Lyapunov instability of large dynamical systems: methods and applications
Final Report Abstract
Important achievements of the project were obtained for the study of fundamental problems, for instance the effective degrees of freedom of dissipative partial di¤erential equations, for the development of conceptual methods, for instance the idenfication of spurious Lyapunov exponents in time series analysis, and for the application of these new methods to physically relevant situations, for example the treatment of supercooled fluids by using Lyapunov correlation functions. Partial differential equations are in principle finite dimensional systems. Owing to the dissipative nature the active dynamics occurs instead on finite dimensional subspace, i.e. the effective degrees of freedom are ofinite number. This is the basic idea behind the so-called concept of inertial manifolds, smootfinite dimensional objects, where the effective dynamics takes place. One important progress achieved in the project is to provide, for the first time, a method which can accurately estimate the dimension of the inertial manifold. This results may on one hand promote the further development of the understanding of the fundamentals of partial differential equations, and would on the other hand be important for the numerical integration of partial differential equations and the controlling issues of continuum systems. The key publication describing this effect (Yang et al. PRL 2009) has received a lot of attention already, and has been cited almost 50 times up to now. Going beyond the linear stability analysis we proposed a projection method which enables to probe the local nonlinear geometric structure of the inertial manifold of partial differential equation systems. Although the method was demonstrated by using a simple prototypical system of space-time chaos, it is valid in general for a large class of spatially extended dissipative dynamical systems. Lyapunov exponents are important characteristic of the underlying nonlinear dynamics of time series. The appearance of spurious Lyapunov exponents owing to the reconstruction procedure is a known issue and also a big hurdle for the application of time series analysis techniques. We proposed in this project to use covariant Lyapunov vectors to identify spurious Lyapunov exponents. The success of our method was demonstrated for not only simple dynamical systems but also for experimental data from a laser system. Valuable information can still be gained by using our new method even with the appearance of measurement noise, which is unavoidable for all real situations. For the study of supercooled fluids we performed systematic investigations of the binary Lennard-Jones systems, both in 1D and 2D, by using the Lyapunov vector correlation functions. It was found that the existence of hydrodynamical Lyapunov modes, the long wave length slowly varying structures in Lyapunov vectors associated with near-zero Lyapunov exponents, is a generic feature for these systems. With varying both the mass ratio and temperature we found that the structure and dynamics of these hydrodynamical Lyapunov modes can be well characterized by using concepts from phonon theory, which implies a certain deep relation between the two sets of modes, an issue which needs to be explored further in the future.
Publications
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Dimensional collapse and fractal properties of a map system with fluctuating delay times
J. Wang, G. Radons, H.-L. Yang
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Complex behavior of simple maps with fluctuating delay times, Eur. Phys. J. B 71, 111 (2009)
G. Radons, H.-L. Yang, J. Wang and J.-F. Fu
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Hyperbolicity and the Effective Dimension of Spatially Extended Dissipative Systems, Phys. Rev. Lett. 102, 074102 (2009)
H.-L. Yang, K. A. Takeuchi, F. Ginelli, H. Chaté, and G. Radons
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Lyapunov modes in extended systems, in: Topics on Nonequilibrium Statistical Mechanics and Nonlinear Physics, Phil. Trans. of the Royal Society A 367, 3197 (2009)
H.-L. Yang and G. Radons
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Comparison between covariant and orthogonal Lyapunov vectors, Phys. Rev. E 82, 046204 (2010)
H.-L. Yang and G. Radons
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Dynamic structure factors and Lyapunov modes in disordered chains, Phys. Rev. E 82, 026206 (2010)
K. Helbig, W. Just, G. Radons, and H.-L. Yang
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Hyperbolic decoupling of tangent space and effective dimension of dissipative systems, Phys. Rev. E 84, 046214 (2011)
K. A. Takeuchi, H.-L. Yang, F. Ginelli, G. Radons, and H. Chaté
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Covariant Lyapunov Vectors from Reconstructed Dynamics: The Geometry behind True and Spurious Lyapunov Exponents, Phys. Rev. Lett. 109, 244101, Editors’Suggestion (2012)
H.-L. Yang, G. Radons, and H. Kantz
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Geometry of Inertial Manifolds Probed via a Lyapunov Projection Method, Phys. Rev. Lett. 108, 154101 (2012)
H.-L. Yang and G. Radons
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Hydrodynamic Lyapunov modes and e¤ective degrees of freedom of extended systems, J. Phys. A 46, 254015 (2013)
H.-L. Yang, G. Radons
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Lyapunov Modes in Extended Systems, p.361-391 in: R. Klages, W. Just, C. Jarzynski (eds.), Nonequilibrium Statistical Physics of Small Systems (Wiley-VCH, Weinheim 2013)
H.-L. Yang, G. Radons
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The problem of spurious Lyapunov exponentes in time series analysis and its solution by covariant Lyapunov vectors, J. Phys. A 46, 254009 (2013)
H. Kantz, G. Radons, H.-L. Yang
