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The dynamical instability of rough hard-disk fluids in two dimensions is characterized through the Lyapunov spectrum and the Kolmogorov-Sinai entropy, $h_{KS}$, for a wide range of densities and moments of inertia $I$. For small $I$ the spectrum separates into translation-dominated and rotation-dominated parts. With increasing $I$ the rotation-dominated part is gradually filled in at the expense of translation, until such a separation becomes meaningless. At any density, the rate of phase-space mixing, given by $h_{KS}$, becomes less and less effective the more the rotation affects the dynamics. However, the degree of dynamical chaos, measured by the maximum Lyapunov exponent, is only enhanced by the rotational degrees of freedom for high-density gases, but is diminished for lower densities. Surprisingly, no traces of Lyapunov modes were found in the spectrum for larger moments of inertia. The spatial localization of the perturbation vector associated with the maximum exponent however persists for any $I$.
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Although many equations of state of hard-disk fluids have been proposed, none is capable of reproducing the currently calculated or estimated values of the first eighteen virial coefficients at the same time as giving very good accuracy when compared with computer simulation values for the compressibility factor over the whole fluid range. A new virial-based expression is here proposed which achieves these aims. For that, we use the fact that the currently accepted estimated values for the highest virial coefficients behave linearly with their order, and also that virial coefficients must have a limiting behaviour that permits the closest packing limit in the compressibility factor to be also adequately reproduced.
Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.
We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in systems with spatiotemporal chaos. We focus on characteristic LVs and compare the results with backward LVs obtained via successive Gram-Schmidt orthonormalizations. Systems of a very different nature such as coupled-map lattices and the (continuous-time) Lorenz `96 model exhibit the same features in quantitative and qualitative terms. Additionally we propose a minimal stochastic model that reproduces the results for chaotic systems. Our work supports the claims about universality of our earlier results [I. G. Szendro et al., Phys. Rev. E 76, 025202(R) (2007)] for a specific coupled-map lattice.
The scaling behavior of the maximal Lyapunov exponent in chaotic systems with time-delayed feedback is investigated. For large delay times it has been shown that the delay-dependence of the exponent allows a distinction between strong and weak chaos, which are the analogy to strong and weak instability of periodic orbits in a delay system. We find significant differences between scaling of exponents in periodic or chaotic systems. We show that chaotic scaling is related to fluctuations in the linearized equations of motion. A linear delay system including multiplicative noise shows the same properties as the deterministic chaotic systems.
We provide a constructive proof on the equivalence of two fundamental concepts: the global Lyapunov function in engineering and the potential function in physics, establishing a bridge between these distinct fields. This result suggests new approaches on the significant unsolved problem namely to construct Lyapunov functions for general nonlinear systems through the analogy with existing methods on potential functions. In addition, we show another connection that the Lyapunov equation is a reduced form of the generalized Einstein relation for linear systems.
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