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On the construction of Lyapunov functions with computer assistance

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.