Riddled Basin



You can expand the area by dragging the field with your mouse.

If you download the ZIP file, you can change the size of this simulator by editing the html file.
If the above application does not start, please install Java from here. >>
Let us treat a two dimensional map written as
xn+1 = xn2 - yn2 -xn - λ yn,
yn+1 = 2xnyn - λ xn + yn.

The variables x and y denote the real part and the imaginary part of the map on the complex plane
zn+1 = zn2 - (1+ λ i) zn*.


In this map, there exists three attractors as shown in the following figure.



The line segments shown in red, green, and blue are the attractors respectively,
and each internal dynamics is chaotic like the logistic map.

The set of points which are attracted to an attractor is called its basin.
This simulator investigates the basins of the above three attractors.
The points in the red area are attracted to the red line segment (attractor), and the whiter color means that the point is attracted to the attractor quickly. The green and blue areas are also painted in the similar manner.

It is observed that the basins of these attractors are intermingled very complexly.
Actually, you can observe the blue and green area by expanding the any region in the red area.

As seen in this simulator, the basin with very complicated structures is called as riddled basin.

For the existence of the riddled basin, it is known that the following conditions are required.
  • There exists an invariant submanifold, and its internal dynamics is chaotic.
  • The Lyapunov exponent normal to the invariant submanifold is negative, i.e., the submanifold is an attractor.
  • There exist other attractors.
These complex properties of the dynamical system are thought to exist ubiquitously in the high-dimensional dynamics.


The simulator in this page is based on the following paper.
J.C.Alexander et al., "Riddled Basins"
International Journal of Bifurcation and Chaos, Vol.2, (1992) 795-813.
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Introduction to Chaos and Nonlinear Dynamics