Investigate scaling behavior of lyap_e

[CSchoel]

The function lyap_e is reliable when it comes to the sign of the largest Lyapunov exponent, but not so much when it comes to the absolute size. Changing the parameters emb_dim, matrix_dim, min_nb, and min_tsep can give wildly different ranges of values.

This could be due to an error in the normalization regarding the number of matrices created, dimensions, and tau. I went through the code and paper again, but could not find anything that would support that. The other explanation could be that the higher exponents that turn up are actually "spurious" exponents that are also described in the original paper.

To investigate this, one would first need a good reference using the algorithm that mentions explicit settings for all the above parameters as well as the input data. The examples in the original paper unfortunately omit some of these. 🙁

ℹ️ I don't plan to take this issue on myself, but only leave it here in case someone is interested.

DoD

• Find a good reference in the literature that reports the expected result of lyap_e and lists all parameters used.
• (optional) If the above cannot be found, use the Lorenz example given by Eckmann et al. (1986) themselves and try to guess the parameters that they didn't mention.
• Replicate the chosen reference exactly and look for deviations.

[dwh2022300890]

hey sir,I also want to calculate the largest Lyapunov exponent for the Kuramoto, i met the same question changing the parameters and can give wildly different ranges of values but i need the precision value, so can you give me some advice to choose the parameters? Thanks a lot

[CSchoel]

Hi. 👋 Unfortunately, I didn't have an opportunity to investigate this further yet. If you only need the largest Lyapunov exponent, I would therefore suggest using lyap_r which is a bit more stable regarding parameter choice. I documented all advice for choosing parameter values that I could find in the literature in the docstrings.

Additionally, you can take a look at the comments I recently made for the new test for lyap_e and lyap_r using the Lorenz system. This is just my personal experience, but it may help to find a procedure to get sane parameter choices.