if( !require('phaseR') ){ install.packages('phaseR') } library(phaseR) my_jacobian = function(x, y) { # returns the Jacobian at the point (x, y) # J = matrix(c(1 - 3*x^2 - y^2, 1 - 2*x*y, -1 - 2*x*y, 1 - x^2 - 3*y^2), nrow=2, ncol=2, byrow=TRUE) } # The critical points of this system: CP = data.frame(x=c(0), y=c(0)) 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] = yt + xt*(1 - xt^2 - yt^2) dy[2] = -xt + yt*(1 - xt^2 - yt^2) list(dy) } diff_eq_params = list() #postscript('../../WriteUp/Graphics/Chapter9/chap_9_sect_3_prob_16_plot.eps', onefile=FALSE, horizontal=FALSE) t.end = 5.0 L = 4 flowField(my_yprime, x.lim = c(-L, +L), y.lim = c(-L, +L), parameters = diff_eq_params, points = 21, add = FALSE, xlab='x', ylab='y') # Plot some trajectories with initial condition near the critical points: # trajectory(my_yprime, y0 = c(-1.5, 1.5), t.end = t.end, parameters = diff_eq_params, col='black', pch=19) trajectory(my_yprime, y0 = c(0.1, -0.1), t.end = t.end, parameters = diff_eq_params, col='blue', pch=19) trajectory(my_yprime, y0 = c(0.1, 1.25), t.end = t.end, parameters = diff_eq_params, col='red', pch=19) trajectory(my_yprime, y0 = c(0.9, 0.1), t.end = t.end, parameters = diff_eq_params, col='green', pch=19) #dev.off()