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¶

\[ \begin{aligned} \dot{y_{0}} &= - a y_{0} + a y_{1} \\ \dot{y_{1}} &= r y_{1} - y_{0} y_{2} \\ \dot{y_{2}} &= - b y_{2} + y_{0} y_{1} \end{aligned} \]

Parameters¶

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

Properties¶

Lyapunov spectrum

$+1.933,\; +0.001941,\; -18.94$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.102$

Divergence ∇·f

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

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics¶

import tsdynamics as ts

sys = ts.systems.RayleighBenard()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference¶

Yanagita & Kaneko (1995), Physica D 82, 288-313

BibTeX

@misc{rayleighbenard,
  title = {RayleighBenard system},
  note = {Yanagita & Kaneko (1995), Physica D 82, 288-313}
}