# Plot f(x),f(y),alpha where f(x) and f(y) are the solutions of a two ODEs system depending on a parameter α [closed]

My question is in the title. For example, the system could be:

\begin{align*}\frac{dx}{dt}&=-y+x[\alpha-(x^2+y^2)]\\\frac{dy}{dt}&=x+y[\alpha-(x^2+y^2)]\end{align*}

And I want to plot this :

For example with {t, 0, 10} and {α, -10, 10, 2}.

I begin to try some experiments, but it seems to be too much difficult for the beginner I am. So, thank you for your educational help.

Pascal77

• My code is not available. Sorry. – Pascal77 Feb 24 at 12:52
• By $f(x)$ do you mean $x(t)$, or is there another function $f$? The image looks like you wish to plot $(x(t), y(t))$. – Michael E2 Feb 24 at 16:48

sol = ParametricNDSolve[{D[x[t], t] == -y[t] +
x[t] (α - (x[t]^2 + y[t]^2)),
D[y[t], t] == x[t] + y[t] (α - (x[t]^2 + y[t]^2)),
x[0] == a, y[0] == b}, {x, y}, {t, 0, 10}, {a, b, α}];
{x0, y0} = {1, 1};
ParametricPlot3D[
Table[{x[x0, y0, α][t], α, y[x0, y0, α][t]} /.
sol, {α, -10, 10, 2}], {t, 0, 10},
AxesLabel -> {x, α, y}, PlotStyle -> Red,
LabelStyle -> Directive[Blue, Bold]]


• Wonderfull! I tried on my own side with ParametricPlot3D but unsuccessfully. I was lost with the Table used in your solution. Thank you. – Pascal77 Feb 24 at 15:52
• I just modified 2 little things : I added : "PlotRange-> All" I replaced "LabelStyle -> Directive[Blue, Bold]" by "LabelStyle -> Directive[Blue, Large]" – Pascal77 Feb 24 at 16:04
• I just want to say that this help website is amazing! I snoop in on it very often, and I always discover some astounding solutions, very original ones, with a big touch of class. So congratulations to all specialists which, cvgmt, you belong. Many thanks! – Pascal77 Feb 24 at 16:28

Here's an approach that is a little non-precise mathematically but maybe more aesthetic:

sol = NDSolve[{
x'[t] == -y[t] + x[t] (α[t] - (x[t]^2 + y[t]^2)),
y'[t] == x[t] + y[t] (α[t] - (x[t]^2 + y[t]^2)),
α'[t] == -0.1,
x[-50] == 1, y[-50] == 1, α[-50] == 25}, {x, y, α}, {t, 0, 400}][[1]];
ParametricPlot3D[{α[t], x[t], y[t]} /. sol, {t, 0, 400}, AxesLabel -> {α, x, y}, ImageSize -> Large]


• Thanks for your answer. Indeed, it is very smooth. It is another approach with alpha considered as a variable, in fact. Interesting. – Pascal77 Feb 24 at 17:22