# Autocorrelation function for deterministic nonlinear dynamical systems

I am quite puzzled with the problem that spectral analysis has been either applied to noisy dynamical systems or to chaotic ones. I was wondering why nobody makes analysis of non-linear dynamical systems based on their autocorrelation? At least for non-linear oscillators which are essentially periodic and seem to suit for such analyses.

For example I simulated Van der Pol oscillator for 500 seconds from initial condition $(1.1,0.1)$. The 2D ODE is as follows: $$\frac{d\textbf{x}}{dt}= \begin{cases} \mu(x_0-1/3 x_0^3-x_1)\\ \frac{x_0}{\mu} \end{cases}$$ Where I have set the $\mu$ to $5$. The plot of the oscillator and the autocorrelation function as defined by Wiener are as follows:

Sorry for the lack of labels. The $x$-label in the second plot is representing lag in seconds and the $y$ axis is $C(\tau)=\frac{1}{T}\lim\limits_{T\to \infty}\int^T_{-T}x_0(t)x_0(t+\tau)dt$.

What would be wrong with such analysis when the autocorrelation function exists? Is this a totally dumb question??

• The autocorrelation you show looks a bit weird to me. I would expect it to be maximum at $\tau=0$ (by definition) and periodic with the same period of your $x$. Also, didn't you mean to write $\dfrac{d\mathbf{x}}{dt}$ in your equation? – Bernhard Sep 7 '14 at 17:52
• @Bernhard, you are completely right. My mistake. I made the corrections. – Cupitor Sep 7 '14 at 18:30
• The main problem with a linear quantity like autocorrelation applied to a non-linear system is, that the autocorrelation function would depend on the amplitude (more generally the total energy) of the system. You wouldn't end up with one graph, but with an infinite set of them that is parametrized by the energy of the system. What are you going to do with that in general? In special cases, e.g. to separate chaotic from non-chaotic regions, it could be a useful way of looking at the system, but I am not sure that it would be more useful than e.g. Lyapunov exponents, limit cycles etc.. – CuriousOne Sep 7 '14 at 21:04
• @CuriousOne The existence of such functions would rather rely on ergodicity of the system and that is independent of the system being linear or non-linear and there are quite a lot of nonlinear system who have constant levels of energy I believe; indeed my main argument was about self sustained oscillators and limit cycles. Maybe I forgot to point that out. And I wonder why such measure is not used these analysis? Regarding your linear argument I should say, is that anything different in nature for correlation integral in being linear? But it is one of the main tools for chaos theory, right? – Cupitor Sep 7 '14 at 22:05
• @Cupitor - Are you working with data or simulation/theory? If data, then it is probably not very useful to use the autocorrelation for many things. It can be useful for pattern matching (e.g., find wave modes in time series that have a specific appearance). However, as CuriousOne said, the possible dependence on the amplitude of the signal will make interpretation of the results all but impossible. I suppose it is possible to do such analysis, but whether it would yield anything physically significant is another question. – honeste_vivere Oct 25 '14 at 15:23