The covariant Lyapunov analysis of chaotic dynamical systems

In this thesis the Lyapunov analysis, as applied to the chaotic dynamics of quasi one dimensional hark disk systems, is extended to the covariant Lyapunov vectors of both equilibrium and deterministic nonequilibrium steady states. Previously, the Lyapunov analysis of chaotic dynamics has solely utilised the backward Lyapunov vectors (BLVs) as calculated via the Benettin scheme. In recent years Ginelli et al. found an effective and realistically employable algorithm to calculate the covariant Lyapunov vectors (CLVs). In this thesis the CLVs are analysed, the evolution of the covariant hydrodynamic Lyapunov modes are derived exactly via the reverse tangent space dynamics and their properties are compared with the well-known BLVs. The converged angle between the hydrodynamic stable and unstable conjugate manifolds of the tangent space are predicted to high accuracy. Intriguing aspects and new interpretations are also presented of the evolution and characteristics of the BLVs, derived directly from the forward tangent space dynamics. The covariant Lyapunov analysis is generalised to systems attached to deterministic thermal reservoirs. The thermal reservoirs create a heat current across the system and perturb it away from equilibrium. In this thesis the augmentation of the Lyapunov exponents as a function of heat current is described and explained. Both the nonequilibrium backward and covariant hydrodynamic Lyapunov modes are analysed and compared for the first time. With the free flight time of the dynamics, the movement of the converged angle between the hydrodynamic stable and unstable conjugate manifolds 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 taking the functional forms of each mode in the positive conjugate mode pair. This in turn leads to the angular distributions between th