Search for: [Abstract = "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 multifiactal, 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\" \{λ\}. Local singularities at a particular phase\-space point can alternatively be described by the local eigenvalues and eigenvectors of the \"dynamical matrix\" D=Əv\/Ər=∆v. D is the matrix of derivates of the equations of motion r=v\(r\). 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."]
