Dynamical properties of two-dimensional patterns generated by spatially extended systems can be described via the characteristics of attractors in the matrix phase space of the associated translation (or translational-evolution) dynamical systems. Questions regarding the possibility of estimating the fractal dimensions of two-dimensional patterns from the fractal dimensions of one-dimensional observables scanning the patterns along a chosen path are investigated. The presented proofs state that the generalized dimensions of the scanning observables are lower bounds for estimating the corresponding generalized dimensions of two-dimensional patterns. Spatial field distributions defined as superposition of planar waves and different spatiotemporal patterns produced by cellular neural networks made of Chua’s circuits are studied numerically. The results of computer experiments confirm the theoretical predictions presented in this paper.
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