Transient Chaos and Fractal Basins of Attraction in Delay Differential Equations

The focus of my research is on chaotic dynamics in infinite-dimensional systems. A relatively simple case is that of systems modeled by delay differential equations (DDE's), e.g. of the form x'(t) = f(x(t),x(t-1)). I've been investigating transient chaos in the "logistic" DDE, x'(t) = -x(t) + a x(t-1) [ 1 - x(t-1) ]. For some values of the parameter a, this DDE exhibits multiple stable periodic solutions. The basins of attraction of these solutions are shown in the images below (actually, the images show just one particular 2-dimensional slice through the full infinite-dimensional phase space, which is the space of continuous functions on the interval [-1,0]).

The first image below corresponds to the value a=5.81. The basins of attraction are complicated, but not fractal. The next three images show the basins of attraction, at increasing levels of magnification, for the case a=6.73 (I think). These basins appear to be fractal. Numerical evidence supports the hypothesis that the basin boundaries lie on the stable manifold of an unstable chaotic set ("chaotic saddle" or "strange repeller"): solutions for initial conditions near the boundary exhibit long chaotic transients before converging on one of the stable periodic solutions. The fifth image shows just the basin boundary (via edge detection) from one of the other images.

Image gallery

Click on an image to view the larger version.

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