# The natural metric of a phase space and the Lyapunov exponent

For me, it seems that there is no apparent metric on a phase space of a dynamical system. Of course one can naively define an Euclidean metric on it, but it seems that this metric has not much to do with the peculiar features of a phase space.

But in many textbooks on dynamical systems, it is the metric employed in the definition of the Lyapunov exponent.

Is this really a nice approach?

• I think this is a great question but I am not good enough to answer it right now. My direct answer would have been "it's not a mertic space, it's a symplectic space", but it doesn't quite answer anything. By searching the web I did find these notes which I cannot fully comprehend yet but might hold some answers for you. – gatsu Sep 8 '14 at 16:22

It has been shown by Eichhorn, Linz and Hänggi in 2000 that the numerical values of Lyapunov exponents are invariant under any invertible variable transform. This is just a reformulation of the fact that they are metric invariant, because the authors presume the norm $|\cdot|$ to be an arbitrary norm in the given coordinates - just it's basic properties such as linearity are enough.
A second way to get an intuition is through the explicit definition of the exponents $\lambda$ via the linear variation $\delta x(t)$ evolved in time: $$\lambda = \lim_{t\to \infty} \frac{\log |\delta x(t)|}{t}$$ Let us assume that $\delta x(t) = e^{\mu t}\delta x(0)$. Then out of the linearity of the norm we have $|\delta x(t)| = e^{\mu t} |\delta x(0)|$ and the limit yields $$\lambda = \lim_{t \to \infty} (\frac{\mu t}{t} + \frac{\log |\delta x(0)|}{t})$$ The second term dies off and we have $\lambda=\mu$ for any positive definite linear $|\cdot|$. I.e. you get the same number with a different norm and thus the Lyapunov exponent gives you something which is connected to a "relative growth rate" independent of the metric. There are some loopholes to this argument and these are covered by the article cited above.
• @Phonon Not really, the best known Lorenz model has a $\mathbf{R}^3$ phase space. The famous kicked oscillators have a formal $\mathbf{T}^2 \times \mathbf{R}^2$ phase space which is also the case of the planar double pendulum. There are some topological considerations in chaos but the phase space only has to have dimension $\geq 3$ to allow for continuous-time chaos. – Void Dec 31 '14 at 13:51