Prove sensitivity to initial conditions numerically?

Question

How can I prove sensitivity to initial conditions numerically? I mean directly from the computed data and neglecting the dynamical system that originated the data.

The data comes from hybrid dynamical system (ode + resets when a value is reached) that can be considered as a discrete dynamical system.

Thanks.

Answer 1

To be precise sensitivity to initial conditions is physicists language for the fact that a small change in the initial conditions grows exponentially in time.

Numerically, just check if for two solutions the distance $|x(t)-y(t)|$ grows exponentially in time $t$ for slightly different initial conditions $y_0=x_0+\epsilon$.

Formally, for $|\epsilon|\to 0$ and $t\to\infty$ the exponent of this dependence, if it exists, is the Lyapunov exponent mentioned in the comments.

Answer 2

In view of the discussion in the comments, I'll post an answer concerning the Liapunov exponent, even though I'm not sure it will be helpful here.

Given a discrete dynamical system, which for my purposes here is a differentiable function $f$, a starting point $x_0$, and a sequence of iterates $x_1,x_2,\dots$ given by $x_{n+1}=f(x_n)$, the Lyapunov exponent of $f$ at $x_0$ is given by $$\lim_{n\to\infty}{1\over n}\sum_{k=1}^n\log|f'(x_k)|$$ provided the limit exists. It is common in applications for the limit to exist and to be independent of the initial value $x_0$. A limit exceeding zero is indicative of sensitive dependence on initial conditions.

In practice, one generally cannot compute this quantity exactly, so one approximates it by $${1\over n}\sum_{k=m+1}^{m+n}\log|f'(x_k)|$$ for some appropriate $m$ and $n$, say, $m=n=500$.