The Arnéodo-Coullet-Tresser system is an autonomous third-order ordinary differential equation with a cubic nonlinearity. Write 
, with primes denoting derivatives
with respect to 
.
A form with real coefficients 
, 
,
and 
is
 |
With 
and 
, this gives the first-order system 
, 
,
and 
(Harrington and Van Gorder 2017). A negative coefficient 
with 
in front of 
reduces to 
by using 
as the dependent variable. When 
, it has the equilibrium
points 
, 
,
and 
. The characteristic
equation of the linearization at 
is
 |
Cubic oscillators of this type provide examples of strange attractors with spiral structure (Arnéodo et al. 1981). They also illustrate chaos near a homoclinic orbit, a trajectory that approaches the same equilibrium point in both time directions. In this example the equilibrium is a saddle focus, with one real eigenvalue and a nonreal complex conjugate pair whose real part has the opposite sign (Arnéodo et al. 1982).
