Are chaotic systems the same as dissipative systems in inverse time?

Question

Lyapunov exponents define whether a system expands or contracts in phase space and can be used to determine whether a dynamical system is chaotic, conservative, or dissipative. If the volume expands in at least one dimension, the system is chaotic. If the volume contracts in all dimensions, it is dissipative. However, if we were to invert time, a contraction in phase space becomes an expansion, and vice versa. Would it be a valid intuition to think of chaotic systems as dissipative systems in inverse time, or conversely of dissipative systems as chaotic systems in inverse time? Or is there a nuance I am missing?

Answer 1

Yes, you are missing something. Looking at the change of phase-space volume ($∇·f$), you get three categories – if you have a constant sign of $∇·f$ (more on the alternative at the end):

• dissipative ($∇·f<0$): shrinking phase-space volume; allows for attractors, including chaotic ones (e.g., Lorenz system);

• conservative ($∇·f=0$): invariant phase-space volume; time-invertible; allows for chaos (e.g., double pendulum), but not for attractors;

• unstable/explosive ($∇·f>0$): expanding phase-space volume; unbounded dynamics; no chaos, attractors, or similar (as those are bounded dynamics per definition).

If you invert time on a dissipative system, you get an unstable one, which cannot be chaotic. If you invert time on a conservative system, you get another conservative system. If you invert time on a chaotic system, the outcome depends on whether that system is dissipative or conservative: A dissipative system will turn into an unstable one; a conservative system will remain a conservative chaotic system, e.g., the double pendulum is chaotic and clearly time-invertible.

Now, how do expansions in the individual direction (Lyapunov exponents) for chaos fit into this? A three-dimensional chaotic system has three Lyapunov exponents: $λ_1>0;$ $λ_2=0$ (the direction of time) and $λ_3<0$ (otherwise you couldn’t get a bounded dynamics). You need to have $λ_1 < -λ_3$ to get a dissipative system and that property does not hold on time inversion (which flips the signs of the Lyapunov exponents). In case of $λ_1=-λ_3,$ you have a conservative system, and nothing changes on time inversion. The argument translates to higher-dimensional systems, only that you have to first summarize several directions.

Finally, note that many non-conservative systems do not have a constant sign of $∇·f$: Some regions of phase space are unstable (e.g., around an unstable fixed point), while others are dissipative (e.g., around a chaotic attractor). In this case, you cannot really say much about what time inversion does as the dynamics may end up in a different region of the phase space.

Answer 2

You're missing a key ingredient in the usual definition of chaos: it's bounded in phase space.

That means that being 'explosive', the opposite of dissipative, won't make it chaotic. Actually, the best known example of chaos are chaotic attractors, such as the (butterfly) Lorenz attractor, which can only attract due to dissipation being present on the system.

As the Stanford Encyclopedia of Knowledge puts it:

In much physics and philosophy literature, something like the following set of conditions seems to be assumed as adequately defining chaos:

1. Trajectories are confined due to some kind of stretching and folding mechanism.
2. Some trajectory orbits are aperiodic, meaning that they do not repeat themselves on any time scales.
3. Trajectories exhibit a strong form of sensitive dependence or a positive global Lyapunov exponent.

where the highlight is mine. Wrzlprmft's answer already mentions it, with less emphasis.

So, for instance, the shift map on the unit line, \begin{equation} x \mapsto 10x \mod 1,\quad x \in [0,1), \end{equation} is chaotic, but only because the $\mod{}$ operation folds the trajectories back to the unit line — without it, the map is simply $x \mapsto 10x$ which is divergent alright (everything but $0$ explodes exponentially), but boringly predictable, not chaotic at all.

Besides, also conservative systems can be (and usually are) chaotic, and remain so under time reversal.