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The covariant Lyapunov analysis is generalized to systems attached to deterministic thermal reservoirs that create a heat current across the system and perturb it away from equilibrium. The change in the Lyapunov exponents as a function of heat current is described and explained. Both the nonequilibrium backward and covariant hydrodynamic Lyapunov modes are analyzed and compared. The movement of the converged angle between the hydrodynamic stable and unstable conjugate manifolds with the free flight time of the dynamics is accurately predicted for any nonequilibrium system simply as a function of their exponent. The nonequilibrium positive and negative LP mode frequencies are found to be asymmetrical, causing the negative mode to oscillate between the two functional forms of each mode in the positive conjugate mode pair. This in turn leads to the angular distributions between the conjugate modes to oscillate symmetrically about π/2 at a rate given by the difference between the positive and negative mode frequencies. [ABSTRACT FROM AUTHOR]

A review of the results obtained for hard disk fluids confined to a quasione- dimensional (QOD) system is presented. One of the main achievements in recent years has been determining the hydrodynamic Lyapunov modes (HLMs) as covariant stable and unstable manifolds defined by k-vector analogues of the zero-exponent subspace of conserved quantities. The tangent space dynamics allows an interpretation of the HLMs as reduced hydrodynamic fields over the perturbations and is expanded upon in this paper. The time evolution of these modes is governed by the dynamics of conjugate pairs of stable and unstable directions. Each pair is seen to interact in a subspace almost completely separated from other vectors; the modes undergo a rotation until they are collinear with the stable or unstable manifold. The angle distributions between these covariant stable and unstable HLMs are also determined, and tangencies are not observed for a generic chaotic trajectory. [ABSTRACT FROM AUTHOR]