Mixed basin boundary structures of chaotic systems

Motivated by recent numerical observations on a four-dimensional continuous -time dynamical system, we consider different types of basin boundary struc tures for chaotic systems. These general structures are essentially mixture s of the previously known types of basin boundaries where the character of the boundary assumes features of the previously known boundary types at dif ferent points arbitrarily finely interspersed in the boundary. For example, we discuss situations where an everywhere continuous boundary that is othe rwise smooth and differentiable at almost every point has an embedded uncou ntable, zero Lebesgue measure set of points at which the boundary curve is nondifferentiable. Although the nondifferentiable set is only of zero Lebes gue measure, the curve's fractal dimension may (depending on parameters) st ill be greater than one. In addition, we discuss bifurcations from such a m ixed boundary to a "pure" boundary that is a fractal nowhere differentiable curve or surface and to a pure nonfractal boundary that is everywhere smoo th. [S1063-651X(99)02401-0].