mjhjmtu

I'm having a bit of trouble making a ParametricPlot from two curves, At fisrt I was having difficulty solving my systems of autonomous ODEs, but then finally got a way to solve it. However, I am now not able to figure out how to get the parametric plot.

solution[t_] = 
 With[n = 1.5, 
 NDSolve[
 q'[t] == 
 ((q[t]^2)/3)*(3 - n)*x[t]^2 - ((q[t]^2)/3)*n*(y[t] - 1) - ((q[t]^2)/3) + (q[t]*x[t]/3)*(3 - n)*x[t]^2 - (q[t]*x[t]/3)*n*(y[t] - 1) + (q[t]*x[t]/3) + (1/3)*x[t]^2 - (1/3) + (1/3)*((n*y[t]/(n - 1))), 
 y'[t] == 
 ((2*y[t]*x[t]^2)/3)*(3 - n)*(x[t] + q[t]) + (2*x[t]*y[t]/3)*((n^2 - 2*n + 2)/(n - 1)) - (2*x[t]*y[t]/3)*n*y[t] + (2*q[t]*n*y[t]*((1 - y[t])/3)), 
 x'[t] == 
 ((x[t]^3)/3)*(3 - n)*(q[t] + x[t]) + ((x[t]^2)/3)*n*(2 - y[t]) - 5*((x[t]^2)/3) + (q[t]*x[t]/3)* n*(1 - y[t]) - (q[t]*x[t]) + (1/3)*((n*(n - 2)*y[t])/(n - 1)) - (1/3)*n + 2/3, 
 q[0] == -1.33, y[0] == 0.88, x[0] == 0.33, 
 q[t], y[t], x[t], t, 0, 10]][[1, All, 2]]

Plot[solution[t], t, 0, 10]

I want to get a parametric plot of q[t] and y[t]. And eventually a 3D plot of q[t], y[t], x[t].

The issue you are facing is to correctly call out the desired dependent variables. So, you need to specify the position of the output. For example, you want to have q[x] then you need to use solution[t][[1]].

For parametric plot, try this out,

ParametricPlot[solution[t][[1]], solution[t][[2]], t, 0, 10]

and then for 3D

ParametricPlot3D[solution[t][[1]], solution[t][[2]], solution[t][[3]], t, 0, 10]

Redefine your solution a little bit to

solution := 
With[n = 1.5, 
NDSolve[q'[t] == ((q[t]^2)/3)*(3 - n)*x[t]^2 - ((q[t]^2)/3)*
 n*(y[t] - 1) - ((q[t]^2)/3) + (q[t]*x[t]/3)*(3 - n)*
 x[t]^2 - (q[t]*x[t]/3)*n*(y[t] - 1) + (q[t]*x[t]/3) + (1/3)*
 x[t]^2 - (1/3) + (1/3)*((n*y[t]/(n - 1))), 
y'[t] == ((2*y[t]*x[t]^2)/3)*(3 - n)*(x[t] + q[t]) + (2*x[t]*
 y[t]/3)*((n^2 - 2*n + 2)/(n - 1)) - (2*x[t]*y[t]/3)*n*
 y[t] + (2*q[t]*n*y[t]*((1 - y[t])/3)), 
x'[t] == ((x[t]^3)/3)*(3 - n)*(q[t] + x[t]) + ((x[t]^2)/3)*
 n*(2 - y[t]) - 
 5*((x[t]^2)/3) + (q[t]*x[t]/3)*
 n*(1 - y[t]) - (q[t]*
 x[t]) + (1/3)*((n*(n - 2)*y[t])/(n - 1)) - (1/3)*n + 2/3, 
q[0] == -1.33, y[0] == 0.88, x[0] == 0.33, q , y , x , t, 0, 
10]] [[1]]

Now you can plot your results

Plot[Evaluate[ q[t], y[t] /. solution ], t, 0, 10]