LorenzCoupled

Systems / ODEs / Chaotic attractors

Two identical Lorenz subsystems diffusively coupled through their x-variables — the canonical testbed for chaos synchronization.

continuous · ODE6 dimensionssynchronization

projection (x, y, z)

projection (x, w, z)

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Definition

\[ \begin{aligned} \dot{y_{0}} &= \kappa \left(- y_{0} + y_{3}\right) + \sigma \left(- y_{0} + y_{1}\right) \\ \dot{y_{1}} &= \rho y_{0} - y_{0} y_{2} - y_{1} \\ \dot{y_{2}} &= - \beta y_{2} + y_{0} y_{1} \\ \dot{y_{3}} &= \kappa \left(y_{0} - y_{3}\right) + \sigma \left(- y_{3} + y_{4}\right) \\ \dot{y_{4}} &= \rho y_{3} - y_{3} y_{5} - y_{4} \\ \dot{y_{5}} &= - \beta y_{5} + y_{3} y_{4} \end{aligned} \]

Parameters

Symbol	Default	Role
beta	2.66667	geometric aspect ratio (shared)
kappa	2.85	diffusive coupling strength between the two subsystems
rho	28	Rayleigh ratio (shared)
sigma	10	Prandtl number (shared)

Properties

Lyapunov spectrum

$+0.6616,\; +0.07139,\; -0.01559,\; -0.4128,\; -14.35,\; -18.98$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 4.021$

Divergence ∇·f

$\nabla\!\cdot f = - 2 \beta - 2 \kappa - 2 \sigma - 2$
constant

Equilibria

9 equilibria
0 stable · 9 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Lorenz (1963), J. Atmos. Sci. 20, 130-141

BibTeX

@misc{lorenzcoupled,
  title = {LorenzCoupled system},
  note = {Lorenz (1963), J. Atmos. Sci. 20, 130-141}
}