SRB measures for partially hyperbolic systems whose central direction is mostly contracting. (English) Zbl 0996.37033

Bonatti, Christian; Viana, Marcelo

Isr. J. Math. 115, 157-193 (2000).

The authors show that synchronization with positive Lyapunov exponents can be observed on numerical simulations of identical, generalized and noise-induced synchronization. They also demonstrate that in all the cases, the synchronization is an outcome of the finite precision in numerical simulations. It is shown that some behaviour and properties of the synchronized system with slightly positive conditional Lyapunov exponents can be understood based on the theory of on-off intermittency.

Reviewer: Messand Efendiev (Berlin)

MSC:

37D30	Partially hyperbolic systems and dominated splittings
37C40	Smooth ergodic theory, invariant measures for smooth dynamical systems

Keywords:

SRB measure; partially hyperbolic system; synchronization; Lyapunov exponents

References:

[1]	[AbSm] R. Abraham and S. Smale,Nongenericity of Ω-stability inGlobal Analysis, Proceedings of Symposia in Pure Mathematics, Vol. 14, American Mathematical Society, 1970. · Zbl 0215.25102
[2]	Afraimovich, V. S.; Bykov, V. V.; Shil’nikov, L. P., On the appearance and structure of the Lorenz attractor, Doklady of the Academy of Sciences of the USSR, 234, 336-339 (1977)
[3]	[Al] J. F. Alves,SRB measures for nonhyperbolic systems with multidimensional expansion, thesis and preprint IMPA, 1997.
[4]	[ABV] J. F. Alves, C. Bonatti and M. Viana,SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly expanding, preprint, 1999. · Zbl 0962.37012
[5]	Benedicks, M.; Carleson, L., The dynamics of the Hénon map, Annals of Mathematics, 133, 73-169 (1991) · Zbl 0724.58042 · doi:10.2307/2944326
[6]	[BeVi1] M. Benedicks and M. Viana,Solution of the basin problem for certain non-uniformly hyperbolic attractors, preprint, 1998. · Zbl 0967.37023
[7]	[BeVi2] M. Benedicks and M. Viana,Random perturbations of Hénon-like maps, in preparation. · Zbl 1131.37033
[8]	Benedicks, M.; Young, L.-S., SBR measures for certain Hénon attractors, Inventiones Mathematicae, 112, 541-576 (1993) · Zbl 0796.58025 · doi:10.1007/BF01232446
[9]	[BeYo2] M. Benedicks and L.-S. Young,Markov extensions and decay of correlations for certain Hénon attractors, Astérisque, to appear. · Zbl 1044.37013
[10]	[Bon] C. Bonatti,A semi-local argument for robust transitivity, Lecture at IMPA’s Dynamical Systems Seminar, August, 1996.
[11]	Bonatti, C.; Díaz, L. J., Persistent nonhyperbolic transitive diffeomorphisms, Annals of Mathematics, 143, 357-396 (1996) · Zbl 0852.58066 · doi:10.2307/2118647
[12]	[BDP] C. Bonatti, L. J. Díaz and E. Pujals,A C^1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, preprint, 1999. · Zbl 1049.37011
[13]	Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms (1975), Berlin: Springer-Verlag, Berlin · Zbl 0308.28010
[14]	Bowen, R.; Ruelle, D., The ergodic theory of Axiom A flows, Inventiones Mathematicae, 29, 181-202 (1975) · Zbl 0311.58010 · doi:10.1007/BF01389848
[15]	Brin, M.; Pesin, Ya., Partially hyperbolic dynamical systems, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskikh, 1, 170-212 (1974) · Zbl 0304.58017
[16]	[Cas] A. Castro,Backward inducing and exponential decay of correlations for partially hyperbolic attractors whose central direction is mostly contracting, thesis and preprint IMPA, 1998.
[17]	Carvalho, M., Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory and Dynamical Systems, 13, 21-44 (1993) · Zbl 0781.58030
[18]	Collet, P.; Tresser, C., Ergodic theory and continuity of the Bowen-Ruelle measure for geometrical Lorenz flows, Fyzika, 20, 33-48 (1988)
[19]	[DPU] L.J. Díaz, E. Pujals and R. Ures,Normal hyperbolicity and persistence of transitivity, Acta Mathematica, to appear.
[20]	[Do] D. Dolgopyat,Statistically stable diffeomorphisms, preprint, 1998.
[21]	Grayson, M.; Pugh, C.; Shub, M., Stably ergodic diffeomorphisms, Annals of Mathematics, 140, 295-329 (1994) · Zbl 0824.58032 · doi:10.2307/2118602
[22]	Guckenheimer, J.; Williams, R. F., Structural stability of Lorenz attractors, Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 50, 307-320 (1979) · Zbl 0436.58018
[23]	Hénon, M., A two dimensional mapping with a strange attractor, Communications in Mathematical Physics, 50, 69-77 (1976) · Zbl 0576.58018 · doi:10.1007/BF01608556
[24]	Hirsch, M.; Pugh, C.; Shub, M., Invariant manifolds (1977), Berlin: Springer-Verlag, Berlin · Zbl 0355.58009
[25]	Kan, I., Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bulletin of the American Mathematical Society, 31, 68-74 (1994) · Zbl 0853.58077
[26]	Kifer, Yu., Random Perturbations of Dynamical Systems (1988), Basel: Birkhäuser, Basel · Zbl 0659.58003
[27]	Lorenz, E. N., Deterministic nonperiodic flow, Journal of Atmospheric Sciences, 20, 130-141 (1963) · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[28]	Mañé, R., Contributions to the stability conjecture, Topology, 17, 383-396 (1978) · Zbl 0405.58035 · doi:10.1016/0040-9383(78)90005-8
[29]	Mañé, R., Smooth Ergodic Theory (1987), Berlin: Springer-Verlag, Berlin
[30]	[Pa] J. Palis,A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, to appear. · Zbl 1044.37014
[31]	Pesin, Ya.; Sinai, Ya., Gibbs measures for partially hyperbolic attractors, Ergodic Theory and Dynamical Systems, 2, 417-438 (1982) · Zbl 0519.58035
[32]	Pesin, Ya., Families of invariant manifolds for diffeomorphisms with non-zero characteristic Lyapunov exponents, Izvestiya Akademii Nauk SSSR, 40, 1332-1379 (1976) · Zbl 0372.58009
[33]	Pesin, Ya., Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties, Ergodic Theory and Dynamical Systems, 12, 123-151 (1992) · Zbl 0774.58029 · doi:10.1017/S0143385700006635
[34]	Pugh, C.; Shub, M., Ergodic attractors, Transactions of the American Mathematical Society, 312, 1-54 (1989) · Zbl 0684.58008 · doi:10.2307/2001206
[35]	Rokhlin, V. A., Lectures on the theory of transformations with invariant measures, Russian Mathematical Surveys, 22, 5, 1-54 (1967) · Zbl 0174.45501 · doi:10.1070/RM1967v022n05ABEH001224
[36]	Ruelle, D., A measure associated with Axiom A attractors, American Journal of Mathematics, 98, 619-654 (1976) · Zbl 0355.58010 · doi:10.2307/2373810
[37]	Ruelle, D.; Takens, F., On the nature of turbulence, Communications in Mathematical Physics, 20, 167-192 (1971) · Zbl 0223.76041 · doi:10.1007/BF01646553
[38]	Sataev, E. A., Invariant measures for hyperbolic maps with singularities, Russian Mathematical Surveys, 471, 191-251 (1992) · Zbl 0795.58036 · doi:10.1070/RM1992v047n01ABEH000864
[39]	Shub, M., Topologically transitive diffeomorphisms on T^4, 39-39 (1971), Berlin: Springer-Verlag, Berlin
[40]	Shub, M., Global Stability of Dynamical Systems (1987), Berlin: Springer-Verlag, Berlin · Zbl 0606.58003
[41]	Sinai, Ya., Gibbs measures in ergodic theory, Russian Mathematical Surveys, 27, 21-69 (1972) · Zbl 0246.28008 · doi:10.1070/rm1972v027n04ABEH001383
[42]	Smale, S., Differentiable dynamical systems, Bulletin of the American Mathematical Society, 73, 747-817 (1967) · Zbl 0202.55202
[43]	Sparrow, C., The Lorenz Equations: Bifurcations, Chaos and Strange Attractors (1982), Berlin: Springer-Verlag, Berlin · Zbl 0504.58001
[44]	Young, L.-S., Stochastic stability of hyperbolic attractors, Ergodic Theory and Dynamical Systems, 6, 311-319 (1986) · Zbl 0633.58023
[45]	Young, L.-S., Large deviations in dynamical systems, Transactions of the American Mathematical Society, 318, 525-543 (1990) · Zbl 0721.58030 · doi:10.2307/2001318