# Plot solutions to a differential sys with initial conditions from loop?

Posted 2 months ago
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 I have been working with potentially chaotic systems and it would be extremely helpful if I could plot many solutions to the equation starting from different initial conditions on the same graph. What I tried to do was create a loop that picks a starting value for each equation on the system and plots a new solution as the loop iterates, but my basic knowledge of this program failed to produce anything that worked: a = .1 M = .2 J = .1 For[i = 0, i < M, i += J, For[u = 0, u < M, u += J, For[o = 0; o < M , o += J, {x[][t], y[][t], z[][t]} /. sol2 sol2 = ParametricNDSolve[{x'[t] == (Sin[y[t]] - b*x[t])* Cos[t]*(-Sin[t]* Cos[ArcTan[a*t]] + (a*Cos[t]*(-Sin[ArcTan[a*t]]))/(1 + a*t^2)), y'[t] == (Sin[x[t]] - b*z[t])* Sin[t]* (Cos[t]* Cos[ArcTan[a*t]] + (a*Sin[t]*(-Sin[ArcTan[a*t]]))/(1 + a*t^2) ), z'[ t] == (Sin[z[t]] - b*y[t])*((-Cos[ArcTan[a*t]]*a)/(1 + a*t^2)), x[0] == i, y[0] == u, z[0] == o}, {x, y, z}, {t, -100, 100}, {a}] ParametricPlot3D[ Evaluate[{x[.1][t], y[.1][t], z[.1][t]} /. sol2], {t, -100, 100}, PlotRange -> All, BoxRatios -> {1, 1, 1}, PlotStyle -> Tube[.0005]]]]] Obviously, my coding is by no means elegant let alone functional but the idea was that M and J would determine how many plots the system would run. Basically, what I have is a mess and was hoping someone could help me out.
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Posted 2 months ago
 The plot is supposed to show many different 3D solutions to the differential equation, so I figure your solution would work if it instead used ParametricPlot3D
 Michael,ParametricNDSolve already does much of what you want. You first need to clear out values you previously set Clear[a, b, i, u, o] Solve your problem: sol = ParametricNDSolve[{x'[t] == (Sin[y[t]] - b*x[t])* Cos[t]*(-Sin[t]* Cos[ArcTan[a*t]] + (a*Cos[t]*(-Sin[ArcTan[a*t]]))/(1 + a*t^2)), y'[t] == (Sin[x[t]] - b*z[t])* Sin[t]*(Cos[t]* Cos[ArcTan[a*t]] + (a*Sin[t]*(-Sin[ArcTan[a*t]]))/(1 + a*t^2)), z'[t] == (Sin[z[t]] - b*y[t])*((-Cos[ArcTan[a*t]]*a)/(1 + a*t^2)), x[0] == i, y[0] == u, z[0] == o}, {x, y, z}, {t, -100, 100}, {a, b, i, u, o}] I chose a,b,i,u,o as parameters, you should probably narrow that list. You can now plot results. I do not understand what you want to show exactly so I made up a plot: Plot[Evaluate@ Table[Tooltip[y[.1, 0.2, 0, 0, s][t] /. sol, s], {s, 0.3, 1.9, 0.1}], {t, 0, 10}] The mouse will give you the parameter value. You can forego the tooltip and make a grid of plots or any other format.Regards,Neil