how to find the time taken by a parametric trajectory to reach a point

Questions
1

I can solve this system of parametric ODE (`u` is the parameter,while `g`,`e` and `zi` are constants depends on the physic of the problem)

    zi = -1/2;
    g = 1;
    e = .2;
    Y[t_] :=  g/(2 zf) y[t] + u z[t];
    Z[t_] := -g + g/zf z[t] - u y[t];
    sol =
      FullSimplify[
        DSolve[{y'[t] == Y[t], z'[t] == Z[t],y[0] == 0 , z[0] == -zf},
          {y[t], z[t]}, t]]

Then, I make the assignement:

   zsol[t_] := Evaluate[z[t] /. sol[[1]][[2]]]
    ysol[t_] := Evaluate[y[t] /. sol[[1]][[1]]]

My problem is to compare the time `t(u)` taken by trajectories `{y[t,u],z[t,u]}` to reach a neighborhood of `zf` whit radius `e`, with the time `trelax`, relative to `u=0` (`zf` is the fixed point of free dynamic). Not for all `u` the problem has solution: in general, for `u` large enough, trajectories reaches some fixed point in an infinity time following a spiral (as I seen plotting the system for various `u`), cause of the exponential-complex solutions, and there’s a set of `u` without solutions (i.e, a set of `u` for which trajectories pass no close to `zf`). Then I have thought to use an iterative costruct, a `do-loop` mixed with an `If`. For simplicity, I’ve started, in particular, with:

   trelax = Log[(-e/(2 zf))]*zf/g;
    Do[Do[
       If[N[Abs[zsol[T]] <= (zf + e) && N[Abs[ysol[T]] <= e,
         If[Nsolve[ysol[t], t] <= trelax],
           Print[{T, u}]]], {T, 0, .9, .01}], {u, 0, 40, .2}]

Remark_1: `trelax` is the correct analytic result of dynamical system for `u=0`.
Remark_2: I expect from this code to obtain a list of couple `{t,u}`, with `t<=trelax` and `u>0`, but really I don’t obtain nothing! I think I’m wrong in the definition of the variable in the function…and in many other things!

Finally I ask you if is it possible to change the loop to obtain directly a plot `{t,u}`, i.e. a curve to compare with constant `trelax`…

Thank you

P.S: If you want to see some plots, I attach the ad hoc code:

  u = 5;
    ParametricPlot[{y[t] /. sol[[1]][[1]], z[t] /. sol[[1]][[2]]}, {t, 0, 10}, PlotRange-> All]
2

Without solving the problem of finding the condition where the trajectory lies within a disk. This modification of your code may help you construct a numerical technique.

sol = DSolve[{y'[t] == g/(2 zf) y[t] + u z[t], 
   z'[t] == -g + g/zf z[t] - u y[t], y[0] == 0, z[0] == -zf}, {y[t], 
   z[t]}, t]

This is your original equation, but without putting in the defintions for Z, etc.

Extract the solutions and simplify.

{{ysol, zsol}} = 
 Simplify[{y[t], z[t]} /. sol, 
  Assumptions -> Element[u, Reals] && Element[zf, Reals] && t > 0 ]

Explore the solutions…

With[{y = ysol, z = zsol},
 Manipulate[ParametricPlot[{y, z}, {t, 0, tend}],
  {{tend, 1}, 0, 100},
  {{g, 1}, 0, 10},
  {{zf, 1}, 0, 10},
  {{u, 1}, 0, 10}
  ]
 ]