Pink noise (1/f) is often cited as a signature of complex or critical systems. Is it possible for a low-dimensional time-independent first-order system to generate pink noise?

Intuitively it seems that the answer should be no since pink noise exhibits non-trivial long-range correlations that, unlike with brown noise, cannot be explained by the transitivity of short-range correlations. This seems to imply the system must have some form of long-term memory which would require higher dimensions.

Is there a formal proof of this or a counter example? I am primarily interested in physically plausible systems. I am less interested in systems that treat real numbers as infinite bit streams using symbolic computation. The systems should generate approximately pink noise even if the internal state is only stored to finite precision.

I do understand that true pink noise is not physically possible since would exhibit both infra-red and ultraviolet catastrophes.

The question is restricted to time-independent systems to rule out summing the Fourier component contributions as a function of time and to the first order since a higher-order systems are really higher-dimensional systems in disguise.


If you want to restrict yourself to physically realistic systems (that are not inherently stochastic), we can restrict ourselves to ordinary differential equations, say

$$\dot{x} = F(x); \qquad x ∈ ℝ^n$$

with $F$ being smooth and not ridiculously complex (e.g., not a polynomial with hundred terms).

The dynamics of this system is governed by invariant manifolds with respect to the phase-space flow given by $F$. In case of a periodic system, the trajectories converge to some of these manifolds; in case of a chaotic system, things are more complicated and the trajectories are governed by an interplay of repulsion from unstable and attraction to stable manifolds, but they still roughly move along them. Now movements along a given part of these manifolds are happening on the same time scale (for each direction). The crucial point is now that in a system as described above, we have only a small number of manifolds and directions of movement and hence time scales.

Now, we obtain some oscillatory behaviour by the dynamics moving along at least some of these manifolds, with the frequency being determined by the aforementioned time scales. These frequencies need not be exactly given (in case of a chaotic system) but they can only be smeared out to a certain extent. Thus, to approximate a noise spectrum spanning many frequency scales, you need many time scales in the system, which is impossible in a low-dimensional system.


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