# Plot solution to damped driven pendulum in $(-\pi,\pi)$

I'm solving the damped driven pendulum:

{γ, g, ω} = {0.5, 1.5, 2/3};
{X0, V0} = {0.6184, 0.};
tmax = 1000;
sol = NDSolve[{x''[t] + γ x'[t] + Sin[x[t]] == g Cos[ω t], x == X0, x' == V0}, x, {t, 0, tmax}, MaxSteps -> Infinity]
sol1[t_] := x[t] /. sol[]
sol2[t_] := x'[t] /. sol[]


I then want to plot the solution $$(x,x')$$ constraining $$x$$ to the interval $$(-\pi,\pi)$$. I came with modulo to make one part $$(-2\pi,0)$$, the other $$(0,2\pi)$$, and then restrict the range to $$(-\pi,\pi)$$. First, it is inelegant. Second, I end with a gap in the middle that even with a big number of PlotPoints I cannot get rid of.

ParametricPlot[{{Mod[sol1[t], -2 π], sol2[t]}, {Mod[sol1[t], 2 π], sol2[t]}}, {t, 0, tmax},
Frame -> True, Axes -> False, PlotStyle -> Black, PlotRange -> {{-π, π}, {-3, 3}}, PlotPoints -> 500] Using an alternative formulation for modulo:

normalize[angle_?(NumericQ[#] && Im[#] == 0 &)] := angle - 2 Pi Floor[(angle + Pi)/(2 Pi)]


to plot

ParametricPlot[{normalize@sol1[t], sol2[t]}, {t, 0, tmax},
Frame -> True, Axes -> False, PlotStyle -> Black, PlotRange -> {{-π, π}, {-3, 3}}, PlotPoints -> 500]


I get redundant horizontal lines: How to make such plot elegantly?

Perhaps it's easier to use NDSolveValue and Mod[...,2 Pi,-Pi] with offset:
{γ, g, ω} = {0.5, 1.5, 2/3}; 