Least Squares and Chaotic Behavior in Initial Value Problems

We describe an unconventional method for treating initial value problems. The system of differential equations is discretized on a fixed interval, the initial value is left unspecified, and the underdetermined system is solved by a least squares minimization procedure. When applied to the celebrated Lorenz equations, it produces a simple smooth curve that terminates at a stationary point rather than producing the ‘butterfly’ attractor. As far as we know this is the first method that produces a non-chaotic orbit.