Abstract
Purpose of this work is to determine regularities of formation of spatial structures in an ensemble of chaotic systems with non-local diffusion couplings and to study how these structures depend on the wave response of the digital filter formed by the ensemble couplings structure. Methods. The study was carried out by numerical simulation of an ensemble of logistic maps, calculation of its typical oscillatory regimes and their spectral analysis. The network was considered as a digital filter with a frequency response depending on the coupling parameters. Correlation between the spatial spectra and the amplitude-frequency response of the coupling filter and the mutual coherence of oscillations when the coupling parameters change were considered. Results. The system of couplings between chaotic maps behaves like a wave filter with selective properties, allowing spatial modes with certain wavelengths to exist and suppressing others. The selection of spatial modes is based on the wave characteristic of the coupling filter, the type of which is determined by the radius and the magnitude of couplings. At strong coupling the wave characteristics for ensembles with local and non-local couplings are qualitatively different, therefore they demonstrate essencially different behavior. Discussion. Using spectral methods for dynamics analysis systems with complex network topologies seems to be a promising approach, especially for research of synchronization and multistability in ensembles of chaotic oscillators and maps. The found regularities generalize the results known for ensembles of maps with local couplings. They also can be applied to ensembles of self-sustained oscillators.
Highlights
Purpose of this work is to determine regularities of formation of spatial structures in an ensemble of chaotic systems with non-local diffusion couplings and to study how these structures depend on the wave response of the digital filter formed by the ensemble couplings structure
The network was considered as a digital filter with a frequency response depending on the coupling parameters
The selection of spatial modes is based on the wave characteristic of the coupling filter, the type of which is determined by the radius and the magnitude of couplings
Summary
1. Графики коэффициента передачи фильтра связей при γ = 0.4 (а) и его АЧХ при γ = 0.9 (b). Из выражения (3) легко получить фазочастотную (ФЧХ) и амплитудно-частотную (АЧХ) характеристики фильтра связей. Поскольку максимальное значение коэффициента передачи достигается на нулевой частоте, форма АЧХ будет соответствовать фильтру нижних частот (ФНЧ); при этом основной лепесток можно считать полосой пропускания, а боковые — полосой заграждения фильтра. Исключением является случай локальных связей, при котором переход γ через критическое значение приводит к смене типа фильтра с НЧ на заграждающий Система симметричных дальнодействующих связей при L > 1 ведет себя как пространственный фильтр нижних частот, ширина полосы пропускания которого определяется радиусом действия связей, а селективные свойства (отношение коэффициентов передачи в полосе пропускания и заграждения) — силой связей γ.
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