Does fractal boundary of basin of attraction has something to do with chaos?
I think fractal boundary must lead to chaos, and how about the other way round?
2026-02-23 03:07:22.1771816042
the relation between chaos and fractal basin
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Fractal basins imply a chaotic transient. To see this, consider any initial condition sufficiently close to the boundary of basins: A small perturbation can make it go to a completely different attractor, hence you clearly have sensitivity to initial conditions in this region.
Fractal basins do not imply a chaotic attractor. For example consider the classical example of three potential wells placed on the corners of a regular triangle: Here the attractors are fixed points, but the basins are fractal. Almost any other standard example for riddled basins also serves as a counterexample.
Chaos does not imply a fractal basin. Here any system that only has one attractor (which is chaotic) serves as a counter-example, since you only have one basin to begin with. Also, there are chaotic systems that are Hamiltonian and thus the concept of an attractor does not make sense at all.