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 (x1, y1, z1)

projection (x1, x2, z1)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x1} &= \kappa \left(- x_{1} + x_{2}\right) + \sigma \left(- x_{1} + y_{1}\right) \\ \dot{y1} &= \rho x_{1} - x_{1} z_{1} - y_{1} \\ \dot{z1} &= - \beta z_{1} + x_{1} y_{1} \\ \dot{x2} &= \kappa \left(x_{1} - x_{2}\right) + \sigma \left(- x_{2} + y_{2}\right) \\ \dot{y2} &= \rho x_{2} - x_{2} z_{2} - y_{2} \\ \dot{z2} &= - \beta z_{2} + x_{2} y_{2} \end{aligned} \]

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)

State variables: x1, y1, z1, x2, y2, z2

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

9 equilibria
0 stable · 9 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.LorenzCoupled()
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

Lorenz (1963), J. Atmos. Sci. 20, 130-141
doi:10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2

BibTeX

@misc{lorenzcoupled,
  title = {LorenzCoupled system},
  note = {Lorenz (1963), J. Atmos. Sci. 20, 130-141},
  doi = {10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2}
}