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Overflow problem implementing Euler's method

I tried to solve the following problem by the recursive method using the following code of the improved Euler's method, but it returns me a message ''Overflow occurred in computation''

$$X'=Z+(Y-\alpha)X$$ $$Y'=1-\beta Y-X^2$$ $$Z'=-X-\gamma Z$$ with initial conditions $(X(0),Y(0),Z(0)=(1,2,3)$.

Where

X: interest rate

Υ:investment demand

Z: price index

$\alpha$: savings, $\beta$: cost per investment, $\gamma$: the absolute value of the elasticity of demand

I tried to solve by the recursive method using the following code of the improved Euler's method

Q[a_, b_, h_, N_] := (u[0] = a; v[0] = b; 
  Do[{u[n + 1] = 
     u[n] + h*
       F[u[n] + (h/2)*F[u[n], v[n]], 
        v[n] + (h/2)*
          G[u[n], v[
            n]]],                                                     \

    v[n + 1] = 
     v[n] + h*
       G[u[n] + (h/2)*F[u[n], v[n]], v[n] + (h/2)*G[u[n], v[n]]]}, {n,
     0, N}])

POSTED BY: Athanasios Paraskevopoulos
15 Replies
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I can not do the comparison of two graphs like that
POSTED BY: Athanasios Paraskevopoulos

Amazing!!! Could I do the same for the

Plot[{X[t],Y[t],Z[t]},{t,0,200},PlotLegends->"Expressions"]
POSTED BY: Athanasios Paraskevopoulos

Here is another idea, using AnimationVideo. Same MakePlot function, but with an explicit PlotRange to keep that constant for any parameter choice:

MakePlot[{\[Alpha]_, \[Beta]_, \[Gamma]_}, {a_, b_, c_}, h_, 
  max_Integer] := Module[{u, if},
  f[{x_, y_, z_}] := {z + (y - \[Alpha])*x, 
    1 - \[Beta]*y - x^2, -x - \[Gamma]*z};
  u[0] = {a, b, c};
  Do[u[n + 1] = u[n] + h*f[u[n] + h/2*f[u[n]]], {n, 0, max}];
  if = Interpolation[Table[{n, u[n]}, {n, 0, max/h}]];
  ParametricPlot3D[if[t], {t, 0, 2000}, 
   PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}]
  ]

Then:

AnimationVideo[
 MakePlot[{0.9, 0.2, t}, {1, 1, 1}, 0.02, 1000],
 {t, 0.0, 2.0}]

Generates this video:

https://www.wolframcloud.com/obj/2d831a2f-4c28-4881-82bc-3bc95c88bbad
POSTED BY: Arnoud Buzing
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