Hurewicz and H. Wallmann, Dimension Theory, Princeton Mathematical Series, 4 Princeton University Press, 1941.

[14]

T. Hüls, Homoclinic trajectories of non-autonomous maps, J. Difference Equ. Appl., 17 (2011), 9-31.  doi: 10.1080/10236190902932742.

[15]

K. Jänich, Vektorraumbündel und der Raum der Fredholmoperatoren, Math. Ann., 161 (1965), 129-142.  doi: 10.1007/BF01360851.

[16]

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.

[17]

H. B. Lawson and M. -L. Michelsohn, Spin Geometry, Princeton Mathematical Series, 38 Princeton University Press, Princeton, NJ, 1989.

[18]

J. P. May, A Concise Course in Algebraic Topology, Chicago University Press, 2nd edition, 1999.

[19]

J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton University Press, 1974.

[20]

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.

[21]

K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported, 1 (1988), 265-306. 

[22]

E. Park, Complex Topological K-theory, Cambridge Studies in Advanced Mathematics, 111 Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511611476.

[23]

J. Pejsachowicz, K-theoretic methods in bifurcation theory, Fixed point theory and its applications (Berkeley, CA, 1986), Contemp. Math, 72 (1988), 193-206.  doi: 10.1090/conm/072/956492.

[24]

J. Pejsachowicz, Index bundle, Leray-Schauder reduction and bifurcation of solutions of nonlinear elliptic boundary value problems, Topol. Methods Nonlinear Anal., 18 (2001), 243-267.  doi: 10.12775/TMNA.2001.033.

[25]

J. Pejsachowicz, Bifurcation of homoclinics, Proc. Amer. Math. Soc., 136 (2008), 111-118.  doi: 10.1090/S0002-9939-07-09088-0.

[26]

J. Pejsachowicz, Bifurcation of homoclinics of Hamiltonian systems, Proc. Amer. Math. Soc., 136 (2008).

[27]

J. Pejsachowicz, Bifurcation of Fredholm maps Ⅰ. The index bundle and bifurcation, Topol. Methods Nonlinear Anal, 38 (2011), 115-168. 

[28]

J. Pejsachowicz, Bifurcation of Fredholm maps Ⅱ. The dimension of the set of bifurcation points, Topol. Methods Nonlinear Anal., 38 (2011), 291-305. 

[29]

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.

[30]

J. Pejsachowicz and R. Skiba, Topology and homoclinic trajectories of discrete dynamical systems, Discrete and Continuous Dynamical Systems, Series S, 6 (2013), 1077-1094.  doi: 10.3934/dcdss.2013.6.1077.

[31]

J. Pejsachowicz, The index bundle and bifurcation from infinity of solutions of nonlinear elliptic boundary value problems, J. Fixed Point Theory Appl., 17 (2015), 43-64.  doi: 10.1007/s11784-015-0237-0.

[32]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.  doi: 10.1007/BF01194662.

[33]

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.

[34]

C. Pötzsche, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961.  doi: 10.3934/cpaa.2011.10.937.

[35]

C. Pötzsche, Bifurcations in nonautonomous dynamical systems: Results and tools in discrete time, Proceedings of the International Workshop Future Directions in Difference Equations, 69 (2011), 163-212. 

[36]

S. Secchi and C. A. Stuart, Global Bifurcation of homoclinic solutions of Hamiltonian systems, Discrete Contin. Dyn. Syst., 9 (2003), 1493-1518.  doi: 10.3934/dcds.2003.9.1493.

[37]

M. Starostka and N. Waterstraat, A remark on singular sets of vector bundle morphisms, Eur. J. Math., 1 (2015), 154-159.  doi: 10.1007/s40879-014-0010-8.

[38]

N. Waterstraat, The index bundle for Fredholm morphisms, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 299-315. 

[39]

N. Waterstraat, A remark on bifurcation of Fredholm maps accepted for publication in Adv. Nonlinear Anal., arXiv: 1602.02320 [math. FA] doi: 10.1515/anona-2016-0067.

[40]

M. G. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov, Banach bundles and linear operators, Russian Math. Surveys, 30 (1975), 101-157. 

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