If you are satisfied with a numerical approach, this is one of the problems where ParametricNDSolve[] is called for:

pf = ParametricNDSolveValue[{x'[t] == Sin[t] - x[t]^3, x[0] == a}, x, {t, 0, 2 π}, a,
                            Method -> "StiffnessSwitching", WorkingPrecision -> 25];

(* bracket obtained from a preliminary plot of pf[a][2 π] *)
aopt = a /. FindRoot[pf[a][2 π] - a, {a, -1, 1}, WorkingPrecision -> 25]
   -0.7156846197712976536871740

xs = NDSolveValue[{x'[t] == Sin[t] - x[t]^3, x[0] == aopt}, x, {t, -2 π, 4 π}];

Plot[xs[t], {t, -2 π, 4 π}]

xs[π Range[-2, 4]]
   {-0.7156821841853735, 0.7156848515290952, -0.7156846197712976,
    0.7156846139367274, -0.7156846712483197, 0.7156846219921508,
    -0.7156846049890743}