Description of the Rigorous Initial Value Problem Solver
This Java applet produces zonotope enclosures for the orbit of initial
value problems. Only planar and discrete maps are considered. A
Flush-When-Full algorithm is used to avoid the wrapping effect. If
f is a planar map, and A0 is an initial set,
then the sequence of sets An+1 = f(An)
constitutes the orbit. The algorithm computes guaranteed enclosure
sets Zn of An. The inclusive sets
Zn are zonotopes of high order. Zonotopes are
special polytopes. For order 1, the zonotopes are always boxes. A
box is a rectangle aligned with the coordinate axes. For order 2, the
zonotope is the Minkowski sum of a box and a parallelepiped, for oder
3 the Minkowski sum of a box and two parallelepipeds, and so on.
The user supplies the following:
- The planar map
Allowed are the operations +, -, *, /,
^ (integer power) and the standard functions sin,
cos, exp and sqrt. Numerical
constants are either floating point constants like 3.141, or intervals
formed by such constants, for example [3.141, 3.142].
- The visible view of the plane defined by the minimal and maximal x
and y values.
- Order of Zonotopes
This is the performance parameter of the algorithm. The quality of
the enclosures increases with the order. An order of 1 always produces
interval enclosures and therefore performs very poorly. An order of 5
is usually appropriate.
- Number of steps
Specifies the number of steps performed each time the step button is pressed.
- Initial set
Specify the (x, y) coordinate and the radius of the initial set.
The radius is with respect to the max-norm, i.e., the initial
set is always a square.
- Initial Value Chooser
Click once on canvas to select a new initial value.
Note that the initial radius can only be selected manually.
- Domain Chooser
Click and drag mouse on canvas to choose new view. To activate that view,
press the set button.
- PostScript Interface
PostScript output can be generated and directed to a file. To
activate, press SHIFT and click on one of the three canvases. Note
that PostScript code is not generated of what you see, but of what is
newly displayed after you active it. For example, you have to press
the erase button to generate the axis. This feature needs improvement.
Last modified by Wolfgang Kühn on
Sunday, 10 October 1999