Red spectra from white and blue noise

• Published in: Proceedings of the Royal Society. B, Biological sciences Vol. 266; no. 1416; pp. 311 - 314
• Main Authors: Balmforth, Neil J., Provenzale, Antonello, Spiegel, Edward A., Martens, M., Tresser, Charles, Chai, Wa Wu
• Format: Journal Article
• Language: English
• Published: The Royal Society 07.02.1999
• Subjects: 1/f Spectra; Acoustic spectra; Dynamic modeling; Dynamical Systems; Economic fluctuations; Environmental Effects; Logistics; Low frequencies; Noise spectra; On-Off Intermittency; Population Dynamics; Time series; Time series models; White noise
• ISSN: 0962-8452; 1471-2954
• DOI: 10.1098/rspb.1999.0639

The value of maps of the interval in modelling population dynamics has recently been called into question because temporal variations from such maps have blue or white power spectra, whereas many observations of real populations show time-series with red spectra. One way to deal with this discrepancy is to introduce chaotic or stochastic fluctuations in the parameters of the map. This leads to on-off intermittency and can markedly redden the spectrum produced by a model that does not by itself have a red spectrum. The parameter fluctuations need not themselves have a red spectrum in order to achieve this effect. Because the power spectrum is not invariant under a change of variable, another way to redden the spectrum is by a suitable transformation of the variables used. The question this poses is whether spectra are the best means of characterizing a fluctuating variable.