Quantized linear systems on integer lattices: a frequency-based approach

Abstract

The roundoff errors in computer simulations of continuous dynamical systems, caused by finiteness of machine arithmetic, can lead to qualitative discrepancies between phase portraits of the resulting spatially discretized systems and the original systems. These effects can be modelled on a multidimensional integer lattice by using a dynamical system obtained by composing the transition operator of the original system with a quantizer. Such models manifest pseudorandomness which can be studied using a rigorous probability theoretic approach. To this end, the lattice $\mathbb{Z}^n$ is endowed with a class of frequency measurable subsets and a spatial frequency functional as a finitely additive probability measure on them. Using a multivariate version of Weyl's equidistribution criterion, we introduce an algebra of frequency measurable quasiperiodic subsets of the lattice. This approach is applied to quantized linear systems with the transition operator $R \circ L$, where $L$ is a nonsingular matrix of the original linear system in $\mathbb{R}^n$, and the map $R$ commutes with the additive group of translations of the lattice. For almost every $L$, the events associated with the deviation of trajectories of the quantized and original systems are frequency measurable quasiperiodic subsets of the lattice whose frequencies involve geometric probabilities on finite-dimensional tori. Using the skew products of measure preserving toral automorphisms, we prove mutual independence and uniform distribution of the quantization errors and investigate statistical properties of invertibility loss for the quantized linear system, extending V.V.Voevodin's results. When $L$ is similar to an orthogonal matrix, we establish a functional central limit theorem for the deviations of trajectories of the quantized and original systems. These results are demonstrated for rounded-off planar rotations.