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Why Nature's Clocks Sometimes Run Fast or Slow ✎

Original: "Stable limit cycles perturbed by noise"
· Stilianos Louca
arXiv:1506.00756 · 2015-06-02 · arXiv non-exclusive · independent analysis · 1 min
Dynamical Systems
Random nudges disrupt oscillation rhythms, but scientists have learned to describe this disruption with simple formulas.

A swing swings evenly until someone gives it a push at a random moment. Each such push slightly shifts the rhythm, and soon the swing is no longer in time. Many systems in nature behave similarly: the heart, animal populations, climate cycles.

A limit cycle is a closed trajectory that a system follows indefinitely if left alone. Like a track on which a toy train runs around and around.

Scientists decomposed the motion into two parts: one along the track, the other across it. The first part accumulates random shifts, like a drunkard wandering in a circle. The second part quickly decays, as if a rubber band pulls the point back onto the track. This allowed them to derive simple formulas that describe how quickly the rhythm is lost.

The method was applied to El Niño, a climate oscillation in the Pacific Ocean. They obtained a period of about 4.2 years. Thus, a swing that is pushed randomly turned out to be described by formulas that help predict natural rhythms.

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El Niño, despite its irregularity, has a characteristic period of about 4 years, which is close to the estimate obtained in this work.

ACV[x](u) = \frac{r^2}{2} \left[1 + NSR^2 e^{-\lambda |u|}\right] \cos(\alpha u) e^{-|u|(\sigma/r)^2/2}
The autocovariance decays exponentially due to phase drift and contains an oscillatory factor at the cycle frequency.
PS[x](\omega) = \frac{2r^2(\sigma/r)^2 [4(\alpha^2+\omega^2)+(\sigma/r)^4]}{[4(\alpha-\omega)^2+(\sigma/r)^4][4(\alpha+\omega)^2+(\sigma/r)^4]} + NSR^2 \cdot \frac{2r^2((\sigma/r)^2+2\lambda)[4(\alpha^2+\omega^2)+((\sigma/r)^2+2\lambda)^2]}{[4(\alpha-\omega)^2+((\sigma/r)^2+2\lambda)^2][4(\alpha+\omega)^2+((\sigma/r)^2+2\lambda)^2]}
The spectrum has a peak shifted from the deterministic frequency and a finite width due to noise.
Concepts
stochastic dynamical system hopf bifurcation biological rhythm modeling El Niño-Southern Oscillation spring predictability barrier statistical noise stochastic self organization natural variability ENSO instability mixed mode oscillations seasonal cycle Southern Oscillation index white noise Dynamical Systems Jin-Timmermann model El Niño index Neural Network Oscillations recharge oscillator model limit