Find basins of attraction by simulation

Attempts to determine which areas of the range are drawn to which attractors by simulation. The attractors must be added to the model before simbasins() is added. The space is discretized into squares, and repeated iteration of the model's function is used to determine which attractor the middle of each square tends towards. The square is then given the color of that attractor. The model is assumed to be well behaved in that every point will eventually move within epsilon of an attractor. There is absolutely no guarantee that all basins will be captured by this approach, even with a fine-grained discretization. It is possible to blur boundaries, crop basins, or miss entire regions.

simbasins(discretize = NULL, xlim = NULL, ylim = NULL, iters = NULL,
  epsilon = NULL, behind = TRUE, tolerance = sqrt(.Machine$double.eps),
  stride = 8, cols = NULL, missingCol = "NA", ...)

Arguments

discretize

The size of each square. If not set, the discretization of the range is used. May be set separately from the discretization of the range without overwriting.
xlim

The range of x values to calculate regions over. Defaults to the limits of the range.
ylim

The range of y values to calculate regions over. Defaults to the limits of the range.
iters

If not set, each point will be iterated indvidually. If set as a number, exactly that many iterations will be used. If set as Inf, will iterate until points stabilize (see tolerance). See details.
epsilon

The distance at which a point is considered to have reached an attractor. Defaults to discretize^2. Not used if iters has a numeric value.
behind

Forces this item to be a background object for the purposes of layering
tolerance

The distance distance at which a point is considered to have stopped moving. Defaults to sqrt(.Machine$double.eps).
stride

The number of times the function is applied before movement is checked: in essence finding the basins for f^stride. For non-periodic dynamical systems, this is merely an efficiency concern. For points that move to a periodic attractor with a period that is a factor of stride, this may color the basins by their parity or rank. Only used when iters has a non-numeric value. Defaults to 8.
cols

The colors to use for the various regions. The colors will be used in the order the attractors were added to the model.
missingCol

The color given to points that stop outside of epsilon of an attractor. Defaults to "NA".
...

Extra graphical parameters for image.

Details

All attractors should be dspoints with the attractor flag set to TRUE, and should already be composed with the model. Attractors may have display=FALSE set to avoid displaying the attractor itself. Their color (or region color, if defined) will be used as the color of the region. If there are no points with the attractor flag set, then all points are used as possible attractors. This is not recommended.

If iters is not set, or is set to NULL, then each point will be individually iterated until within epsilon distance of an attractor, or until it moves less than tolerance between iterations. Points that stop moving further than epsilon of an attractor, are colored missingCol, default "NA".

If iters is given a numeric value, each point is iterated exactly iters times, and tolerance will have no effect. If the final image is within epsilon of an attractor, then the square is colored appropriately. If not, then the point is colored missingCol, default "NA". This will take bounded time, but may give a poorer result.

If iters is given an infinite value, the points are iterated until they move less than tolerance distance. An attractor is chosen only if it falls within epsilon distance, otherwise the point is colored missingCol.

The image function is used to display the results.

See also

dspoint dsregion dspolygon simattractors

Examples

library(dsmodels)

model <- dsmodel(function(X0,Y0) {
list(X0*exp(2.6-X0-6.45/(1+4.5*X0)),
     Y0*exp(2.6-Y0-0.15*X0-6.25/(1+4.5*Y0)))
})
model+dsrange(0:3,0:3,discretize = .08)+
      dspoint(1.9358, 1.5059, attractor=TRUE, col="green",  label = "K12")+
      dspoint(1.9358, 0,      attractor=TRUE, col="magenta",label = "K1")+
      dspoint(0,      1.9649, attractor=TRUE, col="orange", label = "K2")+
      dspoint(0,      0,      attractor=TRUE, col="blue",   display=FALSE)+
      dspoint(0.4419, 0.4416, col="yellow", label="A11")+
      simbasins(discretize=0.05)