What is the most suitable Lyapunov function?

• Algorithm for Hamilton energy function in dynamical system is clarified. • Hamilton energy function is the most suitable Lyapunov function for dynamical control. • Control of energy flow is the most effective way to synchronization control of chaotic systems. Lyapunov function provides feasible estimation and prediction of nonlinear system stability, and useful guidance for adaptive control in chaos and synchronization approach. In case of synchronization and control of chaotic systems, the involvement of adjustable gains in the Lyapunov function can be effective to optimize the convergence of orbits to stability and controllers within finite transient period. As a result, shorter transient period and lower power consumption can be approached by detecting the most suitable gains in the controllers and parameter observers. In this paper, we claim that the most suitable Lyapunov function can be the Hamilton energy for chaotic systems and more nonlinear dynamical systems, and so the parameter region for stability and controllability can be detected exactly, in addition, the reliability of controllers can be confirmed in practical way. Furthermore, the Lorenz and improved Chua oscillators in chaotic states are presented to confirm the dependence of Hamilton energy and stability on the intrinsic parameters and variables. It indicates that control of energy flow can be an effective scheme to control chaos in nonlinear systems and synchronization realization between chaotic systems, neurons and networks.