psol = ParametricNDSolve[{y''[t]+y[t]==3a Sin[y[t]],y[0]==y'[0]==1,α'[t]==Sqrt[1+y[t]^2],α[0]==0},{y,α},{t,0,10},{a}];

GraphicsRow[{Plot[Evaluate@Table[y[a][t] /. psol,{a,0,1,.1}],{t,0,10},ImageSize->Small],Plot[Evaluate[α[a][10] /. psol],{a,0,1},MaxRecursion->0,PlotPoints->100,AxesLabel->{a,α},ImageSize->Small]}]

lmin = FindMinimum[α[a][10] /. psol, {a, 0.72},AccuracyGoal->7]

Show[{Plot[Evaluate@Table[y[a][t] /. psol,{a,.65,.8,.005}],{t,0,10},PlotStyle->Lighter[Gray,0.5]],Plot[Evaluate[y[a][t]/.psol/.lmin[[2]]],{t,0,10},PlotStyle->Thick]},ImageSize->Medium]