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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]]