quasiokresowy

Quasiokresowy is a Polish term corresponding to the concept of quasi-periodic. It describes phenomena, functions, or motions that exhibit regular oscillations without exact repetition. In mathematics, a function f: R → C is called quasiokresowy if it can be written as a finite sum of harmonics with incommensurate frequencies: f(t) = Σ a_k e^{i ω_k t}, where the coefficients a_k are real or complex and the frequencies ω_k are real numbers that are linearly independent over the rational numbers. If the frequencies are rationally dependent, the function becomes periodic with a common period; if they are independent, the function never repeats exactly but shows a regular, structured pattern.

In dynamical systems, quasiokresowy motion refers to trajectories on an invariant torus where each coordinate advances linearly in time with its own frequency. When the frequencies are rationally independent, the path densely fills the torus and does not close. Quasi-periodic behavior often arises as a stable regime under small perturbations of integrable systems, a theme central to the KAM (Kolmogorov–Arnold–Mëri) theory.

Quasiokresowy is related to, but distinct from, almost periodicity. Every quasiokresowy function is almost periodic, though not every almost periodic function is quasiokresowy. In practical applications, quasi-periodic signals appear as sums of oscillations with incommensurate frequencies in physics and engineering, and in celestial mechanics as models with multiple independent orbital frequencies.

Common examples include sums like f(t) = Σ a_k cos(ω_k t) with incommensurate ω_k, illustrating non-repeating but regular harmonic structure.