Two paths to finding the pitchfork bifurcation in motivation dynamics

We perform two different model reduction techniques on a dynamical system recently used to develop mechanisms for switching between low-level control vector fields. Both techniques result in the same reduced model which is shown to be a one-parameter unfolding of the supercritical pitchfork bifurcation. The first technique uses singular perturbation to reduce the original two-dimensional system to a one-dimensional system whose vector field is a rational polynomial. The second technique uses the Lyapunov-Schmidt reduction to find another one-dimensional system. In a singular limit analogous to that of the first technique, the Lyapunov-Schmidt reduced dynamics are identical to the rational polynomial found directly through singular perturbation. A nonlinear time scaling argument then shows that the rational polynomial is equivalent to a normal form for the unfolding of the pitchfork.