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some quick duffing attractor poincare sections:

anim1 - n 5,000,000; h 0.01;
anim2 -
anim3 - n 10,000,000; h 0.1;

can see that there is good definition for anim1 and anim2 (same parameters but different animation parameters (so ignore phi value for this)). but for smaller step size been run too long, the definition is lost as point of intersection with poincare surface drifts from "true" intersection point, because trajectory through phase space has drifted.

anim4 - n 50,000,000; h0.001;
anim5- n 50,000,000; h0.0005;

the definition on the curves is lower for anim4 than anim5. this is due to a halved step size, keeping the trajectory closer to the "true" value for longer.