Hong-liu Yang

Hyperbolicity and the Effective Dimension of Spatially Extended Dissipative Systems

Using covariant Lyapunov vectors, we reveal a split of the tangent space of standard models of one-dimensional dissipative spatiotemporal chaos: A finite extensive set of N dynamically entangled vectors with frequent common tangencies describes all of the physically relevant dynamics and is hyperbolically separated from possibly infinitely many isolated modes representing trivial, exponentially decaying perturbations. We argue that N can be interpreted as the number of effective degrees of freedom, which has to be taken into account in numerical integration and control issues.

The problem of spurious Lyapunov exponents in time series analysis and its solution by covariant Lyapunov vectors

We briefly recall the methods for the determination of Lyapunov exponents from time series data and their limitations. One particular problem is given by the fact that reconstructed phase spaces have usually extra dimensions compared to the true phase space of a dynamical system, leading to extra, so called spurious Lyapunov exponents. Several methods to identify the true ones have been proposed which do not give satisfactory results. We show that the geometric information contained in covariant Lyapunov vectors can be used to identify the true exponents. We illustrate its use and its limitations by applying it to experimental NMR laser data.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Lyapunov analysis: from dynamical systems theory to applications?.

Lyapunov modes in extended systems

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.

Geometry of inertial manifolds probed via a Lyapunov projection method.

A method for determining the dimension and state space geometry of inertial manifolds of dissipative extended dynamical systems is presented. It works by projecting vector differences between reference states and recurrent states onto local linear subspaces spanned by the Lyapunov vectors. A sharp characteristic transition of the projection error occurs as soon as the number of basis vectors is increased beyond the inertial manifold dimension. Since the method can be applied using standard orthogonal Lyapunov vectors, it provides a possible way to also determine experimentally inertial manifolds and their geometric characteristics.

Parallel algorithms for the determination of lyapunov characteristics of large nonlinear dynamical systems

Lyapunov vectors and exponents are of great importance for understanding the dynamics of many-particle systems. We present results of performance tests on different processor architectures of several parallel implementations for the calculation of all Lyapunov characteristics. For the most time consuming reorthogonalization steps, which have to be combined with molecular dynamics simulations, we tested different parallel versions of the Gram-Schmidt algorithm and of QR-decomposition. The latter gave the best results with respect to runtime and stability. For large systems the blockwise parallel Gram-Schmidt algorithm yields comparable runtime results.

Lyapunov Instabilities of Extended Systems

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