Basin boundaries in chaotic dynamical systems can be either smooth or fractal. As a system parameter changes, the structure of the basin boundary also changes. In particular, the dimension of the basin boundary changes continuously except when a basin boundary metamorphosis occurs, at which it can change abruptly. We present numerical experiments to demonstrate such sudden dimension changes. We have also used a one-dimensional analytic calculation and a two-dimensional qualitative model to explain such changes.
Park, B. S., Grebogi, C., & Lai, Y-C. (1992). Abrupt Dimension Changes at Basin Boundary Metamorphoses. International Journal of Bifurcation and Chaos, 2(3), [533-541]. https://doi.org/10.1142/S0218127492000689