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Global-in-time behavior of the solution to a Gierer-Meinhardt system
Partial hyperbolicity and central shadowing
1. | Faculty of Mathematics and Mechanics and Chebyshev laboratory, Saint-Petersburg State University Universitetsky pr., 28, 198504, Peterhof, St. Petersburg, Russian Federation |
2. | Institut fur Mathematik, Freie Universitat Berlin, Arnimallee 3, Berlin, 14195, Germany |
References:
[1] |
F. Abdenur and L. Diaz, Pseudo-orbit shadowing in the $C^1$ topology,, Discrete Contin. Dyn. Syst., 7 (2003), 223.
|
[2] |
D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967).
|
[3] |
D. Bohnet and Ch. Bonatti, Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics,, preprint , (). Google Scholar |
[4] |
Ch. Bonatti, L. J. Diaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,", Springer, (2004).
|
[5] |
Ch. Bonatti, L. Diaz and G. Turcat, There is no shadowing lemma for partially hyperbolic dynamics,, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 587.
doi: 10.1016/S0764-4442(00)00215-9. |
[6] |
R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes Math., 470 (1975).
|
[7] |
M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395.
doi: 10.1017/S0143385702001499. |
[8] |
K. Burns and A. Wilkinson, Dynamical coherence and center bunching,, Discrete and Continuous Dynamical Systems, 22 (2008), 89.
doi: 10.3934/dcds.2008.22.89. |
[9] |
N. Gourmelon, Adapted metric for dominated splitting,, Ergod. Theory Dyn. Syst., 27 (2007), 1839.
doi: 10.1017/S0143385707000272. |
[10] |
F. Rodriguez-Hertz, M. A. Rodriguez-Hertz and R. Ures, A survey of partially hyperbolic dynamics,, Fields Institute Communications, 51 (2007), 35.
|
[11] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Math., 583 (1977).
|
[12] |
Huyi Hu, Yunhua Zhou and Yujun Zhu, Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms,, preprint, (). Google Scholar |
[13] |
A. Morimoto, The method of pseudo-orbit tracing and stability of dynamical systems,, Sem. Note, 39 (1979). Google Scholar |
[14] |
K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications,", Kluwer, (2000).
|
[15] |
S. Yu. Pilyugin, "Shadowing in Dynamical Systems,", Lecture Notes in Math., 1706 (1999).
|
[16] |
S. Yu. Pilyugin, Variational shadowing,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 733.
doi: 10.3934/dcdsb.2010.14.733. |
[17] |
S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing imply structural stability,, Nonlinearity, 23 (2010), 2509.
doi: 10.1088/0951-7715/23/10/009. |
[18] |
C. C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, revisited,, J. of Modern Dynamics, 6 (2012), 79.
doi: 10.3934/jmd.2012.6.79. |
[19] |
C. Robinson, Stability theorems and hyperbolicity in dynamical systems,, Rocky Mount. J. Math., 7 (1977), 425.
|
[20] |
K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms of closed manifolds,, Osaka J. Math., 31 (1994), 373.
|
[21] |
K. Sawada, Extended f-orbits are approximated by orbits,, Nagoya Math. J., 79 (1980), 33.
|
[22] |
J. Schauder, Der fixpunktsatz in funktionalraumen,, Stud. Math., 2 (1930), 171. Google Scholar |
[23] |
S. B. Tikhomirov, Hölder shadowing on finite intervals,, preprint, (). Google Scholar |
show all references
References:
[1] |
F. Abdenur and L. Diaz, Pseudo-orbit shadowing in the $C^1$ topology,, Discrete Contin. Dyn. Syst., 7 (2003), 223.
|
[2] |
D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967).
|
[3] |
D. Bohnet and Ch. Bonatti, Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics,, preprint , (). Google Scholar |
[4] |
Ch. Bonatti, L. J. Diaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,", Springer, (2004).
|
[5] |
Ch. Bonatti, L. Diaz and G. Turcat, There is no shadowing lemma for partially hyperbolic dynamics,, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 587.
doi: 10.1016/S0764-4442(00)00215-9. |
[6] |
R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes Math., 470 (1975).
|
[7] |
M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395.
doi: 10.1017/S0143385702001499. |
[8] |
K. Burns and A. Wilkinson, Dynamical coherence and center bunching,, Discrete and Continuous Dynamical Systems, 22 (2008), 89.
doi: 10.3934/dcds.2008.22.89. |
[9] |
N. Gourmelon, Adapted metric for dominated splitting,, Ergod. Theory Dyn. Syst., 27 (2007), 1839.
doi: 10.1017/S0143385707000272. |
[10] |
F. Rodriguez-Hertz, M. A. Rodriguez-Hertz and R. Ures, A survey of partially hyperbolic dynamics,, Fields Institute Communications, 51 (2007), 35.
|
[11] |
M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Math., 583 (1977).
|
[12] |
Huyi Hu, Yunhua Zhou and Yujun Zhu, Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms,, preprint, (). Google Scholar |
[13] |
A. Morimoto, The method of pseudo-orbit tracing and stability of dynamical systems,, Sem. Note, 39 (1979). Google Scholar |
[14] |
K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications,", Kluwer, (2000).
|
[15] |
S. Yu. Pilyugin, "Shadowing in Dynamical Systems,", Lecture Notes in Math., 1706 (1999).
|
[16] |
S. Yu. Pilyugin, Variational shadowing,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 733.
doi: 10.3934/dcdsb.2010.14.733. |
[17] |
S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing imply structural stability,, Nonlinearity, 23 (2010), 2509.
doi: 10.1088/0951-7715/23/10/009. |
[18] |
C. C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, revisited,, J. of Modern Dynamics, 6 (2012), 79.
doi: 10.3934/jmd.2012.6.79. |
[19] |
C. Robinson, Stability theorems and hyperbolicity in dynamical systems,, Rocky Mount. J. Math., 7 (1977), 425.
|
[20] |
K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms of closed manifolds,, Osaka J. Math., 31 (1994), 373.
|
[21] |
K. Sawada, Extended f-orbits are approximated by orbits,, Nagoya Math. J., 79 (1980), 33.
|
[22] |
J. Schauder, Der fixpunktsatz in funktionalraumen,, Stud. Math., 2 (1930), 171. Google Scholar |
[23] |
S. B. Tikhomirov, Hölder shadowing on finite intervals,, preprint, (). Google Scholar |
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