Large value fluctuations are modeled by a system of nonlinear stochastic equations describing the interacting phase transitions. Under the action of anisotropic white noise, random processes are formed with the 1/f^alpha dependence of the power spectra on frequency at values of the exponent from 0.7 to 1.7. It is shown that fluctuations with 1/f^alpha power spectra in the studied range of changes correspond to the entropy maximum, which indicates the stability of processes with 1/f^alpha power spectra at different values of the exponent alpha.
In the system of two nonlinear differential equations, proposed to explain the physical nature of the 1/f spectra, chaotization of the trajectories is revealed under periodic external action on one of the equations. External noise effects lead to stochastic resonance and low-frequency 1/f behavior of power spectra. Stochastic resonance and 1/f behavior of power spectra corresponds to the maximum information entropy, which indicates the stability of a random process.
Scale-invariant random processes with large fluctuations are modeled by a system of two stochastic nonlinear differential equations describing interacting phase transitions. It is shown that under the action of white noise, a critical state arises, characterized by a turbulent spectrum and a scale-invariant distribution function. The critical state corresponds to the maximum entropy, which indicates the stability of the process. An external harmonic action on a random process with a turbulent spectrum gives rise to a resonant response of scale-invariant functions.
A stochastic differential equation is proposed for a characteristic function whose inverse function describes a self-similar random process with a power-law behavior of power spectra in a wide frequency range and a power-law amplitude distribution function. Gaussian “tails” for the characteristic distribution make it possible to evaluate its stability according to the formulas of classical statistics using the maximum of the Gibbs-Shannon entropy and, therefore, the stability of a random process given by an inverse function.
Экстремальные флуктуации моделируются точечной системой стохастических уравнений, в которой под воздействием белого шума формируются спектры мощности, обратно пропорциональные частоте. Распределение экстремальных флуктуаций соответствует максимуму статистической энтропии, что свидетельствует об их устойчивости. Расчёт спектральной энтропии случайных процессов дает возможность исследовать их устойчивость непосредственно по спектрам мощности, не прибегая к вычислению функций амплитудных распределений. Зависимость спектральной энтропии от амплитуды белого шума имеет минимум. Положение минимума спектральной энтропии соответствует критическому состоянию системы, при котором спектры флуктуирующих величин обратно пропорциональны частоте.
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