Lorenz96

Systems / ODEs / Chaotic attractors

A ring of advectively coupled sites — the standard testbed for atmospheric data assimilation and spatiotemporal chaos.

continuous · ODE20 dimensionsspatiotemporal

spatiotemporal field

Definition

@staticmethod
def _equations(y_sym, t_sym, *, f, N):
    return [
        (y_sym((i + 1) % N) - y_sym((i - 2) % N)) * y_sym((i - 1) % N) - y_sym(i) + f
        for i in range(N)
    ]

Parameters

Symbol	Default	Role
f	8	forcing (drives the chaos onset)
N	20	number of lattice sites

Properties

Lyapunov spectrum

TODO — dim 20 > 6 — full spectrum too slow

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = -20$
uniform over 20 nodes

Equilibria

TODO — dim 20 > 8 — equilibrium search skipped

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Lorenz96()
traj = sys.integrate(final_time=100.0, dt=0.01)

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

Reference

Lorenz (1996), Proc. ECMWF Seminar on Predictability 1, 1-18

BibTeX

@misc{lorenz96,
  title = {Lorenz96 system},
  note = {Lorenz (1996), Proc. ECMWF Seminar on Predictability 1, 1-18}
}