In a 1989 paper, E. Lorenz studied the application of Euler's method to a certain 2-dimensional system of ODEs. As the time step was increased, the corresponding map progressed from exhibiting an attracting fixed point to an invariant circle to "full chaos." Of special interest is the parameter range including the breakup of the invariant circle and the first appearance of sensitivity to initial conditions. The "invariant circle" develops bumps, then cusps, then loops. A follow-up numerical study [Frouzakis, Kevrekidis, P 2003] revealed some details of this transition, including the interaction of the invariant circle with stable and unstable manifolds of saddles for periodic points. Additional investigation is still being performed. This talk will discuss computation of Arnold tongues and (if this is a good week) the paths corresponding to invariant circles with fixed irrational rotation numbers that lie "in between" the tongues. Techniques follow [Schilder and P, 2007].