Scaling of fractal basin boundaries near intermittency transitions to chaos

It is the purpose of this paper to point out that the creation of fractal basin boundaries is a characteristic feature accompanying the intermittency transition to chaos. (Here ``intermittency'' transition is used in the sense of Pomeau and Manneville [Commun. Math. Phys. 74, 189 (1980)]; viz., a chaotic attractor is created as a periodic orbit becomes unstable.) In particular, we are here concerned with type-I and type-III intermittencies. We examine the scaling of the dimension of basin boundaries near these intermittency transitions. We find, from numerical experiments, that near the transition the dimension scales with a system parameter p according to the power law d~=d₀-k||p-p_I||^β with β=(1/2, where d₀ is the dimension at the intermittency transition parameter value p=p_I and k is a scaling constant. Furthermore, for type-I intermittency d₀<D, while for type-III intermittency d₀=D, where D is the dimension of the space. Heuristic analytic arguments supporting the above are presented.