logperiodicznymi

Logperiodicznymi is the Polish term used to describe log-periodic phenomena, or log-periodicity, which refers to patterns that repeat on a logarithmic, rather than linear, scale. This behavior arises in systems with discrete scale invariance, a type of symmetry in which a structure repeats itself under magnification by a fixed ratio. In data, log-periodic signals appear as oscillations whose frequency grows as a system approaches a critical point.

A common mathematical description models a quantity as a power law with a log-periodic modulation: f(t) = A + B (t_c − t)^−α [1 + C cos(ω log(t_c − t) + φ)], where t is time, t_c is a critical time, α is a critical exponent, ω is the log-frequency, and φ is a phase. The log-frequency is related to the preferred scaling ratio λ by ω = 2π / ln λ. This form captures both the overall power-law trend and the oscillatory corrections driven by discrete scale invariance.

Origins of log-periodicity are linked to hierarchical or cascading structures that exhibit self-similarity only at specific magnifications. Such patterns are discussed in various fields, including physics, geophysics, and econophysics, where discrete scale invariance has been proposed as a mechanism behind observed regularities in complex systems.

Applications include modeling critical phenomena in physics, pre-seismic activity in earthquake research, and speculative bubbles in financial markets through log-periodic power-law approaches. Proponents argue these models can aid in detecting approaching critical events, while critics caution about overfitting and limited predictive reliability. Overall, logperiodicznymi reflects a broader interest in how discrete scale invariance manifests as measurable oscillations across disciplines.