Is there an analytic way to tell if a system of ordinary differential equations is conservative?

Question

I don’t know the exact definition of conservative, but we can assume that conservative means the sum of Lyapunov exponents is zero. Is there an analytic method to show that a given system is conservative? I came to this question by reading Sprott’s Elegant Chaos. A system of interest could be e.g.:

$$U''' + U' + \frac{1}{3}U^3 - Uc=0$$

where $c$ is constant, $U(t)$ is a function of time. It can be described as system of three equations:

$$\begin{align} U' &= X \\ X' &= Y \\ Y' &= -X - \frac{1}{3} U^3 +Uc \end{align}$$

Then I think (but cannot prove it) that the trace of the Jacobian $\operatorname{Tr}(J)$ shows the conservativeness.

$$\operatorname{Tr}(J) = \frac{\partial U'}{\partial U} + \frac{\partial X'}{\partial X} + \frac{\partial Y'}{\partial Y} = 0 + 0 + 0 = 0$$

It also means that if $\operatorname{Tr}(J) < 0$, the system is dissipative, and if $\operatorname{Tr}(J) > 0$, the system diverges, i.e., it is not bounded. Can you say if I am right or wrong?

Answer 1

Your hunches are absolutely right. In dynamical-systems theory, systems are often classified via their divergence (i.e., the trace of the Jacobian). In this context, if your dynamics is $\dot{x} = f(x)$, then $\operatorname{div} f = 0$ defines a conservative system; and $\operatorname{div} f < 0$ defines a dissipative one.

The only difficulty is that divergence may be inhomogeneous and thus you have to look at the average over a trajectory, which brings us to the sum of Lyapunov exponents. However, conservative usually implies $\operatorname{div} f = 0$ everywhere. A system with $\operatorname{div} f = 0$ on average would have to be carefully constructed.

This is linked with conservativity in the sense of energy conservation via one of Liouville’s Theorems and Noether’s Theorem: $\operatorname{div} f = 0$ implies a conserved phase-space volume, which imples symmetry with respect to the direction of time, which in turn implies conservation of energy (or a similar quantity). On the other hand, for Hamiltonian systems (which are energy-conserving), you have $\operatorname{div} f = 0$.

if $\operatorname{Tr}(J) > 0$, the system diverges, i.e., it is not bounded.

Exactly. Your phase-space volume of an arbitrary set of initial conditions would always expand and thus you would the dynamics would have to be unbounded.