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November  2016, 21(9): 3191-3207. doi: 10.3934/dcdsb.2016093

Takens theorem for random dynamical systems

Weigu Li 1, and  Kening Lu 2,

1. 

School of Mathematics, Peking University, Beijing 100871
2. 

Department of Mathematics, Brigham Young University, Provo, Utah 84602

Received  October 2015 Revised  January 2016 Published  October 2016

In this paper, we study random dynamical systems with partial hyperbolic fixed points and prove the smooth conjugacy theorems of Takens type based on their Lyapunov exponents.
Keywords: normal forms., random dynamical systems, Multiplicative Ergodic theorem, Takens' theorem.
Mathematics Subject Classification: Primary: 37H99; Secondary: 37C15, 37C40, 37D25, 37D3.
Citation: Weigu Li, Kening Lu. Takens theorem for random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3191-3207. doi: 10.3934/dcdsb.2016093
References:
[1]

L. Arnold, Random Dynamical Systems, Springer, New York, 1998. Google Scholar

[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.  Google Scholar

[3]

V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential. Equations, Springer-Verlag, New York, 1983.  Google Scholar

[4]

V. I. Arnold, Small denominators I. One the mapping of a circle into itself, Izv. Akad. Nauk. Math., 25 (1961), 21-86.  Google Scholar

[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.  Google Scholar

[6]

G. R. Beliskii, Functional equations and the conjugacy of diffeomorphism of finite smoothness class, Functional Anal. Appl., 7 (1973), 17-28.  Google Scholar

[7]

G. R. Beliskii, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys, 33 (1978), 95-155.  Google Scholar

[8]

A. D. Brjuno, Analytic form of differential equations I, II, Trans. Mosc. Math. Soc., 25 (1971), 119-262; 26 (1972), 199-239.  Google Scholar

[9]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-Heidelberg-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. D. Cong, Topological Dynamics of Random Dynamical Systems, Orford Mathematical Monographs, Clarendon Press, Oxford, 1997.  Google Scholar

[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.  Google Scholar

[14]

D. M. Grobman, Topological classification of the neighborhood of a singular point in $n$-dimensional space, Mat. Sb., 56 (1962), 77-94.  Google Scholar

[15]

P. Guo and J. Shen, Smooth invariant manifolds for random dynamical systems,, Rocky Mountain Journal of Mathematics, ().   Google Scholar

[16]

P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexican, 5 (1960), 220-241.  Google Scholar

[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.  Google Scholar

[18]

Yu. S. Ilyashenko and W. Li, Nonlocal Bifurcations. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

W. Li and K. Lu, Poincaré theorems for random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 1221-1236. doi: 10.1017/S014338570400094X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

P.-D. Liu, Random perturbations of Axiom A basic sets, J. Statist. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.  Google Scholar

[25]

P.-D. Liu, Dynamics of Random perturbations: Smooth ergodic theory, Ergod. Th. & Dynam. Sys., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.  Google Scholar

[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.  Google Scholar

[27]

K. R. Meyer, The implicit function theorem and analytic differential equations, Lect. Notes in Math., Springer-verlag, 468 (1975), 191-208.  Google Scholar

[28]

J. K. Moser, A rapidly covergent iteration method and nonlinear differential equations II, Ann. Scuo. Norm. Sup. Pisa., 20 (1966), 499-535.  Google Scholar

[29]

E. Nelson, Topics in Dynamics I. Flows, Princeton University Press, 1969.  Google Scholar

[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.  Google Scholar

[31]

C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363-367. doi: 10.2307/2373513.  Google Scholar

[32]

F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147. doi: 10.1016/0040-9383(71)90035-8.  Google Scholar

[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. Google Scholar

[34]

R. Pérez-Marco, Linearization of holomorphic germs with resonant linear part,, preprint., ().   Google Scholar

[35]

H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique, Acta Math., 13 (1890), 1-270. Google Scholar

[36]

D. Ruelle, Random Smooth Dynamical Systems, Lecture Notes in Rutgers, 1996. Google Scholar

[37]

G. R. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091. doi: 10.2307/2374346.  Google Scholar

[38]

G. R. Sell, Obstacles to linearization, Diff. Urav., 20 (1984), 446-450.  Google Scholar

[39]

C. L. Siegel, Iteration of analytic functions, Ann. Math., 43 (1942), 607-612. doi: 10.2307/1968952.  Google Scholar

[40]

C. L. Siegel, Ober die Normalform analytischer Differentialgleichungen in der Nahe einer Gleichgewichtslosung, Nachr . Akad. Wiss. Gottingen, Math.-phys., 1952 (1952), 21-30.  Google Scholar

[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.  Google Scholar

[42]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824. doi: 10.2307/2372437.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, New York, 1998. Google Scholar

[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.  Google Scholar

[3]

V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential. Equations, Springer-Verlag, New York, 1983.  Google Scholar

[4]

V. I. Arnold, Small denominators I. One the mapping of a circle into itself, Izv. Akad. Nauk. Math., 25 (1961), 21-86.  Google Scholar

[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.  Google Scholar

[6]

G. R. Beliskii, Functional equations and the conjugacy of diffeomorphism of finite smoothness class, Functional Anal. Appl., 7 (1973), 17-28.  Google Scholar

[7]

G. R. Beliskii, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys, 33 (1978), 95-155.  Google Scholar

[8]

A. D. Brjuno, Analytic form of differential equations I, II, Trans. Mosc. Math. Soc., 25 (1971), 119-262; 26 (1972), 199-239.  Google Scholar

[9]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580. Springer-Verlag, Berlin-Heidelberg-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. D. Cong, Topological Dynamics of Random Dynamical Systems, Orford Mathematical Monographs, Clarendon Press, Oxford, 1997.  Google Scholar

[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.  Google Scholar

[14]

D. M. Grobman, Topological classification of the neighborhood of a singular point in $n$-dimensional space, Mat. Sb., 56 (1962), 77-94.  Google Scholar

[15]

P. Guo and J. Shen, Smooth invariant manifolds for random dynamical systems,, Rocky Mountain Journal of Mathematics, ().   Google Scholar

[16]

P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexican, 5 (1960), 220-241.  Google Scholar

[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.  Google Scholar

[18]

Yu. S. Ilyashenko and W. Li, Nonlocal Bifurcations. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

W. Li and K. Lu, Poincaré theorems for random dynamical systems, Ergodic Theory Dynam. Systems, 25 (2005), 1221-1236. doi: 10.1017/S014338570400094X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

P.-D. Liu, Random perturbations of Axiom A basic sets, J. Statist. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.  Google Scholar

[25]

P.-D. Liu, Dynamics of Random perturbations: Smooth ergodic theory, Ergod. Th. & Dynam. Sys., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.  Google Scholar

[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.  Google Scholar

[27]

K. R. Meyer, The implicit function theorem and analytic differential equations, Lect. Notes in Math., Springer-verlag, 468 (1975), 191-208.  Google Scholar

[28]

J. K. Moser, A rapidly covergent iteration method and nonlinear differential equations II, Ann. Scuo. Norm. Sup. Pisa., 20 (1966), 499-535.  Google Scholar

[29]

E. Nelson, Topics in Dynamics I. Flows, Princeton University Press, 1969.  Google Scholar

[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.  Google Scholar

[31]

C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363-367. doi: 10.2307/2373513.  Google Scholar

[32]

F. Takens, Partially hyperbolic fixed points, Topology, 10 (1971), 133-147. doi: 10.1016/0040-9383(71)90035-8.  Google Scholar

[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. Google Scholar

[34]

R. Pérez-Marco, Linearization of holomorphic germs with resonant linear part,, preprint., ().   Google Scholar

[35]

H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique, Acta Math., 13 (1890), 1-270. Google Scholar

[36]

D. Ruelle, Random Smooth Dynamical Systems, Lecture Notes in Rutgers, 1996. Google Scholar

[37]

G. R. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091. doi: 10.2307/2374346.  Google Scholar

[38]

G. R. Sell, Obstacles to linearization, Diff. Urav., 20 (1984), 446-450.  Google Scholar

[39]

C. L. Siegel, Iteration of analytic functions, Ann. Math., 43 (1942), 607-612. doi: 10.2307/1968952.  Google Scholar

[40]

C. L. Siegel, Ober die Normalform analytischer Differentialgleichungen in der Nahe einer Gleichgewichtslosung, Nachr . Akad. Wiss. Gottingen, Math.-phys., 1952 (1952), 21-30.  Google Scholar

[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.  Google Scholar

[42]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824. doi: 10.2307/2372437.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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