We derive an analytical relationship between the parameters of a square-wave forced, nonlinear, two-dimensional ordinary differential equation which determines conditions under which the Poincare map has a horseshoe. This provides an analytical test for chaos for this equation. In doing this we show that the Poincare map has a closed-form expression as a transformation of R-2 of the form FTFT, where F is a flip, i.e., a 180-degree rotation about the origin and T is a twist centered at (a, 0) for a > 0. We show that this derivation is quite general. We also show how to relate our results to ODEs with continuous periodic forcing (e.g., the sinusoidal-forced Duffing equation). Finally, we provide a conjecture as to a sufficient condition for chaos in square-wave forced, nonlinear ODEs.
