Orsdxy

I have been tried to do two things with the solutions from NDSolveValue

• plot the norm of the solutions of a differential equation system versus time.


• plot one component of the solutions of a differential equation system versus time.

but I have been having difficulty it setting up the right syntax to do so.

The problem seems to be (for plotting the norms of the solutions), that Mathematica takes the norm of all the solutions, or tries to find the norm of a function rather than the value of the function at a certain time.

I have created a minimum working example from the original code. The major change is that in the original code set is a random set of $n$ points. The examples are my best guess for the correct syntax for the problems listed above. For context I have included a 3D parametric plot which works as intended.

If you have any questions please don't be afraid to ask.

Minimum Example

(*Simulation Parameters*)

Clear[i, P, B]

Clear[f]

f[P_, B_] := 1/2 P + 10 B/(1 + B);

tmax = 20;

A = {{1/20, 1/4, 1/50}, {1/4, 1/26, 1/40}};

set = {{1.1, 11.2, 0.2}, {5.6, 4.3, 7.8}, {2.3, 3.4, 3.4}};



(*ODE System*)

ODEsys = {i'[t] == f[P[t], B[t]] - i[t],

   P'[t] == 

    P[t] (1 - A[[1, 1]] P[t] - A[[1, 2]] B[t] - A[[1, 3]] i[t]),

   B'[t] == B[t] (1 - A[[2, 2]] B[t] - A[[2, 3]] i[t])};



(* Simulation *)

With[{ttmax = tmax}, 

  sol = ParametricNDSolveValue[{ODEsys, {P[0] == init1, B[0] == init2,

       i[0] == init0}}, {P, B, i}, {t, 0, ttmax}, {init1, init2, 

     init0}]];



(* Plots I am having trouble with *)

(* Cannot plot the first component of multiple solutions. *)

ParametricPlot[{t, 

  Evaluate[Through /@ (sol[#[[1]], #[[2]], #[[3]]][t] & /@ set)][[All,

     1]]}, {t, 0, tmax}, PlotRange -> All]



(* Takes the norm of all solutions. Does not plot the norms of the 

three different solutions. *)

ParametricPlot[{t, 

  Norm[Evaluate[

    Through /@ (sol[#[[1]], #[[2]], #[[3]]][t] & /@ set)]]}, {t, 0, 

  tmax}, PlotRange -> All]



(* This plot works as attended. *)

trajectoriesPlot = 

 ParametricPlot3D[

  Evaluate[Through /@ (sol[#[[1]], #[[2]], #[[3]]][t] & /@ set)], {t, 

   0, tmax}, PlotRange -> All]

• plot the norm of the solutions of a differential equation system versus time.

Plot[Evaluate@(Norm[Through[sol[## & @@ #][t]]] & /@ set), {t, 0, tmax}, 

 PlotRange -> All, AspectRatio -> 1, ImageSize -> Large, 

 PlotLegends -> Placed[ToString /@ set, Top], 

 PlotLabel -> (Norm[{P[t], B[t], i[t]}])]

• plot one component of the solutions of a differential equation system versus time.

Row[ParametricPlot[Evaluate@Thread[{t, (Through[sol[## & @@ #][t]] & /@ set)[[All, #]]}], 

    {t, 0, tmax}, PlotRange -> All, 

    PlotLegends -> Placed[set[[All, #]], Top] , AspectRatio -> 1, 

    ImageSize -> 300, PlotLabel -> ({P[t], B[t], i[t]}[[#]])] & /@ {1,

    2, 3}, Spacer[5]]

Alternatively, you can use Plot:

Row[Plot[Evaluate@(Through[sol[## & @@ #][t]] & /@ set)[[All, #]], {t, 0, tmax}, 

    PlotRange -> All, AspectRatio -> 1, ImageSize -> 300, 

    PlotLegends -> Placed[set[[All, #]], Top] , 

    PlotLabel -> ({P[t], B[t], i[t]}[[#]])] & /@ {1, 2, 3}, Spacer[5]]

sol = ParametricNDSolveValue[{ODEsys, {P[0] == init1, B[0] == init2,i[0] == init0}}, {P[t], B[t], i[t]}, {t, 0, tmax}, {init1, init2,init0}]

plot of solutions:

Plot[Table[# &[ Apply[sol, set[[i]]]]  , {i, 1, Length[set]}] , {t, 0,tmax}, PlotRange -> {0, Automatic}]

plot of euclidean norm:

Plot[Table[Sqrt[ #.#] &[ Apply[sol, set[[i]]] ], {i, 1, Length[set]}] , {t, 0,tmax}, PlotRange -> {0, Automatic}]