A limit cycle is an isolated periodic orbit of a dynamical system. An attracting limit cycle is approached by nearby trajectories, while a repelling limit cycle is approached in reverse time. The Oregonator provides an example of an attracting cycle in three-dimensional phase space.
A planar example in Cartesian coordinates 
and 
,
with primes denoting derivatives with respect to time
, is
 |  |  |
(1)
|
 |  |  |
(2)
|
In polar coordinates with 
,
it becomes
 |  |  |
(3)
|
 |  |  |
(4)
|
Thus 
increases for 
and decreases for 
.
The unit circle is an attracting cycle. Reversing
the sign of both right-hand sides reverses time, making the same periodic
orbit repelling.
Qin and Sun (2026) proved that the maximum number 
of limit cycles of a real planar polynomial vector field of degree
at most 
satisfies 
as 
, where 
is big-Omega notation.
This is a lower bound, not an upper bound or an exact value. Qin and Sun (2026) report
using GPT-5.6 Sol for proof strategies and Lean formalization.
