Maximal stochastic transport in the Lorenz equations

Abstract

We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh-Bénard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected, the stochastic upper bounds are larger than the deterministic counterpart of Souza and Doering [1], but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations and the numerical convergence of the noise correlations. The numerical convergence of both the ensemble and time averages of the noise correlations is sufficiently slow that it is the limiting aspect of the realization of these bounds. Finally, we note that the full solutions of the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos.

Stochastic Chaos and Markov Blankets

“…The ensuing equations were then subject to a spectral Galerkin approximation, to produce a set of three coupled, nonlinear ordinary differential equations [ 18 ]. This system has been studied extensively and is used to model many systems in the physical and life sciences, e.g., [ 39 , 40 ].…”

Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions

“…The Lorenz-63 attractor is a three dimensional, chaotic system that was originally derived by applying a severe Galerkin approximation to the equations of motion (EOMs) for Rayleigh-Bénard convection with stress-free boundary conditions [8]. We study an extension with additive stochastic forcing [10][11][12][13][14] that obeys Eq. 3.…”

“…We illustrate this new method by applying it to the Lorenz-63 system [8] with additive stochastic forcing. Although a phenomenological FPE has been applied for a quantum system without the addition of stochastic forcing [9], we follow previous work [10][11][12][13][14] and add small additive white noise to wash out fractal structure below the lattice scale.…”

Attractor and bifurcation of forced Lorenz-84 system

“…However, random noise is ubiquitous and unavoidable in the time or frequency domain. Random noise has a great impact on the dynamics of the system [17][18][19][20][21][22][23][24][25][26][27][28]. On the one hand, Lorenz [5] once pointed out that F and G should be allowed to vary periodically during a year.…”