Lyapunov exponents - good for what?

David M writes:

Q: I am working in nonlinear dynamics in the area of Sports Science and was intrigued by your comments in ChaosBook.org (version14.4.1), Chapter 6 Lyapunov exponents:

In short: Lyapunov exponents are an abomination. We are doubtful of their utility as means of predicting any observables of physical significance, but that is the minority position - in the literature one encounters many provocative speculations, especially in the context of foundations of statistical mechanics (`hydrodynamic' modes) and the existence of a Lyapunov spectrum in the thermodynamic limit of spatiotemporal chaotic systems.

Lyapunov exponents are used extensively in the literature of nonlinear systems as an indicator of chaos.  Do you have any suggestions as what should be used as an indicator in place of Lyapunov exponents - is it a number of multiple factors that indicate Chaos?

A: thanks for pointing this out, I thought I had censored the 'abomination' already - it was stated so emphatically because I cannot get my students deprogrammed from 'Lyapunov exponents' no matter how I rewrite the stability chapters, but it gets colleagues' hair up. How about this revision:

In short: we do not recommend that you evaluate Lyapunov exponents; compute stability exponents and the associated covariant vectors instead. Cost less and gets you more insight. Whatever you call your exponents, please state clearly how are they being computed. While the Lyapunov exponents are a diagnostic for chaos, we are doubtful of their utility as  means of predicting any observables of physical significance. This is the minority position - in the literature one encounters many provocative speculations, especially in the context of foundations of statistical mechanics (`hydrodynamic' modes) and the existence of a Lyapunov spectrum in the thermodynamic limit of spatiotemporal chaotic systems.

'Stability exponents' and the associated 'covariant vectors' are defined in the preceding "Local stability" and "Cycle stability" chapters. The "Lyapunov exponents" chapter I had to write because it is only very recently that people (see  Ginelli et al. Covariant Lyapunov vectors, J. Phys. A 46, 254005 (2013),  arXiv.org/abs/1212.3961) do the right thing.