Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.3646

Symmetries and Dynamics of Discrete Systems

Vladimir V. Kornyak
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program constructs and investigates phase portraits of discrete dynamical systems modulo groups of their symmetries, searches dynamical systems possessing specific properties, e.g., reversibility, computes microcanonical partition functions and searches phase transitions in mesoscopic systems. Some computational results and observations are presented. In particular, we explain formation of moving soliton-like structures similar to ``spaceships'' in cellular automata.

Explore related subjects

math-phmath.DSmath.MPnlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vladimir V. Kornyak. 2007-10-08. Symmetries and Dynamics of Discrete Systems. https://arxiv.org/abs/0706.3646

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Wave scattering by a transversal defect in a discrete waveguide

We study wave scattering by a finite transversal strip in a discrete square-lattice waveguide with Dirichlet boundary conditions imposed on the strip and the waveguide walls. The setting is motivated as a discrete analogue of the classical continuous waveguide problem with a screen. The corresponding Wiener--Hopf formulation leads to an equation with a $4 \times 4$ matrix kernel, which reduces to a $2 \times 2$ matrix kernel under some symmetry assumptions. The prospects for factorising this kernel are discussed, but this approach is not pursued. Instead, an exact analytical solution is obtained using the pole removal technique. This contrasts with the continuous case, where only approximate solutions are currently available. The reflection and transmission coefficients resulting from an incident duct mode are computed with an accuracy up to $10^{-13}$, showing consistency with theoretical predictions from continuous waveguide theory. In particular, full reflection and zero transmission are recovered as the frequency approaches the cut-off value for the incident mode. Finally, the solution is validated against a numerical computation of the diffraction problem via the Boundary Algebraic Equations method with a tailored lattice Green's function.

math-ph↗
arXiv2026

Heat Kernel and Resurgence

We study the resurgent structure of short--time heat kernel asymptotics from the viewpoint of Picard--Lefschetz theory. For a real analytic Riemannian manifold, we show the heat kernel admits a 1-- Gevrey small--time expansion whose Borel transform detects complex--geometric data beyond the real geodesic sector. We formulate an infinite--dimensional Picard--Lefschetz problem of Morse--Floer type for the holomorphic energy functional on the complexified path space, and propose a heat--kernel analogue of the Picard--Lefschetz/Alien correspondence. In this framework, pointed alien operators acting on the asymptotic expansion associated with the real geodesic are predicted to produce the formal heat--kernel sectors associated with other holomorphic geodesics, with coefficients given by signed counts of connecting trajectories of the Morse flow. We perform a confirming test of this proposal on the hyperbolic plane $\mathbb{H}_{\R}^2$.

math-ph↗
arXiv2026

Cantor Spectrum via a Reducibility-Duality Bridge for the Mosaic Almost Mathieu Operator

We study the mosaic Almost Mathieu operator, a quasiperiodic model that naturally admits a singular strip-Jacobi representation. By establishing a duality framework and extending the correspondence between the integrated density of states and the fibered rotation number to this setting, we obtain an effective reduction to $SL(2,\mathbb{R})$ cocycles. As a consequence, combining Aubry duality, reducibility theory, and the Moser--Pöschel argument, we prove that the spectrum is a Cantor set for all noncritical parameters.

math-ph↗