The order-disorder transition observed in periodically driven Rayleigh-Bénard convection is studied by extending the generalized Lorenz model introduced by Ahlers, Hohenberg, and Lücke ͓Phys. Rev. A 32, 3493 ͑1985͔͒ to include the effects of thermal noise. It is shown that this stochastic Lorenz model predicts, for thermal noise intensities, an order-disorder transition line much closer to the experimental values than the prediction of previous models. This result makes clear that a dynamical description allowing for inertial effects is needed to account for the behavior of systems dynamically forced to cross an instability threshold.
Starting from 80 families of low-energy fast periodic transfer orbits in the Earth-Moon planar circular Restricted Three Body Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the Sun-Earth-Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic months, their energies are among the lowest possible to achieve an EarthMoon transfer, and they show a diversity of circumlunar trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close approaches to the last.
We solve the Fokker-Planck equation for an overdamped Brownian particle in a periodically forced bistable potential by means of a path integral method, obtaining the propagators in the steepest-descent (small-noise) approximation. We compute the long-times asymptotic probability distribution, the asymptotic correlation functions, and the time-averaged spectral density, which allows us the immediate calculation of the signal to noise ratio, a directly measurable quantity useful to characterize the phenomenon of stochastic resonance. Our numerical algorithm is fast and runs on a desktop computer, and the results agree with experiments and with former theoretical calculations of the amplification factor; in addition it allows us to calculate the experimentally more accessible signal to noise ratio.
We present a new model for periodically driven Rayleigh-Bénard convection with thermal noise, derived as a truncated vertical mode expansion of a mean field approximation to the Oberbeck-Boussinesq equations. The resulting model includes the continuous dependence on the horizontal wave number, and preserves the full symmetries of the hydrodynamic equations as well as their inertial character. The model is shown to reduce to a Swift-Hohenberg-like equation in the same limiting cases in which the Lorenz model reduces to an amplitude equation. The order-disorder transition experimentally observed in the recurrent pattern formation near the convective onset is studied by using both the present model and its above-mentioned limiting form, as well as a generalization of the amplitude equation for modulated driving introduced by Schmitt and Lücke ͓Phys. Rev. A 44, 4986 ͑1991͔͒ and the generalized Lorenz model previously introduced by the authors ͓Phys. Rev. E 55, R3824 ͑1997͔͒. We show that all these models agree with the experimental data much closer than previous models like the Swift-Hohenberg equation or the amplitude equation, though thermal noise alone still seems insufficient to lead to a precise fit. The relationship between these models is discussed, and it is shown that the inclusion of the continuous wave-number dependence, a consistent treatment of the driving time dependence, and the inclusion of inertial effects are all relevant to the formulation of a model describing equally well both the time-periodic and static driving cases. ͓S1063-651X͑97͒09812-7͔
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