If statement within WhenEvent

Question

I'm trying to integrate an equation with a periodic solution using NDsolve. I want to stop the integration after the derivative of my solution has become zero for the $n$-th time (in my example code, $n=5$). For this purpose I included a "counter" variable $i$ in WhenEvent. In principle everything works fine, except that here "StopIntegration" is not recognized. I guess it has to do something with the fact that "StopIntegration" is wrapped within an If statement. However, in principle "StopIntegration" should be read by Mathematica (as Print["Integration stopped at t=", tend] is) and NDSolve should stop? Below is a simple minimal working example of my Problem.

Module[{i = 0}, 
First@NDSolve[{D[x[t], t] == 2 π y[t], 
D[y[t], t] == -2 π x[t], x[0] == 0, y[0] == 1, 
WhenEvent[y[t] == 0, 
 If[i >= 4, {tend = t, "StopIntegration", 
   Print["Integration stopped at t=", tend]}, i += 1]], 
WhenEvent[t == 10, tend = t]}, {x, y}, {t, 0, 10}, 
Method -> "LSODA"]]

Any help would be highly appreciated.

Answer 1

The string "StopIntegration" needs to be the result of the If evaluation (You are returning it as part of a list. This seems to work:

sol=Module[{i = 0}, 
     First@NDSolve[{D[x[t], t] == 2 \[Pi] y[t], 
          D[y[t], t] == -2 \[Pi] x[t], x[0] == 0, y[0] == 1, 
        WhenEvent[y[t] == 0, 
           If[i >= 4, tend = t; Print["Integration stopped at t=", tend]; 
            "StopIntegration", i += 1]], WhenEvent[t == 10, tend = t]},
                  {x, y}, {t, 0, 10}, Method -> "LSODA"]]

Aside, The terminating time is captured in the "Domain" of the solution, so you don't need to capture tend like that.

(x /. sol)["Domain"]

{{0., 2.25}}

Answer 2

I believe this will also work. Use i as a DiscreteVariable:

sols = First@NDSolve[
  {D[x[t], t] == 2 π y[t], D[y[t], t] == -2 π x[t]
   , x[0] == 0, y[0] == 1
   , i[0] == 0
   , WhenEvent[y[t] == 0, i[t] -> i[t] + 1]
   , WhenEvent[i[t] == 4, {"StopIntegration", Print[tend = t]}]
   , WhenEvent[t == 10, tend = t]
  }
  , {x, y, i}
  , {t, 0, 10}
  , Method -> "LSODA"
  , DiscreteVariables -> {i}
 ]
 Plot[{i[t], y[t]} /. % // Evaluate, {t, 0, tend}]

Update

george2079 points out that the WhenEvents can be rolled into one via

WhenEvent[y[t] == 0, If[i[t] < 4, i[t] -> i[t] + 1, Print[t]; "StopIntegration"]]

which is cleaner. Then, as he pointed out in his solution, one can use "Domain" to extract tend.

Answer 3

I would rather do it in vector form:

i = 0;
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
sols = NDSolveValue[{s'[t] == 2 Pi RotationMatrix[-Pi/2].s[t], s@0 == {0, 1}, 
                    WhenEvent[Last@s[t] == 0, If[++i > 4, "StopIntegration"]]}, 
                    s, {t, 0, 10}]

Plot[sols[t], {t, 0, #}, PlotLabel -> "Integrated up to " <> ToString@#] &@
    (InterpolatingFunctionDomain@ sols // Flatten // Last)