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 and Continuous Dynamical Systems, 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), 2431-2485. doi: 10.1090/S0002-9947-08-04595-9.
[2]	Ch. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopedia of Mathematical Sciences (Mathematical Physics), 102, Mathematical physics, III. Springer Verlag, 2005.
[3]	T. N. Dowker, The mean and transitive points of homeomorphisms, Ann. of Math., 58 (1953), 123-133. doi: 10.2307/1969823.
[4]	J. Guckenheimer, A strange, strange attractor, in The Hopf bifurcation and its applications, ( eds. J. E. Marsden and M. McCracke), Springer-Verlag, New York, (1976), 368-381.
[5]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.
[6]	F. Hofbauer, Kneading invariants and Markov diagrams, in Ergodic theory and related topics (Vitte, 1981), Math. Res., Akademie-Verlag, Berlin, 12 (1982), 85-95.
[7]	T. Jordan, V. Naudot and T. Young, Higher order Birkhoff averages, Dyn. Syst., 24 (2009), 299-313. doi: 10.1080/14689360802676269.
[8]	S. Kiriki and T. Soma, Takens' last problem and existence of non-trivial wandering domains,, preprint, ().
[9]	I. S. Labouriau and A. A. P. Rodrigues, On Takens' Last Problem: Tangencies and time averages near heteroclinic networks,, preprint, ().
[10]	E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141.
[11]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401. doi: 10.1090/S0002-9939-99-04936-9.
[12]	Y. Nakano, Historic behaviour for quenched random expanding maps on the circle,, preprint, ().
[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, Cambridge, 1993.
[14]	C. Robinson, Differentiability of the stable foliation for the model Lorenz equations, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), pp. 302-315, Lecture Notes in Math., 898, Springer, Berlin-New York, 1981.
[15]	D. Ruelle, Historical behaviour in smooth dynamical systems, in Global Analysis of Dynamical Systems (eds. H. W. Broer et al), Inst. Phys., Bristol, 2001, 63-66.
[16]	F. Takens, Heteroclinic attractors: Time averages and moduli of topological stability, Bol. Soc. Bras. Mat., 25 (1994), 107-120. doi: 10.1007/BF01232938.
[17]	F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity, 21 (2008), T33-T36. doi: 10.1088/0951-7715/21/3/T02.
[18]	W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117. doi: 10.1007/s002080010018.
[19]	R. Williams, The structure of Lorenz attractors, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977), Lecture Notes in Math., Springer, 615 (1977), 94-112.
[20]	R. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 73-99.

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), 2431-2485. doi: 10.1090/S0002-9947-08-04595-9.
[2]	Ch. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopedia of Mathematical Sciences (Mathematical Physics), 102, Mathematical physics, III. Springer Verlag, 2005.
[3]	T. N. Dowker, The mean and transitive points of homeomorphisms, Ann. of Math., 58 (1953), 123-133. doi: 10.2307/1969823.
[4]	J. Guckenheimer, A strange, strange attractor, in The Hopf bifurcation and its applications, ( eds. J. E. Marsden and M. McCracke), Springer-Verlag, New York, (1976), 368-381.
[5]	J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 59-72.
[6]	F. Hofbauer, Kneading invariants and Markov diagrams, in Ergodic theory and related topics (Vitte, 1981), Math. Res., Akademie-Verlag, Berlin, 12 (1982), 85-95.
[7]	T. Jordan, V. Naudot and T. Young, Higher order Birkhoff averages, Dyn. Syst., 24 (2009), 299-313. doi: 10.1080/14689360802676269.
[8]	S. Kiriki and T. Soma, Takens' last problem and existence of non-trivial wandering domains,, preprint, ().
[9]	I. S. Labouriau and A. A. P. Rodrigues, On Takens' Last Problem: Tangencies and time averages near heteroclinic networks,, preprint, ().
[10]	E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141.
[11]	C. A. Morales, M. J. Pacifico and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999), 3393-3401. doi: 10.1090/S0002-9939-99-04936-9.
[12]	Y. Nakano, Historic behaviour for quenched random expanding maps on the circle,, preprint, ().
[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, Cambridge, 1993.
[14]	C. Robinson, Differentiability of the stable foliation for the model Lorenz equations, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), pp. 302-315, Lecture Notes in Math., 898, Springer, Berlin-New York, 1981.
[15]	D. Ruelle, Historical behaviour in smooth dynamical systems, in Global Analysis of Dynamical Systems (eds. H. W. Broer et al), Inst. Phys., Bristol, 2001, 63-66.
[16]	F. Takens, Heteroclinic attractors: Time averages and moduli of topological stability, Bol. Soc. Bras. Mat., 25 (1994), 107-120. doi: 10.1007/BF01232938.
[17]	F. Takens, Orbits with historic behaviour, or non-existence of averages, Nonlinearity, 21 (2008), T33-T36. doi: 10.1088/0951-7715/21/3/T02.
[18]	W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117. doi: 10.1007/s002080010018.
[19]	R. Williams, The structure of Lorenz attractors, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977), Lecture Notes in Math., Springer, 615 (1977), 94-112.
[20]	R. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 73-99.

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