sentences:  Finding the solution curves leaving a
  singular point is called branch switching, and uses techniques from bifurcation theory (singularity theory, catastrophe theory).For finite-dimensional systems (as defined above) the Lyapunov-Schmidt decomposition may be used to produce two systems to which the Implicit Function Theorem applies.The Lyapunov-Schmidt decomposition uses the restriction of the system to the complement of the null space of the Jacobian and the range of the Jacobian.