if( !require('phaseR') ){
    install.packages('phaseR')
}
library(phaseR)

save_plots = FALSE


diff_eq_params = list(alpha=2.0)

# The critical points of this system:
#
CP = data.frame(
    x=c(1, 2),
    y=c(3, 4))

my_jacobian = function(x, y, parameters=diff_eq_params) {
    # returns the Jacobian at the point (x, y)
    #
    J = matrix(c(-1, 1, -4 + 2*x, 1), nrow=2, ncol=2, byrow=TRUE)
}

As = mapply(my_jacobian, CP$x, CP$y, SIMPLIFY=FALSE)
for (ii in 1:length(As)){
    es = eigen(As[[ii]])
    print(sprintf('CP: x= %f; y= %f', CP$x[ii], CP$y[ii]))
    print('Jacobian=')
    print(As[[ii]])
    print('eigenvalues=')
    print(es$values)
}

my_yprime = function(t, y, parameters) {
    xt = y[1]
    yt = y[2]
    dy = rep(NA, length(y))
    dy[1] = -parameters$alpha - yt + xt
    dy[2] = -4*xt + yt + xt^2
    list(dy)
}

# Part (c):
#
if( save_plots ){ postscript('../../WriteUp/Graphics/Chapter9/chap_9_sect_4_prob_16_plot.eps', onefile=FALSE, horizontal=FALSE) }

par(mfrow=c(1, 3))

t.end = 5.0
flowField(my_yprime, x.lim = c(0, +4), y.lim = c(0, +6), parameters = diff_eq_params, points = 30, add = FALSE, xlab='x', ylab='y', main=sprintf('alpha= %.3f', diff_eq_params$alpha))

# Plot some trajectories with initial condition near the critical points:
#
trajectory(my_yprime, y0 = c(2, 5), t.end = t.end, parameters = diff_eq_params, col='black', pch=8)
trajectory(my_yprime, y0 = c(3.5, 4.0), t.end = t.end, parameters = diff_eq_params, col='blue', pch=8)
trajectory(my_yprime, y0 = c(2, 1.0), t.end = t.end, parameters = diff_eq_params, col='red', pch=8)
trajectory(my_yprime, y0 = c(3, 1.0), t.end = t.end, parameters = diff_eq_params, col='green', pch=8)
points(CP$x, CP$y, pch=19)



# Part (d):
#
alpha_0 = 9/4
diff_eq_params = list(alpha=alpha_0)

CP = data.frame(
    x=c(2),
    y=c((3/2)*alpha_0))


t.end = 5.0
flowField(my_yprime, x.lim = c(0, +4), y.lim = c(0, +6), parameters = diff_eq_params, points = 30, add = FALSE, xlab='x', ylab='y', main=sprintf('alpha= alpha_0= %.3f', alpha_0))

# Plot some trajectories with initial condition near the critical points:
#
trajectory(my_yprime, y0 = c(2, 5), t.end = t.end, parameters = diff_eq_params, col='black', pch=8)
trajectory(my_yprime, y0 = c(3.5, 4.0), t.end = t.end, parameters = diff_eq_params, col='blue', pch=8)
trajectory(my_yprime, y0 = c(2, 1.0), t.end = t.end, parameters = diff_eq_params, col='red', pch=8)
trajectory(my_yprime, y0 = c(3, 1.0), t.end = t.end, parameters = diff_eq_params, col='green', pch=8)
points(CP$x, CP$y, pch=19)


# Part (e):
#
alpha = 3
diff_eq_params = list(alpha=alpha)

t.end = 5.0
flowField(my_yprime, x.lim = c(0, +4), y.lim = c(0, +6), parameters = diff_eq_params, points = 30, add = FALSE, xlab='x', ylab='y', main=sprintf('alpha= %.3f', alpha))

# Plot some trajectories with initial condition near the critical points:
#
trajectory(my_yprime, y0 = c(2, 5), t.end = t.end, parameters = diff_eq_params, col='black', pch=8)
trajectory(my_yprime, y0 = c(3.5, 4.0), t.end = t.end, parameters = diff_eq_params, col='blue', pch=8)
trajectory(my_yprime, y0 = c(2, 1.0), t.end = t.end, parameters = diff_eq_params, col='red', pch=8)
trajectory(my_yprime, y0 = c(3, 1.0), t.end = t.end, parameters = diff_eq_params, col='green', pch=8)

if( save_plots ){ dev.off() }

par(mfrow=c(1, 1))