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Takens theorem for random dynamical systems
1. | School of Mathematics, Peking University, Beijing 100871 |
2. | Department of Mathematics, Brigham Young University, Provo, Utah 84602 |
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer, New York, 1998. |
[2] |
L. Arnold and K. Xu, Normal forms for random differential equations, Journal of Differential Equations, 116 (1995), 484-503.
doi: 10.1006/jdeq.1995.1045. |
[3] |
V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential. Equations, Springer-Verlag, New York, 1983. |
[4] |
V. I. Arnold, Small denominators I. One the mapping of a circle into itself, Izv. Akad. Nauk. Math., 25 (1961), 21-86. |
[5] |
A. Banyaga, R. de la Llave and C. E. Wayne, Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem, J. Geom. Anal., 6 (1996), 613-649.
doi: 10.1007/BF02921624. |
[6] |
G. R. Beliskii, Functional equations and the conjugacy of diffeomorphism of finite smoothness class, Functional Anal. Appl., 7 (1973), 17-28. |
[7] |
G. R. Beliskii, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys, 33 (1978), 95-155. |
[8] |
A. D. Brjuno, Analytic form of differential equations I, II, Trans. Mosc. Math. Soc., 25 (1971), 119-262; 26 (1972), 199-239. |
[9] |
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-Heidelberg-New York, 1977. |
[10] |
K. T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer.J. Math., 85 (1963), 693-722.
doi: 10.2307/2373115. |
[11] |
S.-N. Chow, K. Lu and Y.-Q. Shen, Normal form and linearization for quasiperiodic systems, Trans. Amer. Math. Soc., 331 (1992), 361-376.
doi: 10.1090/S0002-9947-1992-1076612-1. |
[12] |
N. D. Cong, Topological Dynamics of Random Dynamical Systems, Orford Mathematical Monographs, Clarendon Press, Oxford, 1997. |
[13] |
M. ElBialy, Linearization of vector fields near resonant hyperbolic rest points, J. Differential Equations, 118 (1995), 336-370.
doi: 10.1006/jdeq.1995.1076. |
[14] |
D. M. Grobman, Topological classification of the neighborhood of a singular point in $n$-dimensional space, Mat. Sb., 56 (1962), 77-94. |
[15] |
P. Guo and J. Shen, Smooth invariant manifolds for random dynamical systems,, Rocky Mountain Journal of Mathematics, ().
|
[16] |
P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexican, 5 (1960), 220-241. |
[17] |
P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc., 14 (1963), 568-573.
doi: 10.1090/S0002-9939-1963-0152718-3. |
[18] |
Yu. S. Ilyashenko and W. Li, Nonlocal Bifurcations. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999. |
[19] |
Yu. S. Ilyashenko and S. Yu. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields, (Russian) Uspekhi Mat. Nauk, 46 (1991), 3-39; translation in Russian Math. Surveys 46 (1991), 1-43.
doi: 10.1070/RM1991v046n01ABEH002733. |
[20] |
Y. Kifer, Random Perturbations of Dynamical Systems, Progress in Probability and Statistics, 16, Birkhäuser Boston, Inc., Boston, MA, 1988.
doi: 10.1007/978-1-4615-8181-9. |
[21] |
W. Li and K. Lu, Poincaré theorems for random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 1221-1236.
doi: 10.1017/S014338570400094X. |
[22] |
W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988.
doi: 10.1002/cpa.20083. |
[23] |
W. Li and K. Lu, A Siegel theorem for random dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 635-642.
doi: 10.3934/dcdsb.2008.9.635. |
[24] |
P.-D. Liu, Random perturbations of Axiom A basic sets, J. Statist. Phys., 90 (1998), 467-490.
doi: 10.1023/A:1023280407906. |
[25] |
P.-D. Liu, Dynamics of Random perturbations: Smooth ergodic theory, Ergod. Th. & Dynam. Sys., 21 (2001), 1279-1319.
doi: 10.1017/S0143385701001614. |
[26] |
P. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995.
doi: 10.1007/BFb0094308. |
[27] |
K. R. Meyer, The implicit function theorem and analytic differential equations, Lect. Notes in Math., Springer-verlag, 468 (1975), 191-208. |
[28] |
J. K. Moser, A rapidly covergent iteration method and nonlinear differential equations II, Ann. Scuo. Norm. Sup. Pisa., 20 (1966), 499-535. |
[29] |
E. Nelson, Topics in Dynamics I. Flows, Princeton University Press, 1969. |
[30] |
M. Nagumo and K. Isé, On the normal forms of differential equations in the neighborhood of an equilibrium point, Osaka Math. J., 9 (1957), 221-234. |
[31] |
C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363-367.
doi: 10.2307/2373513. |
[32] |
F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147.
doi: 10.1016/0040-9383(71)90035-8. |
[33] |
K. Palmer, Qualitative behavior of a system of ODE near an equilibrium point, A generalization of the Hartman- Grobman Theorem, Technical Report, Institute fuer Angewandte Mathematik, University Bonn. 1980. |
[34] |
R. Pérez-Marco, Linearization of holomorphic germs with resonant linear part,, preprint., ().
|
[35] |
H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique, Acta Math., 13 (1890), 1-270. |
[36] |
D. Ruelle, Random Smooth Dynamical Systems, Lecture Notes in Rutgers, 1996. |
[37] |
G. R. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.
doi: 10.2307/2374346. |
[38] |
G. R. Sell, Obstacles to linearization, Diff. Urav., 20 (1984), 446-450. |
[39] |
C. L. Siegel, Iteration of analytic functions, Ann. Math., 43 (1942), 607-612.
doi: 10.2307/1968952. |
[40] |
C. L. Siegel, Ober die Normalform analytischer Differentialgleichungen in der Nahe einer Gleichgewichtslosung, Nachr . Akad. Wiss. Gottingen, Math.-phys., 1952 (1952), 21-30. |
[41] |
S. Sternberg, On the behavior of invariant curves near a hyberbolic point of a surface transformation, Amer. J. Math., 77 (1955), 526-534.
doi: 10.2307/2372639. |
[42] |
S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824.
doi: 10.2307/2372437. |
[43] |
T. Wanner, Linearization of random dynamical systems, In C. Jones, U. Kirchgraber and H. O. Walther, editors, Dynamics Reported, Springer-Verlag, New York, 4 (1995), 203-269. |
[44] |
E. Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel, Lect. Notes in Math., Springer-Verlag, 597 (1977), 855-866. |
[45] |
W. Zhang and W. Weinian, Sharpness for C1 linearization of planar hyperbolic diffeomorphisms, J. Differential Equations, 257 (2014), 4470-4502.
doi: 10.1016/j.jde.2014.08.014. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer, New York, 1998. |
[2] |
L. Arnold and K. Xu, Normal forms for random differential equations, Journal of Differential Equations, 116 (1995), 484-503.
doi: 10.1006/jdeq.1995.1045. |
[3] |
V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential. Equations, Springer-Verlag, New York, 1983. |
[4] |
V. I. Arnold, Small denominators I. One the mapping of a circle into itself, Izv. Akad. Nauk. Math., 25 (1961), 21-86. |
[5] |
A. Banyaga, R. de la Llave and C. E. Wayne, Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem, J. Geom. Anal., 6 (1996), 613-649.
doi: 10.1007/BF02921624. |
[6] |
G. R. Beliskii, Functional equations and the conjugacy of diffeomorphism of finite smoothness class, Functional Anal. Appl., 7 (1973), 17-28. |
[7] |
G. R. Beliskii, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys, 33 (1978), 95-155. |
[8] |
A. D. Brjuno, Analytic form of differential equations I, II, Trans. Mosc. Math. Soc., 25 (1971), 119-262; 26 (1972), 199-239. |
[9] |
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-Heidelberg-New York, 1977. |
[10] |
K. T. Chen, Equivalence and decomposition of vector fields about an elementary critical point, Amer.J. Math., 85 (1963), 693-722.
doi: 10.2307/2373115. |
[11] |
S.-N. Chow, K. Lu and Y.-Q. Shen, Normal form and linearization for quasiperiodic systems, Trans. Amer. Math. Soc., 331 (1992), 361-376.
doi: 10.1090/S0002-9947-1992-1076612-1. |
[12] |
N. D. Cong, Topological Dynamics of Random Dynamical Systems, Orford Mathematical Monographs, Clarendon Press, Oxford, 1997. |
[13] |
M. ElBialy, Linearization of vector fields near resonant hyperbolic rest points, J. Differential Equations, 118 (1995), 336-370.
doi: 10.1006/jdeq.1995.1076. |
[14] |
D. M. Grobman, Topological classification of the neighborhood of a singular point in $n$-dimensional space, Mat. Sb., 56 (1962), 77-94. |
[15] |
P. Guo and J. Shen, Smooth invariant manifolds for random dynamical systems,, Rocky Mountain Journal of Mathematics, ().
|
[16] |
P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexican, 5 (1960), 220-241. |
[17] |
P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc., 14 (1963), 568-573.
doi: 10.1090/S0002-9939-1963-0152718-3. |
[18] |
Yu. S. Ilyashenko and W. Li, Nonlocal Bifurcations. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999. |
[19] |
Yu. S. Ilyashenko and S. Yu. Yakovenko, Finitely smooth normal forms of local families of diffeomorphisms and vector fields, (Russian) Uspekhi Mat. Nauk, 46 (1991), 3-39; translation in Russian Math. Surveys 46 (1991), 1-43.
doi: 10.1070/RM1991v046n01ABEH002733. |
[20] |
Y. Kifer, Random Perturbations of Dynamical Systems, Progress in Probability and Statistics, 16, Birkhäuser Boston, Inc., Boston, MA, 1988.
doi: 10.1007/978-1-4615-8181-9. |
[21] |
W. Li and K. Lu, Poincaré theorems for random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 1221-1236.
doi: 10.1017/S014338570400094X. |
[22] |
W. Li and K. Lu, Sternberg theorems for random dynamical systems, Comm. Pure Appl. Math., 58 (2005), 941-988.
doi: 10.1002/cpa.20083. |
[23] |
W. Li and K. Lu, A Siegel theorem for random dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 635-642.
doi: 10.3934/dcdsb.2008.9.635. |
[24] |
P.-D. Liu, Random perturbations of Axiom A basic sets, J. Statist. Phys., 90 (1998), 467-490.
doi: 10.1023/A:1023280407906. |
[25] |
P.-D. Liu, Dynamics of Random perturbations: Smooth ergodic theory, Ergod. Th. & Dynam. Sys., 21 (2001), 1279-1319.
doi: 10.1017/S0143385701001614. |
[26] |
P. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes in Mathematics, 1606. Springer-Verlag, Berlin, 1995.
doi: 10.1007/BFb0094308. |
[27] |
K. R. Meyer, The implicit function theorem and analytic differential equations, Lect. Notes in Math., Springer-verlag, 468 (1975), 191-208. |
[28] |
J. K. Moser, A rapidly covergent iteration method and nonlinear differential equations II, Ann. Scuo. Norm. Sup. Pisa., 20 (1966), 499-535. |
[29] |
E. Nelson, Topics in Dynamics I. Flows, Princeton University Press, 1969. |
[30] |
M. Nagumo and K. Isé, On the normal forms of differential equations in the neighborhood of an equilibrium point, Osaka Math. J., 9 (1957), 221-234. |
[31] |
C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363-367.
doi: 10.2307/2373513. |
[32] |
F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147.
doi: 10.1016/0040-9383(71)90035-8. |
[33] |
K. Palmer, Qualitative behavior of a system of ODE near an equilibrium point, A generalization of the Hartman- Grobman Theorem, Technical Report, Institute fuer Angewandte Mathematik, University Bonn. 1980. |
[34] |
R. Pérez-Marco, Linearization of holomorphic germs with resonant linear part,, preprint., ().
|
[35] |
H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique, Acta Math., 13 (1890), 1-270. |
[36] |
D. Ruelle, Random Smooth Dynamical Systems, Lecture Notes in Rutgers, 1996. |
[37] |
G. R. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.
doi: 10.2307/2374346. |
[38] |
G. R. Sell, Obstacles to linearization, Diff. Urav., 20 (1984), 446-450. |
[39] |
C. L. Siegel, Iteration of analytic functions, Ann. Math., 43 (1942), 607-612.
doi: 10.2307/1968952. |
[40] |
C. L. Siegel, Ober die Normalform analytischer Differentialgleichungen in der Nahe einer Gleichgewichtslosung, Nachr . Akad. Wiss. Gottingen, Math.-phys., 1952 (1952), 21-30. |
[41] |
S. Sternberg, On the behavior of invariant curves near a hyberbolic point of a surface transformation, Amer. J. Math., 77 (1955), 526-534.
doi: 10.2307/2372639. |
[42] |
S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824.
doi: 10.2307/2372437. |
[43] |
T. Wanner, Linearization of random dynamical systems, In C. Jones, U. Kirchgraber and H. O. Walther, editors, Dynamics Reported, Springer-Verlag, New York, 4 (1995), 203-269. |
[44] |
E. Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel, Lect. Notes in Math., Springer-Verlag, 597 (1977), 855-866. |
[45] |
W. Zhang and W. Weinian, Sharpness for C1 linearization of planar hyperbolic diffeomorphisms, J. Differential Equations, 257 (2014), 4470-4502.
doi: 10.1016/j.jde.2014.08.014. |
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