Can probe trajectories to compute Lyapunov exponents get "stuck in more regular orbits" after rescaling?

Question

I am computing Lyapunov exponents, and there is something that I do not understand about the data. The model has a regime for $\delta \approx 1$ (in some units) where it is fully chaotic, and the usual way (evolving probe trajectories or volume elements) of computing exponents behaves as expected.

However, for a small to medium $\delta > 0$ (the model is not integrable in the limit delta = 0), the rescaling of exponents leads to a weird behavior...

For example, I am initializing a probe trajectory $t_0(t)$ for a given test trajectory at an initial distance of $10^{-11}$ and evolve both in time. If it exceeds the upper bound of $10^{-10}$, I am rescaling the distance to $10^{-11}$ and continue the evolution. The first time interval until $t_1$, where it exceeds the upper bound, is reached quickly, predicting an exponent of O($1$). However, after the first (or second) rescaling, the distance grows extremely slowly, predicting a wide range of different exponents. If I respawn a new (random) probe trajectory at $t_1$, the distance between the new trajectory $t_1$ grows extremely fast while the rescaled one $t_0$ grows extremely slow. Somehow, the rescaling procedure is biased towards projecting the probe trajectory into a more regular orbit. This is somehow counterintuitive to me as I thought that (regardless of the rescaling) the probe trajectory aligns along the direction of maximal growth (?)

Attached is a plot visualizing the scenario. The red curve is the probe trajectory $t_0(t)$. If it exceeds the upper bound, I rescale it and respawn a new random trajectory $t_1(t)$ shown in green. The green (randomly initialized) trajectory always shows a rapid increase in distance. However, the growth in both cases seems to be linear, which is presumably attributed to a finite tolerance of the ODE solver.

This behavior occurs for all points in phase space, and I find the same response when I use the Jacobian to evolve a volume... Somehow, the trajectories do not align along the maximal growth direction but "get stuck" in more regular orbits after the rescaling.

Did someone experience a similar problem, or can point me to any reference? Can it be related to boundary effects between chaotic and regular manifolds?

Answer 1

I believe the following is going on: The phase space is mixed, and there are regular trajectories where the perturbation grows linearly in time $d(t) = d_0 \alpha t + d_0$, where $d_0$ is the initial perturbation. $\alpha$ is of $\mathcal{O}(1)$. This can be understood from action-angle variables.

Rescaling affects the integrals of motion, reducing $\alpha$, which yields a smaller growth.