RayleighBenard¶

Systems / ODEs / Climate & geophysics

A three-mode Saltzman truncation of thermal convection heated from below — the convection family from which the Lorenz attractor descends.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= - a x + a y \\ \dot{y} &= r y - x z \\ \dot{z} &= - b z + x y \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`30`	Prandtl-like diffusion ratio
`b`	`5`	geometric aspect-ratio damping
`r`	`18`	reduced Rayleigh number (convective driving)

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a - b + r$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.RayleighBenard()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

Yanagita & Kaneko (1995), Physica D 82, 288-313
doi:10.1016/0167-2789(94)00233-g

BibTeX

@misc{rayleighbenard,
  title = {RayleighBenard system},
  note = {Yanagita & Kaneko (1995), Physica D 82, 288-313},
  doi = {10.1016/0167-2789(94)00233-g}
}