Distinguishing between chaos and multiperiodic oscillations from the Fourier spectrum

Question

Consider a system which exhibits multiperiodicity, say with oscillations of the form $x(t) = \sum_{n=0} c_n \cos(n \omega_0 t)$, $\lim_{n \to \infty} c_n = 0$. The Fourier transform $\tilde{x}(\omega)$ will then contain peaks at $n \omega_0$ with strength proportional to $c_n$ (for simplicity, we can add a small decaying factor $e^{-\gamma t}$ to $x(t)$ to avoid delta-functions).

Restricting to deterministic systems, chaos appears as a continuous Fourier spectrum (as opposed to a line spectra). In our example, suppose the Fourier peaks are governed by some continuous envelope function $f(\omega)$ such that $\tilde{x}(n \omega_0) = f(n\omega_0)$. In the limit of $\omega_0 \to 0$, the peaks become infinitesimally close. Can this then be interpreted as a chaotic motion?

Answer 1

Any process that occurs in a limited time, $\Delta t$, will have a continuous spectrum when examined at a resolution $\Delta f << 1/\Delta t$. Also, the spectrum of a simple random series, e.g. the clicks of a Geiger counter, is continuous. So, no, you cannot conclude that a continuous spectrum indicates chaos.