We show that systems with a first integral (i.e., a constant of motion) or a Lyapunov function can be written as “linear-gradient systems,” $\dot{x}=L\left(x\right)▽V\left(x\right)$, for an appropriate matrix function $L$, with a generalization to several integrals or Lyapunov functions. The discrete-time analog, $\Delta x/\Delta t=L\overline{▽}V$, where $\overline{▽}$ is a “discrete gradient,” preserves $V$ as an integral or Lyapunov function, respectively.

• Received 21 April 1998

DOI:https://doi.org/10.1103/PhysRevLett.81.2399