Defines a model object encapsulating a dynamical system

To begin, define a two dimensional function that outputs a two-dimensional list. For instance: fun <- function(x,y) list(x,y). Then dsmodel(fun) will initialize the model. Make sure that the function defining your model indeed outputs a list. The model is used to hold the data for graphics. To display a desired graphic, add the corresponding feature of type "dsproto" to your model.

dsmodel(fun, title = "", display = TRUE)

Arguments

fun	Function with two inputs and two outputs which defines the dynamical system. The output should be a list, preferably with field names x and y.
title	A string title for the graph. Text can be input in the form of pseudo-LaTeX code within quotes. See TeX for more details.
display	If set to FALSE, the model will be drawn only when the user calls `MODELNAME`$display(). Otherwise, the model will be drawn with every dsmodels object added after the dsrange is added.

Details

Models are constructed incrementally using the + operator to add features to the existing dsmodel object. A dsrange must be one of the objects added to a model.

Methods

dsmodel objects support the following methods, which may be helpful for advanced users. dsmodel$points(format="list", filter="all") returns a list of the points in the model.

filter

Valid values are "all", "fixed", "attractor", and "sim". "sim" returns only points generated by simattractors.

format

If "list", return a list with x, y, col, and ind fields holding the coordinates, color, and index. Other formats are "objects", returning a vector of dspoint objects, and "pairs", returning a list of pairs of coordinates.

dsmodel$display() forces the model to re-render the plot from scratch. Primarily useful if display=TRUE was set. dsmodel$basins() returns a list of which fixed points have a basin. This requires simbasins() to have been composed with the model, and is primarily useful when testing if a dynamical system is globally stable. In that case, the method will return a list of length 1. The list will contain the indices of the fixed points, as given in dsmodel$points(format="list", filter="attractor"). An index of 0 means that some points never moved within epsilon of an attractor. dsmodel$sim.is.stable() attempts to determine if the system is stable by simulation. If no attractors have been composed with the model, simattractors() is composed with defaults. If simbasins has not been composed with the model, it is be composed with defaults. If every point is drawn to a single attractor, the system has been deemed stable. Note that boundary points on the range will not be tested.

See also

dsrange

Examples

library(dsmodels)

fun <- function(X,Y) {
  list(
    x = X/exp(Y),
    y = Y/exp(X)
  )
}
# Add dsRange to see the actual range.
model <- dsmodel(fun)

dsmodel(function(x,y) {
  list(
    x = x^2,
    y = x/(y+1)
  )
}, title = "Another function showing $f(x)=x^{\\alpha}$!")