A periodic orbit of a dynamical system is the trajectory of a state that returns to itself after a positive time. For a flow 
, a nonconstant periodic orbit through
satisfies
 |
for all 
and some 
.
The least such 
is its least period. An equilibrium
point is a constant trajectory and is usually distinguished from a nonconstant
periodic orbit. For a discrete-time map 
, a periodic point of least
period 
produces the finite orbit 
(Guckenheimer and Holmes 1983).
A limit cycle is an isolated periodic orbit. Periodicity alone does not imply isolation. For example, the solutions of 
, 
lie on circles, and every nonconstant
orbit has period 
.
