seem (version 1.0)

Stability: Stability analysis and plots (didactic purpose)

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

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.

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.

References

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

See Also

Matrix operations eigen, solve

Examples

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)