The Langford system is a three-dimensional polynomial dynamical system introduced to study bifurcations involving
invariant tori (Langford 1984, Fleurantin and Mireles James
2020). It is also called the Aizawa system (Romeo et al. 2025), an attribution
questioned by Fleurantin and Mireles James (2020). For the variables 
, 
, and 
, primes denote derivatives
with respect to time 
. The system is given by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
, 
, 
, 
, 
, and 
are real parameters.
In polar coordinates in the 
-plane, the radius 
satisfies 
, and the angle 
satisfies 
for 
. The 
-axis is invariant and the restriction to it is the scalar
ordinary differential equation 
.
For 
, 
, 
, 
, and 
, Fleurantin and Mireles James (2020) numerically
studied the emergence of an attracting invariant torus
from a periodic orbit as 
increases. They also found numerical evidence of chaos
after the torus breaks up. Capiński et al. (2020)
proved the existence of a continuous attracting invariant torus
for 
with the same remaining parameters.
The Wolfram ordinary differential equation catalog gives a five-parameter variant under the name Aizawa attractor equations. This variant, also printed by Dumont (2025, p. 24), retains the first two equations and replaces the third by
 |
(4)
|
where the catalog uses 
for the coefficient denoted by 
here.
