Ergodicity of one-dimensional systems coupled to the logistic thermostat

We analyze the ergodicity of three one-dimensional Hamiltonian systems, with harmonic, quartic and Mexican-hat potentials, coupled to the logistic thermostat. As criteria for ergodicity we employ: the independence of the Lyapunov spectrum with respect to initial conditions; the absence of visual "holes" in two-dimensional Poincar\'e sections; the agreement between the histograms in each variable and the theoretical marginal distributions; and the convergence of the global joint distribution to the theoretical one, as measured by the Hellinger distance. Taking a large number of random initial conditions, for certain parameter values of the thermostat we find no indication of regular trajectories and show that the time distribution converges to the ensemble one for an arbitrarily long trajectory for all the systems considered. Our results thus provide a robust numerical indication that the logistic thermostat can serve as a single one-parameter thermostat for stiff one-dimensional systems.