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?
Chaos Theory – Are Chaotic Systems the Same as Dissipative Systems in Inverse Time?
arrow-of-timechaos theorycomplex systemsdissipationintuition
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However, what if the time series is multi-dimensional, indeed of the same dimension as the phase space, to begin with?
Well, how would you know that your time series is of the same dimension as the phase space? Usually, because you already know the dynamical equations for your system (as for your pendulum). If you observe a real-life complex system, however, you might be able to obtain a multivariate time series, but there is no way to say whether its dimension corrensponds to the actual dimension of the phase space, since you cannot know the latter. Therefore I am addressing two cases separately:
- You know the dynamical equations for your system. Be very careful to assume this unless your system is simulated.
- You have obtained a multivariate time series from an unknown system.
1. You can simulate the system
Roughly speaking, you determine the largest Lyapunov exponent (and also the others) by looking at how quick two trajectories diverge after passing through two points that are close in phase space. If you only have a reconstructed phase space of your system from a time series, the only way to obtain two such nearby trajectories is to look for two points that are close to each other in your reconstructed phase space. However, if you can simulate your system, you can generate such points for yourself simply by applying a slight perturbation to the state of your simulated system. Apart from this, the method is basically the same (and is described in section 3 of the paper by Wolf et al. for example).
Also, there are some cases where you can determine the Lyapunov exponents analytically.
2. You have a multivariate time series
Estimating the Lyapunov exponents from a time series happens roughly in two steps:
- Reconstructing the phase space from the time series.
- Estimating the Lyapunov exponent from this reconstructed phase space.
Step 2 does not care about how you reconstructed the phase space – given that you do it properly and that the attractor is maximally unfolded. And in step 1, having more than one observable from your system is usually a huge benefit. A simple approach would be to start with your multivariate time series and add delayed embeddings (as described for example in your quote from Packard et al.) of your component time series until you are confident that you have unfolded the attractor. Keep in mind however, that some of your observables might not be independent or at least strongly correlated. Little surprisingly, there are more sophisticated methods for this (as a start, a quick search yielded this paper).
Best Answer
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.