Lyapunov modes in extended systems

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 367; no. 1901; pp. 3197 - 3212
• Main Authors: Yang, Hong-Liu, Radons, Günter
• Format: Journal Article
• Language: English
• Published: England The Royal Society 28.08.2009; The Royal Society Publishing
• Subjects: Acoustic modes; Branch Splitting; Chaos theory; Crystal lattices; Effective Degrees Of Freedom; Hydrodynamics; Hyperbolicity; Lyapunov Modes; Mass spectroscopy; Mathematical lattices; Mathematical manifolds; Phonons; Radon; Statistical mode
• ISSN: 1364-503X; 1471-2962
• DOI: 10.1098/rsta.2009.0067

Hydrodynamic Lyapunov modes, which have recently been observed in many extended systems with translational symmetry, such as hard sphere systems, dynamic XY models or Lennard-Jones fluids, are nowadays regarded as fundamental objects connecting nonlinear dynamics and statistical physics. We review here our recent results on Lyapunov modes in extended system. The solution to one of the puzzles, the appearance of good and 'vague' modes, is presented for the model system of coupled map lattices. The structural properties of these modes are related to the phase space geometry, especially the angles between Oseledec subspaces, and to fluctuations of local Lyapunov exponents. In this context, we report also on the possible appearance of branches splitting in the Lyapunov spectra of diatomic systems, similar to acoustic and optical branches for phonons. The final part is devoted to the hyperbolicity of partial differential equations and the effective degrees of freedom of such infinite-dimensional systems.