Plot on fundamental square of a torus [duplicate]

I have line on the 2d plane, given by say f(t). I want to plot it in the torus by plotting Mod[f(t),2\[Pi]]. However, there are some sharp horizontal and vertical lines that I do not want. Is there a way to do this?

I get the line by solving a complicated ODE, thus I give below a trial code that reproduces the effect that I do not want.

s = NDSolve[{Derivative[1][x][t] == -3 (x[t] - y[t]), Derivative[1][y][t] == -x[t] z[t] + 26.5 x[t] - y[t], Derivative[1][z][t] == x[t] y[t] - z[t], x[0] == z[0] == 0, y[0] == 1}, {x, y, z}, {t, 0, 1}, MaxSteps -> \[Infinity]];
ParametricPlot[Evaluate[Mod[{x[t], y[t]}, 3] /. s], {t, 0, 1}]


In the worst case I can take a discrete sample and do a list plot, but is there any elegant solution?

s = NDSolve[{Derivative[1][x][t] == -3 (x[t] - y[t]), Derivative[1][y][t] == -x[t] z[t] + 26.5 x[t] - y[t], Derivative[1][z][t] == x[t] y[t] - z[t], x[0] == z[0] == 0, y[0] == 1}, {x, y, z}, {t, 0, 1}, MaxSteps -> \[Infinity]];