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+---
+title: "Randomness and Uncertainty: from random noise to predictable oscillations via differential equations"
+date: 2023-10-20
+tags: diffeq
+toc: false
+---
+
+This [[https://www.lms.ac.uk/sites/default/files/inline-files/NLMS_505_for%20web.pdf#page=24][article (PDF)]] by Nick Trefethen in the [[https://www.lms.ac.uk/publications/lms-newsletter][/London Mathematical
+Society Newsletter/]] demonstrates an interesting relationship between
+what we perceive as "randomness" and what we perceive as "certainty".
+
+There are many ways to generate pseudo-random numbers that look
+perfectly "random" but are actually the output of fully deterministic
+processes. Trefethen gives an example of a chaotic system based on a
+logistic equation.
+
+But more interesting (to me) and more original may be the other way
+around: how to get certainty from randomness. There are ordinary
+differential equations that can take random noise as input, and whose
+solution is very stable, oscillating between two possible
+values. Given a function $f$ approximating random white noise, the
+solution to the ODE
+\[ y' = y - y^3 + Cf(t) \]
+is "bistable" and remains always around -1 and 1. The parameter $C$
+allows to control the half-life of the transitions.
+
+To explore this behaviour, I replicated Trefethen's experiments in
+Python with the [[https://docs.kidger.site/diffrax/][Diffrax]] library (a differential equations solver based
+on [[https://jax.readthedocs.io/][JAX]]). The full code is [[https://gist.github.com/dlozeve/4924e71097e1d86933e8d5528cd2f6b4][in this Gist]].
+
+It suffices to define a simple function for the vector field and to
+give it to a solver:
+#+begin_src python
+def f(t, y, args):
+    return y - y**3 + 0.4 * args[t.astype(int)]
+#+end_src
+where ~args~ will be the input, as a simple array of
+normally-distributed random values. $C$ is hardcoded as 0.4 as in the
+article, but could be passed through ~args~ as well (it can be a
+dictionary).
+
+[[../images/randomness_uncertainty.png]]