MATLAB: Plotting lissajous (phase figure) gives wrong results

diff()differential equationssecond order differential equation

Plotting lissajous (phase figure) I obtained the plot in photo 1, but I shoould have the plot of figure 2. Could someone please help me figure out the mistake?
   c = 1;
   b = 0.1;
   a = 0.45;
   d = 1; %-1
   w=1.3;
   %obtaining the solution to second order differential equation x'' + bx' − x + x^3 = a sin (wt). this is duffing holmes equation
   %the equation was converted to a system of 2 equations y=x' and y' = d*x - c*x^3 − by + a sin(wt)
   %this link was used to solve 2 order ODE https://www.mathworks.com/help/symbolic/solve-differential-equation-numerically-1.html
   syms y(t)
   [V] = odeToVectorField(diff(y, 2) == d*y - c*y^3-b*diff(y) + a*sin(w*t));
   M = matlabFunction(V,'vars', {'t','Y'});
   sol = ode45(M,[0 100],[2 2]);
   figure (1)
   fplot(@(x)deval(sol,x, 1), [0 100]);
   title('position vs time');
   xlabel('time t');
   ylabel ('position x');
   
   %differentiating with respect to time and plotting velocity vs position
   dY = diff(sol.y);
   dYY=dY(:,1:end-1);
   dX = diff(sol.x);
   der = dYY./dX;
   disp = sol.y;
   disp2 = disp(:,1:end-1);
   figure (2)
   plot(disp2, der, 'r-');
   title('Lissajous Figure');
   xlabel('position');
   ylabel ('velocity');

Best Answer

Use a slightly different ode45 call, and then plot the columns of ‘y’ against each other: 
[t,y] = ode45(M,[0 100],[2 2]);
figure
plot(y(:,1), y(:,2))
grid
axis('equal')
title('Lissajous Figure');
xlabel('position');
ylabel ('velocity');
producing:
Other that that, your code is correct, so no other changes are necessary.
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