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August  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 and Continuous Dynamical Systems - S, 2013, 6 (4) : 1077-1094. doi: 10.3934/dcdss.2013.6.1077
References:
[1]

Math. Z., 243 (2003), 525-562. doi: 10.1007/s00209-002-0473-z.

[2]

Studia Math., 177 (2006), 113-131. doi: 10.4064/sm177-2-2.

[3]

Benjamin, New York, 1967.

[4]

Nonlinear Analysis, 17 (1991), 313-331. doi: 10.1016/0362-546X(91)90074-B.

[5]

Mathematical Notes, 67 (2000), 690-698. doi: 10.1007/BF02675622.

[6]

Ann. Sci. Math. Québec, 22 (1998), 131-148.

[7]

Nonlinear Analysis Forum, 6 (2001), 35-47.

[8]

J. Functional Analysis, 8 (1971), 321-340.

[9]

Contemporary Mathematics, 72 (1988). doi: 10.1090/conm/072/956479.

[10]

J. Difference Equ. Appl., 10 (2004), 257-297. doi: 10.1080/10236190310001634794.

[11]

Springer Verlag, 1975.

[12]

Grundlehren der mathematischen Wissenschaften, 132, Springer, Berlin etc., 1980.

[13]

Graduate Text in Mathematics, 160, Springer-Verlag, 1995. doi: 10.1007/978-1-4612-4182-9.

[14]

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

[15]

Proc. Amer. Math. Soc., 136 (2008), 2055-2065. doi: 10.1090/S0002-9939-08-09342-8.

[16]

Topol. Methods Nonlinear Anal., 38 (2011), 115-168.

[17]

Journal d'Analyse Mathematique, 76 (1998), 289-319. doi: 10.1007/BF02786939.

[18]

Central European Journal of Mathematics, 10 (2012), 2088-2109.

[19]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 739-776. doi: 10.3934/dcdsb.2010.14.739.

[20]

Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 941-973. doi: 10.3934/dcds.2011.31.941.

[21]

Communications in Pure and Applied Analysis, 10 (2011), 937-961. doi: 10.3934/cpaa.2011.10.937.

[22]

J. Difference Equ. Appl., 12 (2006), 297-312. doi: 10.1080/10236190500489400.

[23]

J. Diff. Eq., 33 (1979), 368-405. doi: 10.1016/0022-0396(79)90072-X.

show all references

References:
[1]

Math. Z., 243 (2003), 525-562. doi: 10.1007/s00209-002-0473-z.

[2]

Studia Math., 177 (2006), 113-131. doi: 10.4064/sm177-2-2.

[3]

Benjamin, New York, 1967.

[4]

Nonlinear Analysis, 17 (1991), 313-331. doi: 10.1016/0362-546X(91)90074-B.

[5]

Mathematical Notes, 67 (2000), 690-698. doi: 10.1007/BF02675622.

[6]

Ann. Sci. Math. Québec, 22 (1998), 131-148.

[7]

Nonlinear Analysis Forum, 6 (2001), 35-47.

[8]

J. Functional Analysis, 8 (1971), 321-340.

[9]

Contemporary Mathematics, 72 (1988). doi: 10.1090/conm/072/956479.

[10]

J. Difference Equ. Appl., 10 (2004), 257-297. doi: 10.1080/10236190310001634794.

[11]

Springer Verlag, 1975.

[12]

Grundlehren der mathematischen Wissenschaften, 132, Springer, Berlin etc., 1980.

[13]

Graduate Text in Mathematics, 160, Springer-Verlag, 1995. doi: 10.1007/978-1-4612-4182-9.

[14]

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

[15]

Proc. Amer. Math. Soc., 136 (2008), 2055-2065. doi: 10.1090/S0002-9939-08-09342-8.

[16]

Topol. Methods Nonlinear Anal., 38 (2011), 115-168.

[17]

Journal d'Analyse Mathematique, 76 (1998), 289-319. doi: 10.1007/BF02786939.

[18]

Central European Journal of Mathematics, 10 (2012), 2088-2109.

[19]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 739-776. doi: 10.3934/dcdsb.2010.14.739.

[20]

Discrete Contin. Dyn. Syst. Ser. A, 31 (2011), 941-973. doi: 10.3934/dcds.2011.31.941.

[21]

Communications in Pure and Applied Analysis, 10 (2011), 937-961. doi: 10.3934/cpaa.2011.10.937.

[22]

J. Difference Equ. Appl., 12 (2006), 297-312. doi: 10.1080/10236190500489400.

[23]

J. Diff. Eq., 33 (1979), 368-405. doi: 10.1016/0022-0396(79)90072-X.

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