Bifurcations and the positiveP-representation

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The role of quantum fluctuations in dynamical systems can be described conveniently in terms of quasi-probabilities, since this concept bears a strong but formal analogy to classical statistical mechanics. At a closer look, however, only the method of the positiveP-representation has the potential of associating a classical stochastic process with a nonlinear quantum mechanical problem. The doubling of phase space required by this concept not only introduces new and unphysical dimensions, but also doubles the dimensions of the attractors of the associated deterministic system. A focus of the classical process remains a focus in the extended phase space, but a limit cycle is turned into a two-dimensional manifold, which due to the analytic properties of the method is hyperbolic in nature and extends to infinity. A strange attractor with its broken dimensionality also doubles its dimensions, since the Lyapunov exponents of the deterministic evolution in the extended phase space come in pairs as well. The fluctuating forces distribute the probability density about the attractors. In case of a focus, the probability density remains confined to the neighbourhood of the point of attraction, while for a limit cycle the distribution continues to spread over the entire two-dimensional manifold and a stationary solution for the probability density in the doubled phase space does not exist.