Sporadically fractal basin boundaries of chaotic systems

We demonstrate a new type of basin boundary for typical chaotic dynamical s ystems. For the case of a two dimensional map, this boundary has the charac ter of the graph of a function that is smooth and differentiable except on a set of fractal dimensions less than one. In spite of the basin boundary b eing smooth "almost everywhere," its fractal dimension exceeds one (implyin g degradation of one's ability to predict the attractor an orbit approaches in the presence of small initial condition uncertainty). We call such a bo undary sporadically fractal. [S0031-9007(99)09061-4].