Sensitivity to initial conditions and topological transitivity

Question

Forgive the possibly naive question, but I'm returning to a basic undergrad-level study of chaotic dynamical systems and can't quite seem to reconcile the notion that in order for a dynamical system to be considered chaotic, it must have positive global Lyapunov exponents. Such exponents, of course, drive the divergence of trajectories. However, topological transitivity implies that trajectories can get arbitrarily close as well over some finite time. It seems that positive Lyapunov exponents would disallow this mixing. Is there a point at which these positive exponents no longer describe the system, and the mixing behavior begins?

Answer 1

Is there a point at which these positive exponents no longer describe the system, and the mixing behavior begins?

Yes. The exponential divergence described by the Lyapunov exponents only affects trajectories whose separation is sufficiently close. Also, it only holds on average. For example, you can define the largest Lyapunov exponents as follows:

$$ λ_1 := \lim_{τ→∞} \; \lim_{|x(t)-y(t)|→0}\; \frac{1}{τ} \ln\left(\frac{|x(t+τ)-y(t+τ)|}{|x(t)-y(t)|}\right) $$

where $x$ and $y$ are two trajectories of your dynamics. Here, the limit $|x(t)-y(t)|→0$ ensures that you only consider the separation of infinitesimally close trajectories. (Also, see this question on Physics.)

Once the separation of trajectories becomes too large, it is not exponentially growing but governed by the large-scale dynamics of the system. It then has some value that is of the order of magnitude of the size of the attractor (or invariant set) of your dynamics. E.g., here is the temporal evolution of two trajectories of the Lorenz system (attractor diameter $\approx 30$) starting off close by:

Now, you can see, that once the separation has reached the order of the magnitude of the attractor, it occasionally shrinks again (growing roughly exponentially afterwards). So the trajectories can come close to each other again. This happens along a direction corresponding to a negative Lyapunov exponent. This direction is orthogonal to the direction of largest separation which causes the positive Lyapunov exponents. This shrinking can be arbitrarily small and is the backbone of the topological transitivity.