  The covariant Lyapunov analysis is generalised 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 analysed 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 $\pi/2$ at a rate given by the difference between
the positive and negative mode frequencies.
