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April  2013, 6(4): 1077-1094. doi: 10.3934/dcdss.2013.6.1077

Topology and homoclinic trajectories of discrete dynamical systems

1. 

Dipartamento di Matematica

2. 

Politecnico di Torino

3. 

Corso Duca Degli Abruzzi 24, 10129 Torino

4. 

Faculty of Mathematics and Computer Science

5. 

Nicolaus Copernicus University

6. 

Chopina 12/18, 87-100 Toru?

Received  August 2011 Revised  October 2011 Published  December 2012

We show that nontrivial homoclinic trajectories ofa family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circlebifurcate from a stationary solution if the asymptotic stable bundles $E^s(+\infty)$ and$E^s(-\infty)$ of the linearization at the stationary branch are twisted in different ways.
Citation: Jacobo Pejsachowicz, Robert Skiba. Topology and homoclinic trajectories of discrete dynamical systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1077-1094. doi: 10.3934/dcdss.2013.6.1077
References:
[1]

A. Abbondandolo and P. Majer, Ordinary differential operators and Fredholm pairs,, Math. Z., 243 (2003), 525. doi: 10.1007/s00209-002-0473-z. Google Scholar

[2]

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

[3]

M. F. Atiyah, "K-Theory,", Benjamin, (1967). Google Scholar

[4]

T. Bartsch, The global structure of the zero set of a family of semilinear Fredholm maps,, Nonlinear Analysis, 17 (1991), 313. doi: 10.1016/0362-546X(91)90074-B. Google Scholar

[5]

A. G. Baskakov, Invertibility and the Fredholm property of difference operators,, Mathematical Notes, 67 (2000), 690. doi: 10.1007/BF02675622. Google Scholar

[6]

P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree,, Ann. Sci. Math. Québec, 22 (1998), 131. Google Scholar

[7]

P. Benevieri and M. Furi, Bifurcation results for families of Fredholm maps of index zero between Banach spaces,, Nonlinear Analysis Forum, 6 (2001), 35. Google Scholar

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. Google Scholar

[9]

P. M. Fitzpatrick and J. Pejsachowicz, The fundamental group of the space of linear fredholm operators and the global analysis of semilinear equations,, Contemporary Mathematics, 72 (1988). doi: 10.1090/conm/072/956479. Google Scholar

[10]

Y. Hamaya, Bifurcation of almost periodic solutions in difference equations,, J. Difference Equ. Appl., 10 (2004), 257. doi: 10.1080/10236190310001634794. Google Scholar

[11]

D. Husemoller, "Fibre Bundles,", Springer Verlag, (1975). Google Scholar

[12]

T. Kato, "Perturbation Theory for Linear Operators,", Grundlehren der mathematischen Wissenschaften, 132 (1980). Google Scholar

[13]

S. Lang, "Differential and Riemannian Manifolds,", Graduate Text in Mathematics, 160 (1995). doi: 10.1007/978-1-4612-4182-9. Google Scholar

[14]

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

[15]

J. Pejsachowicz, Bifurcation of homoclinics of Hamiltonian systems,, Proc. Amer. Math. Soc., 136 (2008), 2055. doi: 10.1090/S0002-9939-08-09342-8. Google Scholar

[16]

J. Pejsachowicz, Bifurcation of Fredholm maps I; Index bundle and bifurcation,, Topol. Methods Nonlinear Anal., 38 (2011), 115. Google Scholar

[17]

J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$-Fredholm mappings of index $0$,, Journal d'Analyse Mathematique, 76 (1998), 289. doi: 10.1007/BF02786939. Google Scholar

[18]

J. Pejsachowicz and R. Skiba, Global bifurcation of homoclinic trajectories of discrete dynamical systems,, Central European Journal of Mathematics, 10 (2012), 2088. Google Scholar

[19]

C. Pötzsche, Nonautonomus bifurcation of bounded solutions I: A Lyapunov-Schmidt approach,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 739. doi: 10.3934/dcdsb.2010.14.739. Google Scholar

[20]

C. Pötzsche, Nonautonomous bifurcation of bounded solutions II: A shovel bifurcation pattern,, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 941. doi: 10.3934/dcds.2011.31.941. Google Scholar

[21]

C. Pötzsche, Nonautonomus continuation of bounded solutions,, Communications in Pure and Applied Analysis, 10 (2011), 937. doi: 10.3934/cpaa.2011.10.937. Google Scholar

[22]

M. Rasmussen, Towards a bifurcation theory for nonautonomous difference equation,, J. Difference Equ. Appl., 12 (2006), 297. doi: 10.1080/10236190500489400. Google Scholar

[23]

R. J. Sacker, The splitting index for linear differential systems,, J. Diff. Eq., 33 (1979), 368. doi: 10.1016/0022-0396(79)90072-X. Google Scholar

show all references

References:
[1]

A. Abbondandolo and P. Majer, Ordinary differential operators and Fredholm pairs,, Math. Z., 243 (2003), 525. doi: 10.1007/s00209-002-0473-z. Google Scholar

[2]

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

[3]

M. F. Atiyah, "K-Theory,", Benjamin, (1967). Google Scholar

[4]

T. Bartsch, The global structure of the zero set of a family of semilinear Fredholm maps,, Nonlinear Analysis, 17 (1991), 313. doi: 10.1016/0362-546X(91)90074-B. Google Scholar

[5]

A. G. Baskakov, Invertibility and the Fredholm property of difference operators,, Mathematical Notes, 67 (2000), 690. doi: 10.1007/BF02675622. Google Scholar

[6]

P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree,, Ann. Sci. Math. Québec, 22 (1998), 131. Google Scholar

[7]

P. Benevieri and M. Furi, Bifurcation results for families of Fredholm maps of index zero between Banach spaces,, Nonlinear Analysis Forum, 6 (2001), 35. Google Scholar

[8]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. Google Scholar

[9]

P. M. Fitzpatrick and J. Pejsachowicz, The fundamental group of the space of linear fredholm operators and the global analysis of semilinear equations,, Contemporary Mathematics, 72 (1988). doi: 10.1090/conm/072/956479. Google Scholar

[10]

Y. Hamaya, Bifurcation of almost periodic solutions in difference equations,, J. Difference Equ. Appl., 10 (2004), 257. doi: 10.1080/10236190310001634794. Google Scholar

[11]

D. Husemoller, "Fibre Bundles,", Springer Verlag, (1975). Google Scholar

[12]

T. Kato, "Perturbation Theory for Linear Operators,", Grundlehren der mathematischen Wissenschaften, 132 (1980). Google Scholar

[13]

S. Lang, "Differential and Riemannian Manifolds,", Graduate Text in Mathematics, 160 (1995). doi: 10.1007/978-1-4612-4182-9. Google Scholar

[14]

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

[15]

J. Pejsachowicz, Bifurcation of homoclinics of Hamiltonian systems,, Proc. Amer. Math. Soc., 136 (2008), 2055. doi: 10.1090/S0002-9939-08-09342-8. Google Scholar

[16]

J. Pejsachowicz, Bifurcation of Fredholm maps I; Index bundle and bifurcation,, Topol. Methods Nonlinear Anal., 38 (2011), 115. Google Scholar

[17]

J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$-Fredholm mappings of index $0$,, Journal d'Analyse Mathematique, 76 (1998), 289. doi: 10.1007/BF02786939. Google Scholar

[18]

J. Pejsachowicz and R. Skiba, Global bifurcation of homoclinic trajectories of discrete dynamical systems,, Central European Journal of Mathematics, 10 (2012), 2088. Google Scholar

[19]

C. Pötzsche, Nonautonomus bifurcation of bounded solutions I: A Lyapunov-Schmidt approach,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 739. doi: 10.3934/dcdsb.2010.14.739. Google Scholar

[20]

C. Pötzsche, Nonautonomous bifurcation of bounded solutions II: A shovel bifurcation pattern,, Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 941. doi: 10.3934/dcds.2011.31.941. Google Scholar

[21]

C. Pötzsche, Nonautonomus continuation of bounded solutions,, Communications in Pure and Applied Analysis, 10 (2011), 937. doi: 10.3934/cpaa.2011.10.937. Google Scholar

[22]

M. Rasmussen, Towards a bifurcation theory for nonautonomous difference equation,, J. Difference Equ. Appl., 12 (2006), 297. doi: 10.1080/10236190500489400. Google Scholar

[23]

R. J. Sacker, The splitting index for linear differential systems,, J. Diff. Eq., 33 (1979), 368. doi: 10.1016/0022-0396(79)90072-X. Google Scholar

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