Analytical approach to estimating the dimension of attractors

P. Constantin and C. Foias, Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations, Comm. Pure Appl. Math. 38 (1985), 1–27.Google Scholar

P. Constantin, C. Foias, and R. Temam, Attractors Representing Turbulent Flows, Mem. Amer. Math. Soc., Vol. 53, No. 314, American Mathematical Society, Providence, RI, 1985Google Scholar

A. Eden, C. Foias, and R. Temam, Local and global Lyapunov exponents, J. Dyn. Differential Equations 3 (1991), 133–177.Google Scholar

J. D. Farmer, E. Ott, and J. A. Yorke, The dimension of chaotic attractors, Physica 7D (1983), 153–180.Google Scholar

A. C. Fowler, J. D. Gibbon, and M. J. McGuinness, The complex Lorenz equations, Physica 4D (1982), 139–163.Google Scholar

J. D. Gibbon and M. J. McGuinness, The real and complex Lorenz equations in rotating fluids and lasers, Physica 5D (1982), 108–122.Google Scholar

P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Physica 9D (1983), 189–208.Google Scholar

P. Grassberger and I. Procaccia, Dimensions and entropies of strange attractors from a fluctuating dynamics approach, Physica 13D (1984), 34–54.Google Scholar

Y. Hattori, N. Ishimura, I. Ohnishi, and M. Umeki, Dimension estimate of the global attractor for forced oscillation systems, Japan J. Indust. Appl. Math. 10 (1993), 351–366.Google Scholar

N. Ishimura and M. Nakamura, On the simplified magnetic Bénard problem—dimension estimate of the attractor, Adv. Math. Sci. Appl. 4 (1994), 241–247.Google Scholar

J. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Difference Equations and Approximation of Fixed Points, Lecture Notes in Mathematics, Vol. 730, Springer-Verlag, Berlin, 1979, pp. 204–227.Google Scholar

F. Ledrappier, Some relations between dimension and Lyapunov exponents, Comm. Math. Phys. 81 (1981), 229–238.Google Scholar

E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. 20 (1963), 130–141.Google Scholar

M. Sano and Y. Sawada, Measurement of the Lyapunov spectrum from a chaotic time series, Phys. Rev. Lett. 55 (1985), 1082–1085.Google Scholar

I. Shimada and T. Nagashima, A numerical approach to ergodic problems of dissipative dynamical systems, Progr. Theoret. Phys. 61 (1979), 1605–1616.Google Scholar

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Berlin, 1988.Google Scholar

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica 16D (1985), 285–317.Google Scholar

L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems 2 (1982), 109–124.Google Scholar