  Computer assisted procedures of Lyapunov functions defined in given
neighborhoods of fixed points for flows and maps are discussed. We provide a
systematic methodology for constructing explicit ranges where quadratic
Lyapunov functions exist in two stages; negative definiteness of associating
matrices and direct approach. We note that the former is equivalent to the
procedure of cones describing enclosures of the stable and the unstable
manifolds of invariant sets, which gives us flexible discussions of asymptotic
behavior not only around equilibria for flows but also fixed points for maps.
Additionally, our procedure admits a re-parameterization of trajectories in
terms of values of Lyapunov functions. Several verification examples are shown
for discussions of applicability.
