# Stability in dynamical systems

Lyapunov exponents and ilk

May 21, 2019 — February 22, 2022

A placeholder.

Informally, I am admitting as “stable” any dynamical system which does not explode super-polynomially fast; We can think of these as systems where if the system is not stationary then at least the rate of change might be.

Here I would like to think how to parameterize stable systems, and how to discover if systems are stable. This in a general context, which can be extremely hard in interesting systems. But often stability questions can be simpler in the context of linear systems, where is about polynomial root-finding (perversely, polynomial root-finding is itself a famously chaotic system).

In a general setting should probably look at stuff like Lyapunov exponents to quantify stability.

Interesting connections here — we can also think about the relationships between stability and ergodicity, and criticality. Considering stability of neural networks turns out to produce some nice ideas.

## 1 References

*2015 European Control Conference (ECC)*.

*arXiv:1709.03698 [Cs, Stat]*.

*Advances in Neural Information Processing Systems*.

*Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach.*

*Advances in Neural Information Processing Systems*.

*The Annals of Probability*.

*Chaos: An Interdisciplinary Journal of Nonlinear Science*.

*arXiv:2106.10165 [Hep-Th, Stat]*.

*Nonlinear Dynamics and Statistics*.