Logperiodic

Logperiodic refers to patterns or functions that exhibit periodicity on a logarithmic scale. A function f is log-periodic on a domain x > 0 if there exists a period T > 0 such that f(x e^T) = f(x) for all x in the domain, equivalently f(x) = g(log x) where g is periodic with period T. This implies invariance under discrete scale transformations x -> λ x with λ = e^T, a property called discrete scale invariance.

A classic simple example is the function sin(log x), which is periodic in log x with period 2π, corresponding to a scaling factor λ = e^{2π}. More complex models combine a power-law trend with log-periodic oscillations, such as f(t) = A + B(tc - t)^m [1 + C cos(ω log(tc - t) − φ)], where t approaches a critical time tc. Here ω fixes the log-frequency of the oscillations and the term cos(ω log(...)) produces the log-periodic fluctuations.

Applications of log-periodic structure appear in multiple disciplines. In physics, log-periodicity arises in systems exhibiting discrete scale invariance near critical points or in hierarchical models. In finance, log-periodic patterns have been used to model speculative bubbles and crashes via LPPL-type formulations. Similar oscillatory behavior linked to log periodicity has been reported in seismology and other complex systems, where events cluster with a characteristic scale across orders of magnitude.

Detection and analysis often involve transforming data by taking logarithms, applying spectral methods to the log-scale, or fitting LPPL-type models to time-series data. See also discrete scale invariance, fractals, log-periodic oscillations, and LPPL.