  How do we search for the entire family tree without unwanted random guesses,
starting from a high-index and high-energy stationary state on the energy
landscape? Here we introduce a general numerical method that constructs the
pathway map clearly connecting all stationary points branched from a single
parent state. The map guides our understanding of how a physical system moves
on the energy landscape. In particular, the method allows us to identify the
transition state between energy minima and the energy barrier associated with
such a state. As an example, we solve the Landau-de Gennes energy incorporating
the Dirichlet boundary conditions to model a liquid crystal confined in square
box; we illustrate the basic concepts by examining the multiple stationary
solutions and the connected pathway maps of the model.
