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The fractal property of the Lorenz attractor. (English) Zbl 1041.37013
Summary: In a 1963 paper [Deterministic non-periodic flows, J. Atoms. Sci. 20, 130–141 (1963)], E. N. Lorenz inferred that the Lorenz attractor must be an infinite complex of surfaces. We investigate this fractal property of the Lorenz attractor in two ways. Firstly, we obtain explicit plots of the fractal structure of the Lorenz attractor using symbolic dynamics and multiple precision computations of periodic orbits. The method we derive for multiple precision computation is based on iterative refinement and can compute even highly unstable periodic orbits with long symbol sequences with as many as 100 digits of accuracy. Ordinary numerical integrations are much too crude to show even the coarsest splitting of surfaces, and there appear to be no other explicit computations of the fractal structure in the extensive literature about the Lorenz attractor. Secondly, we apply a well known formula that gives the Hausdorff dimension of the Lorenz attractor in terms of the characteristic multipliers of its unstable periodic orbits. The formula converges impressively and the Hausdorff dimension of the Lorenz attractor appears to be 2.0627160. We use a comparison with explicit computations of the fractal structure and discuss the accuracy of this formula and its applicability to the Lorenz equations. Additionally, we apply periodic orbit theory to the Lorenz attractor and exhibit its spectral determinant and compute its Lyapunov exponent.
##### MSC:
 37D45 Strange attractors, chaotic dynamics 37B10 Symbolic dynamics 37C45 Dimension theory of dynamical systems 65L15 Eigenvalue problems for ODE (numerical methods) 65P99 Numerical problems in dynamical systems
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