Time-delay Embeddings of Ifs Attractors

A modified type of iterated function system (IFS) has recently been shown to generate images qualitatively similar to “classical” chaotic attractors. Here, we use time-delay embedding reconstructions of time-series from this system to generate three-dimentional projections of IFS attractors. These reconstructions may be used to access the topological structure of the periodic orbits embedded within the attractor. This topological characterization suggests an approach by which a rigorous comparison of IFS attractors and classical chaotic systems may be attained. A modified iterated function system (IFS) has recently been shown to generate images resembling the classical chaotic attractors generated by nonlinear dynamical systems. These “IFS attractors” exhibit dynamical structure and are classifiable by symbolic dynamics.1 In the present paper, we illustrate how their three-dimensional structures may be accessed by a time-delay embedding reconstruction. A typical IFS consists of N affine transformations of the form xn+1 = aixn + biyn + ci yn+1 = dixn + eiyn + fi (1) where i = 1, . . . ,N , and ai, bi, ci, di, ei and fi are constants, and each transformation is assigned a probability pi. A fractal image is generated from an IFS by iterating the mapping according to the scheme described by Barnsley2 in which each new point is generated from the previous point by the application of one of the N transformations. The particular transformation to be employed in each transformation is selected at random, with probability pi. An IFS attractor is generated by a similar algorithm, except that in place of the affine transformations in (1), we have ∗E-mail: [email protected]