Local symmetry dynamics in one-dimensional aperiodic lattices: a numerical study

• Published in: Nonlinear dynamics Vol. 78; no. 1; pp. 71 - 91
• Main Authors: Morfonios, C., Schmelcher, P., Kalozoumis, P. A., Diakonos, F. K.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.10.2014; Springer Nature B.V
• Subjects: Automotive Engineering; Axes (reference lines); Classical Mechanics; Control; Dynamic tests; Dynamical Systems; Dynamics; Electrons; Engineering; Lattices; Lattices (mathematics); Long range order; Mechanical Engineering; Nonlinear dynamics; Orbits; Original Paper; Properties (attributes); Reflection; Self-similarity; Sequences; Symmetry; Vibration; Symbolic manipulation; Aperiodic lattices; Local symmetries; Discrete dynamics; Long-range order
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-014-1422-1

A unifying description of lattice potentials generated by aperiodic one-dimensional sequences is proposed in terms of their local reflection or parity symmetry properties. We demonstrate that the ranges and axes of local reflection symmetry possess characteristic distributional and dynamical properties, determined here numerically for certain lattice types. A striking aspect of such a property is given by the return maps of sequential spacings of local symmetry axes, which typically traverse few-point symmetry orbits. This local symmetry dynamics allows for a description of inherently different aperiodic lattices according to fundamental symmetry principles. Illustrating the local symmetry distributional and dynamical properties for several representative binary lattices, we further show that the renormalized axis-spacing sequences follow precisely the particular type of underlying aperiodic order, revealing the presence of dynamical self-similarity. Our analysis thus provides evidence that the long-range order of aperiodic lattices can be characterized in a compellingly simple way by its local symmetry dynamics.