LaSalle_principle_example.png
Summary
Description LaSalle principle example.png |
English:
A vector field
in the plane. The function
satisfies
, which satisfies the LaSalle's invariance principle, showing that the origin is asymptotically stable.
Plotted in Julia by the following code. ```julia using CairoMakie, LinearAlgebra
df(x, y) = [-y - x^3; x^5] lyapf(x, y) = x^6 + 3y^2 using Interact
scene = Figure(resolution = (1600, 3200)) Axis(scene[1,1], backgroundcolor = "black")
xmin, xmax, xres = -1, 1, 41 ymin, ymax, yres = -1, 1, 41 x = range(xmin, stop = xmax, length = xres) y = range(ymin, stop = ymax, length = yres) xs = repeat(x, outer=length(y)) ys = repeat(y, inner=length(x))
vectors = df.(xs, ys) us = map((x) -> x[1], vectors) vs = map((x) -> x[2], vectors) us /= 5 vs /= 5 n = vec(norm.(vectors)) n /= maximum(n) / 10 arrows!(xs, ys, us, vs, arrowsize = n, linecolor=n, arrowcolor = :white) Axis(scene[2,1], backgroundcolor = "black")
arrows!(xs, ys, us, vs, arrowsize = n, linecolor=n, arrowcolor = :white)
zs = lyapf.(xs, ys) Makie.contour!(xs, ys, zs, levels = 30, linewidth = 2, colormap = :grayC)
save("LaSalle.png", scene) ``` |
Date | |
Source | Own work |
Author | Cosmia Nebula |
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