stability.phase.plot: Stability analysis and plots (didactic purpose)

In seem: Simulation of Ecological and Environmental Models

Description

For a 2D linear dynamical system, function sta.plot is used to plot state trajectories vs. time. Function sta.plot is used to plot phase-plane and illustrate orbit by arrow. Function root.plot just plots the roots of A. Function root.locus.quad plots is used to plot the locus of the root by sweeping a parameter. It calculates eigenvalues in four quadrants, to be used by functions ReIm.plot and D.plot.

Usage

sta.plot(A, x0, t)
phase.plot(A, x0, t, long = 20)
root.plot(A)
root.locus.quad(det, Tr)
ReIm.plot(x, j, k)
D.plot(x, i)

Arguments

A	matrix
x0	initial condition
t	time sequence
long	determine size of arrows
det	determinant of matrix
Tr	trace of matrix
x	list containing sequences of det and Tr and their discriminant, and eigenvalues (see output from root.locus.quad)
j	quadrant number
i	quadrant number
k	position in array

Details

Given matrix A of 2D linear dynamical system, function sta.plot is used to plot X trajectories vs. time. Function sta.plot is used to plot phase-plane and illustrate orbit by arrow. Determinant, trace, discriminant, and eigenvalues are printed on the plot.

Function root.plot just plots the roots of A. Function root.locus.quad calculates eigenvalues in four quadrants, to be used by plotting functions ReIm.plot and D.plot.

Value

Function root.locus.quad:

det	sequence of det values
Tr	sequence of trace values
D	discriminant for each pair of det and Tr
ReL	real part of eigenvalues for each pair of det and Tr
ImL	imaginary part of eigenvalues for each pair of det and Tr

Other functions do not return values but produce plots.

Note

For didactic purposes.

Author(s)

Miguel F. Acevedo [e-mail]

References

Acevedo M.F. 2012. Simulation of Ecological and Environmental Models. CRC Press.

See Also

Matrix operations eigen, solve

Examples

## Not run: 
t <- seq(0,100,0.1)
# pos det, neg trace, pos D
A <- matrix(c(-0.1,-0.1,-0.2,-0.3),2,byrow=T);x0 <-c(10,5)
sta.plot(A,x0,t)

# pos det, neg trace, pos D
A <- matrix(c(-0.1,-0.1,-0.2,-0.3),2,byrow=T);x0 <-c(10,5)
t <- seq(0,10,0.1)
phase.plot(A,x0,t,long=12)

x <- list()
#pos det, neg trace
det <- seq(0,1,0.5);Tr <- seq(-3,0,0.001) 
x[[1]] <- root.locus.quad(det,Tr)
# pos det, pos trace
det <- seq(0,1,0.5);Tr <- seq(0,3,0.001) 
x[[3]] <- root.locus.quad(det,Tr)
#neg det, neg trace
det <- seq(-1,0,0.5);Tr <- seq(-3,0,0.001) 
x[[2]] <- root.locus.quad(det,Tr)
#neg det, pos trace
det <- seq(-1,0,0.5);Tr <- seq(0,3,0.001) 
x[[4]] <- root.locus.quad(det,Tr)

 mat <- matrix(1:4,2,2,byrow=T)
 nf <- layout(mat, widths=rep(7/2,2), heights=rep(7/2,2), TRUE)
 par(mar=c(4,4,1,.5),xaxs="i",yaxs="i")
for(i in 1:4) D.plot(x[[i]],i)

 mat <- matrix(1:4,2,2,byrow=T)
 nf <- layout(mat, widths=rep(7/2,2), heights=rep(7/2,2), TRUE)
 par(mar=c(4,4,1,.5),xaxs="r",yaxs="r")

  j=1
  ReIm.plot(x[[j]],j,1)
  ReIm.plot(x[[j]],j,2)
  j=3
  ReIm.plot(x[[j]],j,1)
  ReIm.plot(x[[j]],j,2)
  j=2
  ReIm.plot(x[[j]],j,1)
  ReIm.plot(x[[j]],j,2)
  j=4
  ReIm.plot(x[[j]],j,1)
  ReIm.plot(x[[j]],j,2)

## End(Not run)

seem documentation built on April 14, 2017, 9:12 p.m.