Dynamical instabilities, manifolds, and local Lyapunov spectra far from equilibrium

We consider an harmonic oscillator in a thermal gradient far from equilibrium. The motion is made ergodic and fully time-reversible through the use of two thermostat variables. The equations of motion contain both linear and quadratic terms. The dynamics is chaotic. The resulting phase-space distribution is not only complex and multifractal, but also ergodic, due to the time-reversibility property. We analyze dynamical time series in two ways. We describe local, but comoving, singularities in terms of the "local Lyapunov spectrum" {gamma}. Local singularities at a particular phase-space point can alternatively be described by the local eigenvalues and eigenvectors of the "dynamical matrix"D identical to partial differential ni/partial differential r identical to Nabla ni. D is the matrix of derivatives of the equations of motion r=ni(r) . We pursue this eigenvalue-eigenvector description for the oscillator. We find that it breaks down at a dense set of singular points, where the four eigenvectors span only a three-dimensional subspace. We believe that the concepts of stable and unstable global manifolds are problematic for this simple nonequilibrium system. PACS numbers: 05.45.A, 05.10, 07.05.T