Generalized dimensions of Feigenbaum’s attractor from renormalization-group functional equations

A method is suggested for the computation of the generalized dimensions of fractal attractors at the period-doubling transition to chaos. The approach is based on an eigenvalue problem formulated in terms of functional equations, with a coefficient expressed in terms of Feigenbaum’s universal fixed-point function. The accuracy of the results is determined only by precision of the representation of the universal function. PACS numbers: 05.45.Df, 05.45.-a, 05.10.Cc The multifractal or thermodynamic formalism is an important tool for description of strange sets arising in dynamical systems in different contexts. Its basic ideas have been clearly formulated e.g. in the paper of Halsey et al. [1]. Some of the examples presented by these and other authors relate to the fractal attractors that occur at the onset of chaos via period doubling and quasiperiodicity [2]–[7]. The multifractal analysis reveals global scaling properties of these attractors, such as the generalized dimensions and the f(α) spectra. They are of principal interest because of their universality for systems of different nature. Moreover, they allow a measurement in physical experiments [7]. One of the well-studied multifractal objects is the Feigenbaum attractor, which occurs at the period-doubling transition to chaos in unimodal one-dimensional maps with quadratic extremum and in a wide class of more general nonlinear dissipative systems [8, 2, 9]. Beside the original procedure of Halsey et al. (namely the construction and analysis of the partition functions defined as sums over some natural covering of the attractor), several other approaches to the computation of the multifractal characteristics have been developed. Bensimon et al. [3] used a method based on a break up of a partition sum into two components with subsequent use of the scaling property. Kovács [4] suggested a procedure of extracting the dimensions from the eigenvalue problem for the Frobenius-Perron operator. Christiansen et al. [5] exploited the idea of approximating the strange sets by periodic orbits and expressed the desired quantities in terms of cycle expansions. (To our knowledge, the calculation of the Hausdorff dimension of the Feigenbaum attractor in Ref. [5] remains the most precise to date.) In some sense, the global description of scaling properties in the multifractal formalism seems opposite to the local description in terms of the Feigenbaum renormalization group approach [8]. The latter is based on the solution of the functional fixed-point equation and associated with scaling relations for the evolution operators in a neighborhood of the extremum of the map under consideration.