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

Geometric Lorenz flows with historic behavior

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.
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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