Prove sensitivity to initial conditions numerically?

ordinary-differential-equations dynamical-systems chaos-theory

1,081

Solution 1

There is an whole field of study related with the extraction of Lyapunov exponents from time-series, for instance to distinguish noise from chaotic behavior.

A definite starting point is this paper from H. Kantz.

Solution 2

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.

1,081

Ambesh

Updated on June 07, 2020

Comments

Ambesh over 2 years

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.
- Admin almost 9 years
  
  A simple thing to try would be to tweak the initial conditions.
- Ambesh almost 9 years
  
  Yes. But how do I measure in some significant way that the differences while changing the i.c. mean sensitivity?
- Gerry Myerson almost 9 years
  
  For discrete dynamical systems, one can compute an estimate of the Liapunov exponent from the data, and if that estimate is positive then one can take it as evidence of sensitive dependence on initial conditions. But by the differential-equations tag, I take it you are interested in continuous dynamical systems. Maybe there is an analogous computation.
- Ambesh almost 9 years
  
  @GerryMyerson My case is a continous system with discrete resets when a value is reached. Therefore it can be considered as discrete. It would be of enormous help to get a reference on how to perform that estimate of Liapunov!
Ambesh almost 9 years

If the data is bounded and oscillating, I don't see how to implement this.
flonk almost 9 years

Chaotic motion is also possible in a bounded space. The exponential depence (if existing) can only be observed for a finite time, say $T$, then it saturates. For smaller $\epsilon$ this duration $T$ grows. Only for an infinitesimal displacement $\epsilon$ the exponential growth would last forever, $T\to\infty$. This follows from the two limits competing against each other as mentioned in the answer. In more illustrative words: in a finite interval, there is room for exponential growth, as long as you choose the thing you want to grow to be small enough.
Ambesh almost 9 years

Well true. The problem is when we don't know $f$. My question is if there is anyways to approximate this without knowing $f$. Just with the values $x_1,x_2\dots$.
Gerry Myerson almost 9 years

Sorry, then, I'm out of ideas.
nonlinearism almost 9 years

You can normalize the distances after a few steps (say, anytime it becomes bigger than 1/100 of the domain size). And keep repeating your numerical experiment.