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The finite dimensional global attractor for the 3D viscous Primitive Equations
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 |
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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