DynamicModule[{αpr, Tmax, μpr},
Manipulate[

Module[{u, t, ϕ},
sol = NDSolve[
{
u''[t] - (1 - μpr) u[t] ((ϕ'[t])^2)+ μpr == 0, u[t] ϕ''[t] + 2 u'[t] ϕ'[t] == 0, 
u[0] == 1, 
u'[0] == 0,
ϕ[0] ==0, 
ϕ'[0] == αpr
}, 
{u, ϕ}, 
{t, 0, tmax = 100}
];
];
usol = sol[[1]][[1]][[2]];
ϕsol = sol[[1]][[2]][[2]];
ParametricPlot[usol[t] {Cos[ϕsol[t]], Sin[ϕsol[t]]}, {t, 0, Tmax}], 
{{αpr, 0.429}, 0.001, 10, 0.001, Appearance->"Open"},
{{Tmax, 21.511}, 0.001, tmax, 0.01, Appearance->"Open"},
{{μpr, 0.5}, 0.001, 1, 0.001, Appearance->"Open"},
TrackedSymbols:> {αpr, Tmax, μpr}
]
]