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# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2, or (at your option)
# any later version.
#
# This program is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# A copy of the GNU General Public License is available via WWW at
# http://www.gnu.org/copyleft/gpl.html. You can also obtain it by
# writing to the Free Software Foundation, Inc., 59 Temple Place,
# Suite 330, Boston, MA 02111-1307 USA.
# Copyrights (C)
# for this R-port:
# 1999 - 2007, Diethelm Wuertz, GPL
# Diethelm Wuertz <wuertz@itp.phys.ethz.ch>
# info@rmetrics.org
# www.rmetrics.org
# for the code accessed (or partly included) from other R-ports:
# see R's copyright and license files
# for the code accessed (or partly included) from contributed R-ports
# and other sources
# see Rmetrics's copyright file
################################################################################
# FUNCTION: CHAOTIC TIME SERIES MAPS:
# tentSim Simulates series from Tent map
# henonSim Simulates series from Henon map
# ikedaSim Simulates series from Ikeda map
# logisticSim Simulates series from Logistic map
# lorentzSim Simulates series from Lorentz map
# roesslerSim Simulates series from Roessler map
# .rk4 Internal Funtion - Runge-Kutta Solver
################################################################################
tentSim =
function(n = 1000, n.skip = 100, parms = c(a = 2), start = runif(1),
doplot = FALSE)
{ # A function implemented by Diethelm Wuertz
# Description:
# Simulate Data from Tent Map
# Arguments:
# n - number of points x, y
# n.skip - number of transients discarded
# start - initial x
# Details:
# Creates iterates of the Tent map:
# * x(n+1) = a * x(n) if x(n) < 0.5
# * x(n+1) = a * ( 1 - x(n)) if x(n) >= 0.5
# FUNCTION:
# Simulate Map:
a = parms[1]
if (a == 2) a = a - .Machine$double.eps
x = rep(0, times = (n+n.skip))
i = 1
x[i] = start
for ( i in 2:(n+n.skip) ) {
x[i] = (a/2) * ( 1 - 2*abs(x[i-1]-0.5) )
}
x = x[(n.skip+1):(n.skip+n)]
# Plot Map:
if (doplot) {
# Time Series Plot:
# plot(x = x, type = "l", xlab = "n", ylab = "x[n]",
# main = paste("Tent Map \n a =", as.character(a)),
# col = "steelblue")
# abline(h = 0.5, col = "grey", lty = 3)
# Delay Plot:
plot(x[-n], x[-1], xlab = "x[n]", ylab = "x[n+1]",
main = paste("Tent Map\n a =", as.character(a)),
cex = 0.25, col = "steelblue")
}
# Return Value:
ts(x)
}
# ------------------------------------------------------------------------------
henonSim =
function(n = 1000, n.skip = 100, parms = c(a = 1.4, b = 0.3),
start = runif(2), doplot = FALSE)
{ # A function implemented by Diethelm Wuertz
# Description:
# Simulate Data from Henon Map
# Arguments:
# n - number of points x, y
# n.skip - number of transients discarded
# a - parameter a
# b - parameter b
# start[1] - initial x
# start[2] - initial y
# Details:
# Creates iterates of the Henon map:
# * x(n+1) = 1 - a*x(n)^2 + b*y(n)
# * y(n+1) = x(n)
# FUNCTION:
# Simulate Map:
a = parms[1]
b = parms[2]
x = rep(0, times = (n+n.skip))
y = rep(0, times = (n+n.skip))
x[1] = start[1]
y[1] = start[2]
for ( i in 2:(n+n.skip) ) {
x[i] = 1 - a*x[i-1]^2 + b*y[i-1]
y[i] = x[i-1] }
x = x[(n.skip+1):(n.skip+n)]
y = y[(n.skip+1):(n.skip+n)]
# Plot Map:
if (doplot) {
# Time Series Plot:
# ...
# Delay Plot:
plot(x = x, y = y, type = "n", xlab = "x[n]", ylab = "y[n]",
main = paste("Henon Map \n a =", as.character(a),
" b =", as.character(b)) )
points(x = x, y = y, col = "steelblue", cex = 0.25)
}
# Return Value:
ts(cbind(x, y))
}
# ------------------------------------------------------------------------------
ikedaSim =
function(n = 1000, n.skip = 100, parms = c(a = 0.4, b = 6.0, c = 0.9),
start = runif(2), doplot = FALSE)
{ # A function written by Diethelm Wuertz
# Description:
# Simulate Ikeda Map Data
# Arguments:
# n - number of points z
# n.skip - number of transients discarded
# a - parameter a
# b - parameter b; 6.0
# c - parameter c; 0.9
# start[1] - initial Re(z)
# start[2] - initial Im(z)
# Details:
# Prints iterates of the Ikeda map (Re(z) and Im(z)):
# i*b
# z(n+1) = 1 + c*z(n)* exp( i*a - ------------ )
# 1 + |z(n)|^2
# FUNCTION:
# Simulate Map:
A = a = parms[1]
B = b = parms[2]
C = c = parms[3]
a = complex(real = 0, imag = a)
b = complex(real = 0, imag = b)
z = rep(complex(real = start[1], imag = start[2]), times = (n+n.skip))
for ( i in 2:(n+n.skip) ) {
z[i] = 1 + c*z[i-1] * exp(a-b/(1+abs(z[i-1])^2)) }
z = z[(n.skip+1):(n.skip+n)]
# Plot Map:
if (doplot) {
x = Re(z)
y = Im(z)
plot(x, y, type = "n", xlab = "x[n]", ylab = "y[n]",
main = paste("Ikeda Map \n", "a =", as.character(A),
" b =", as.character(B), " c =", as.character(C)) )
points(x, y, col = "steelblue", cex = 0.25)
x = Re(z)[1:(length(z)-1)]
y = Re(z)[2:length(z)]
plot(x, y, type = "n", xlab = "x[n]", ylab = "x[n+1]",
main = paste("Ikeda Map \n", "a =", as.character(A),
" b =", as.character(B), " c =", as.character(C)) )
points(x, y, col = "steelblue", cex = 0.25) }
# Return Value:
ts(cbind(Re = Re(z), Im = Im(z)))
}
# ------------------------------------------------------------------------------
logisticSim =
function(n = 1000, n.skip = 100, parms = c(r = 4), start = runif(1),
doplot = FALSE)
{ # A function written by Diethelm Wuertz
# Description:
# Simulate Data from Logistic Map
# Arguments:
# n - number of points x, y
# n.skip - number of transients discarded
# r - parameter r
# start - initial x
# Details:
# Creates iterates of the Logistic Map:
# * x(n+1) = r * x[n] * ( 1 - x[n] )
# FUNCTION:
# Simulate Map:
r = parms[1]
x = rep(0, times = (n+n.skip))
x[1] = start
for ( i in 2:(n+n.skip) ) {
x[i] = r * x[i-1] * ( 1 - x[i-1] ) }
x = x[(n.skip+1):(n.skip+n)]
# Plot Map:
if (doplot) {
plot(x = x[1:(n-1)], y = x[2:n], type = "n", xlab = "x[n-1]",
ylab = "x[n]", main = paste("Logistic Map \n r =",
as.character(r)) )
points(x = x[1:(n-1)], y = x[2:n], col = "steelblue", cex = 0.25) }
# Return Value:
ts(x)
}
# ------------------------------------------------------------------------------
lorentzSim =
function(times = seq(0, 40, by = 0.01), parms = c(sigma = 16, r = 45.92, b = 4),
start = c(-14, -13, 47), doplot = TRUE, ...)
{ # A function written by Diethelm Wuertz
# Description:
# Simulates a Lorentz Map
# Notes:
# Requires rk4 from R package "odesolve"
# FUNCTION:
# Requirements:
# BUILTIN - require(odesolve)
# Settings:
sigma = parms[1]
r = parms[2]
b = parms[3]
# Attractor:
lorentz =
function(t, x, parms) {
X = x[1]
Y = x[2]
Z = x[3]
with(as.list(parms), {
dX = sigma * ( Y - X )
dY = -X*Z + r*X - Y
dZ = X*Y - b*Z
list(c(dX, dY, dZ))})
}
# Classical RK4 with fixed time step:
s = .rk4(start, times, lorentz, parms)
# Display:
if (doplot) {
xylab = c("x", "y", "z", "x")
for (i in 2:4)
plot(s[, 1], s[, i], type = "l",
xlab = "t", ylab = xylab[i-1], col = "steelblue",
main = paste("Lorentz \n", "sigma =", as.character(sigma),
" r =", as.character(r), " b =", as.character(b)), ...)
k = c(3, 4, 2)
for (i in 2:4) plot(s[, i], s[, k[i-1]], type = "l",
xlab = xylab[i-1], ylab = xylab[i], col = "steelblue",
main = paste("Lorentz \n", "sigma =", as.character(sigma),
" r =", as.character(r), " b =", as.character(b)), ...)
}
# Result:
colnames(s) = c("t", "x", "y", "z")
# Return Value:
ts(s)
}
# ------------------------------------------------------------------------------
roesslerSim =
function(times = seq(0, 100, by = 0.01), parms = c(a = 0.2, b = 0.2, c = 8.0),
start = c(-1.894, -9.920, 0.0250), doplot = TRUE, ...)
{ # A function written by Diethelm Wuertz
# Description:
# Simulates a Lorentz Map
# Notes:
# Requires contributed R package "odesolve"
# FUNCTION:
# Settings:
a = parms[1]
b = parms[2]
c = parms[3]
# Attractor:
roessler = function(t, x, parms) {
X = x[1]; Y = x[2]; Z = x[3]
with(as.list(parms), {
dX = -(Y+Z)
dY = X + a*Y
dZ = b + X*Z -c*Z
list(c(dX, dY, dZ))}) }
# Classical RK4 with fixed time step:
s = .rk4(start, times, roessler, parms)
# Display:
if (doplot) {
xylab = c("x", "y", "z", "x")
for (i in 2:4) plot(s[, 1], s[, i], type = "l",
xlab = "t", ylab = xylab[i-1], col = "steelblue",
main = paste("Roessler \n", "a = ", as.character(a),
" b = ", as.character(b), " c = ", as.character(c)), ...)
k = c(3, 4, 2)
for (i in 2:4) plot(s[, i], s[, k[i-1]], type = "l",
xlab = xylab[i-1], ylab = xylab[i], col = "steelblue",
main = paste("Roessler \n", "a = ", as.character(a),
" b = ", as.character(b), " c = ", as.character(c)), ...)
}
# Result:
colnames(s) = c("t", "x", "y", "z")
# Return Value:
ts(s)
}
# ------------------------------------------------------------------------------
.rk4 =
function(y, times, func, parms)
{
# Description:
# Classical Runge-Kutta-fixed-step-integration
# Autrhor:
# R-Implementation by Th. Petzoldt,
# Notes:
# From Package: odesolve
# Version: 0.5-12
# Date: 2004/10/25
# Title: Solvers for Ordinary Differential Equations
# Author: R. Woodrow Setzer <setzer.woodrow@epa.gov>
# Maintainer: R. Woodrow Setzer <setzer.woodrow@epa.gov>
# Depends: R (>= 1.4.0)
# License: GPL version 2
# Packaged: Mon Oct 25 14:59:00 2004
# FUNCTION:
# Checks:
if (!is.numeric(y)) stop("`y' must be numeric")
if (!is.numeric(times)) stop("`times' must be numeric")
if (!is.function(func)) stop("`func' must be a function")
if (!is.numeric(parms)) stop("`parms' must be numeric")
# Dimension:
n = length(y)
# Call func once to figure out whether and how many "global"
# results it wants to return and some other safety checks
rho = environment(func)
tmp = eval(func(times[1], y,parms), rho)
if (!is.list(tmp)) stop("Model function must return a list\n")
if (length(tmp[[1]]) != length(y))
stop(paste("The number of derivatives returned by func() (",
length(tmp[[1]]),
"must equal the length of the initial conditions vector (",
length(y),")", sep = ""))
Nglobal = if (length(tmp) > 1) length(tmp[[2]]) else 0
y0 = y
out = c(times[1], y0)
for (i in 1:(length(times)-1)) {
t = times[i]
dt = times[i+1] - times[i]
F1 = dt * func(t, y0, parms)[[1]]
F2 = dt * func(t+dt/2, y0 + 0.5 * F1, parms)[[1]]
F3 = dt * func(t+dt/2, y0 + 0.5 * F2, parms)[[1]]
F4 = dt * func(t+dt , y0 + F3, parms)[[1]]
dy = (F1 + 2 * F2 + 2 * F3 + F4)/6
y1 = y0 + dy
out<- rbind(out, c(times[i+1], y1))
y0 = y1
}
nm = c("time",
if (!is.null(attr(y, "names"))) names(y)
else as.character(1:n))
if (Nglobal > 0) {
out2 = matrix(nrow=nrow(out), ncol = Nglobal)
for (i in 1:nrow(out2))
out2[i,] = func(out[i,1], out[i,-1], parms)[[2]]
out = cbind(out, out2)
nm = c(nm,
if (!is.null(attr(tmp[[2]],"names"))) names(tmp[[2]])
else as.character((n+1) : (n + Nglobal)))
}
dimnames(out) = list(NULL, nm)
# Return Value:
out
}
################################################################################
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