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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

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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