Fractional Generalized Inverse Gaussian Process for Population Dynamics of Phase Singularities

In this paper, a generalized inverse Gaussian (GIG) process is studied in conjunction with the population dynamics of phase singularities (PSs). Special attention is paid to a stochastic analysis of PSs based on combined methods with (i) the theory of information geometry, (ii) the eigenvalue problem related to the double confluent Heun equation, (iii) classification of the statistics (sub-Poisson, Poisson, and super-Poisson), and (iv) fractional generalization to introduce a memory effect. The present theoretical method is applied to describe two-dimensional (2D) spiral wave turbulence in CO oxidation on a Pt surface and in the Aliev–Panfilov model. It is demonstrated that the fractional GIG process with the fractional index μ (μ = 0.5) can capture the profile of the PS number distribution and the scaling law of ω−1.5 in the power spectral density at large frequencies, which have been observed in real experiments and numerical simulations.