Geometric Lorenz flows with historic behavior

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December 2016, 36(12): 7021-7028. doi: 10.3934/dcds.2016105

Geometric Lorenz flows with historic behavior

Shin Kiriki ^1, , Ming-Chia Li ^2, and Teruhiko Soma ^3,

1.	Department of Mathematics, Tokai University, 4-1-1 Kitakaname, Hiratuka Kanagawa, 259-1292, Japan
2.	Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan
3.	Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397

Received December 2015 Revised August 2016 Published October 2016

Abstract
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We will show that, in the the geometric Lorenz flow, the set of initial states which give rise to orbits with historic behavior is residual in a trapping region.

Keywords: Lorenz map., Historic behavior, geometric Lorenz flow.

Mathematics Subject Classification: Primary: 37A05, 37C10, 37C40, 37D3.

Citation: Shin Kiriki, Ming-Chia Li, Teruhiko Soma. Geometric Lorenz flows with historic behavior. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7021-7028. doi: 10.3934/dcds.2016105

References:

[1]	V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic,, Trans. Amer. Math. Soc., 361 (2009), 2431. doi: 10.1090/S0002-9947-08-04595-9. Google Scholar
[2]	Ch. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity,, Encyclopedia of Mathematical Sciences (Mathematical Physics), 102 (2005). Google Scholar
[3]	T. N. Dowker, The mean and transitive points of homeomorphisms,, Ann. of Math., 58 (1953), 123. doi: 10.2307/1969823. Google Scholar
[4]	J. Guckenheimer, A strange, strange attractor,, in The Hopf bifurcation and its applications, (1976), 368. Google Scholar
[5]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59. Google Scholar
[6]	F. Hofbauer, Kneading invariants and Markov diagrams,, in Ergodic theory and related topics (Vitte, 12 (1982), 85. Google Scholar
[7]	T. Jordan, V. Naudot and T. Young, Higher order Birkhoff averages,, Dyn. Syst., 24 (2009), 299. doi: 10.1080/14689360802676269. Google Scholar
[8]	S. Kiriki and T. Soma, Takens' last problem and existence of non-trivial wandering domains,, preprint, (). Google Scholar
[9]	I. S. Labouriau and A. A. P. Rodrigues, On Takens' Last Problem: Tangencies and time averages near heteroclinic networks,, preprint, (). Google Scholar
[10]	E. N. Lorenz, Deterministic non-periodic flow,, J. Atmos. Sci., 20 (1963), 130. Google Scholar
[11]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems,, Proc. Amer. Math. Soc., 127 (1999), 3393. doi: 10.1090/S0002-9939-99-04936-9. Google Scholar
[12]	Y. Nakano, Historic behaviour for quenched random expanding maps on the circle,, preprint, (). Google Scholar
[13]	J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Fractal dimensions and infinitely many attractors, Cambridge Studies in Advanced Mathematics, 35,, Cambridge University Press, (1993). Google Scholar
[14]	C. Robinson, Differentiability of the stable foliation for the model Lorenz equations,, Dynamical systems and turbulence, 898* (1980), 302. Google Scholar*
[15]	D. Ruelle, Historical behaviour in smooth dynamical systems,, in Global Analysis of Dynamical Systems (eds. H. W. Broer et al), (2001), 63. Google Scholar
[16]	F. Takens, Heteroclinic attractors: Time averages and moduli of topological stability,, Bol. Soc. Bras. Mat., 25 (1994), 107. doi: 10.1007/BF01232938. Google Scholar
[17]	F. Takens, Orbits with historic behaviour, or non-existence of averages,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/3/T02. Google Scholar
[18]	W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comput. Math., 2 (2002), 53. doi: 10.1007/s002080010018. Google Scholar
[19]	R. Williams, The structure of Lorenz attractors,, Turbulence Seminar (Univ. Calif., 615* (1977), 94. Google Scholar*
[20]	R. Williams, The structure of Lorenz attractors,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 73. Google Scholar

show all references

References:

[1]	V. Araujo, M. J. Pacifico, E. R. Pujals and M. Viana, Singular-hyperbolic attractors are chaotic,, Trans. Amer. Math. Soc., 361 (2009), 2431. doi: 10.1090/S0002-9947-08-04595-9. Google Scholar
[2]	Ch. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity,, Encyclopedia of Mathematical Sciences (Mathematical Physics), 102 (2005). Google Scholar
[3]	T. N. Dowker, The mean and transitive points of homeomorphisms,, Ann. of Math., 58 (1953), 123. doi: 10.2307/1969823. Google Scholar
[4]	J. Guckenheimer, A strange, strange attractor,, in The Hopf bifurcation and its applications, (1976), 368. Google Scholar
[5]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59. Google Scholar
[6]	F. Hofbauer, Kneading invariants and Markov diagrams,, in Ergodic theory and related topics (Vitte, 12 (1982), 85. Google Scholar
[7]	T. Jordan, V. Naudot and T. Young, Higher order Birkhoff averages,, Dyn. Syst., 24 (2009), 299. doi: 10.1080/14689360802676269. Google Scholar
[8]	S. Kiriki and T. Soma, Takens' last problem and existence of non-trivial wandering domains,, preprint, (). Google Scholar
[9]	I. S. Labouriau and A. A. P. Rodrigues, On Takens' Last Problem: Tangencies and time averages near heteroclinic networks,, preprint, (). Google Scholar
[10]	E. N. Lorenz, Deterministic non-periodic flow,, J. Atmos. Sci., 20 (1963), 130. Google Scholar
[11]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems,, Proc. Amer. Math. Soc., 127 (1999), 3393. doi: 10.1090/S0002-9939-99-04936-9. Google Scholar
[12]	Y. Nakano, Historic behaviour for quenched random expanding maps on the circle,, preprint, (). Google Scholar
[13]	J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Fractal dimensions and infinitely many attractors, Cambridge Studies in Advanced Mathematics, 35,, Cambridge University Press, (1993). Google Scholar
[14]	C. Robinson, Differentiability of the stable foliation for the model Lorenz equations,, Dynamical systems and turbulence, 898* (1980), 302. Google Scholar*
[15]	D. Ruelle, Historical behaviour in smooth dynamical systems,, in Global Analysis of Dynamical Systems (eds. H. W. Broer et al), (2001), 63. Google Scholar
[16]	F. Takens, Heteroclinic attractors: Time averages and moduli of topological stability,, Bol. Soc. Bras. Mat., 25 (1994), 107. doi: 10.1007/BF01232938. Google Scholar
[17]	F. Takens, Orbits with historic behaviour, or non-existence of averages,, Nonlinearity, 21 (2008). doi: 10.1088/0951-7715/21/3/T02. Google Scholar
[18]	W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comput. Math., 2 (2002), 53. doi: 10.1007/s002080010018. Google Scholar
[19]	R. Williams, The structure of Lorenz attractors,, Turbulence Seminar (Univ. Calif., 615* (1977), 94. Google Scholar*
[20]	R. Williams, The structure of Lorenz attractors,, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 73. Google Scholar

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