After downloading riddledbasin.jar
, please execute it by double-clicking, or typing "java -jar riddledbasin.jar".
You can expand the area by dragging the field with your mouse.
If the above application does not start, please install OpenJDK from adoptium.net
Android app of this application is also available. >>
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.
These complex properties of the dynamical system
are thought to exist ubiquitously in the high-dimensional dynamics.
- 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.
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.