Stability in dynamical systems

Lyapunov exponents and ilk

May 21, 2019 — February 22, 2022

dynamical systems
functional analysis

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 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.

Figure 1

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

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Lawrence, Loewen, Forbes, et al. 2020. Almost Surely Stable Deep Dynamics.” In Advances in Neural Information Processing Systems.
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Pathak, Lu, Hunt, et al. 2017. Using Machine Learning to Replicate Chaotic Attractors and Calculate Lyapunov Exponents from Data.” Chaos: An Interdisciplinary Journal of Nonlinear Science.
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