Takens theorem for random dynamical systems

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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

Abstract
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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 and 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.
[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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