Divergence is usually used to determine the dissipation of a dynamical system, but some researchers have noticed that it can lead to elusive contradictions. In this article, a criterion, dissipative power, beyond divergence for judging the dissipation of a system is presented, which is based on the knowledge of classical mechanics and a novel dynamic structure by Ao. Moreover, the relationship between the dissipative power and potential function (or called Lyapunov function) is derived, which reveals a very interesting, important, and apparently new feature in dynamical systems: to classify dynamics into dissipative or conservative according to the change of “energy function” or “Hamiltonian,” not according to the change of phase space volume. We start with two simple examples corresponding to two types of attractors in planar dynamical systems: fixed points and limit cycles. In judging the dissipation by divergence, these two systems have both the elusive contradictions pointed by researchers and new ones noticed by us. Then, we analyze and compare these two criteria in these two examples, further consider the planar linear systems with the coefficient matrices being the four types of Jordan’s normal form, and find that the dissipative power works when divergence exhibits contradiction. Moreover, we also consider another nonlinear system to analyze and compare these two criteria. Finally, the obtained relationship between the dissipative power and the Lyapunov function provides a reasonable way to explain why some researchers think that the Lyapunov function does not coexist with the limit cycle. Those results may provide a deeper understanding of the dissipation of dynamical systems.
Reducibility by Moments of Linear Impulse Markov Dynamical Systems with Almost Constant Coefficients
