Mixed Basin Boundary Structures of Chaotic Systems

Phys. Rev. E 59, 343 (1999)https://ireap.umd.edu/10.1103/PhysRevE.59.3431999Journal ArticleComplex and Emergent Systems

Motivated by recent numerical observations on a four-dimensional continuous-time dynamical system, we consider different types of basin boundary structures for chaotic systems. These general structures are essentially mixtures of the previously known types of basin boundaries where the character of the boundary assumes features of the previously known boundary types at different points arbitrarily finely interspersed in the boundary. For example, we discuss situations where an everywhere continuous boundary that is otherwise smooth and differentiable at almost every point has an embedded uncountable, zero Lebesgue measure set of points at which the boundary curve is nondifferentiable. Although the nondifferentiable set is only of zero Lebesgue measure, the curve’s fractal dimension may (depending on parameters) still be greater than one. In addition, we discuss bifurcations from such a mixed boundary to a “pure” boundary that is a fractal nowhere differentiable curve or surface and to a pure nonfractal boundary that is everywhere smooth.