A closed trajectory in the phase space of an
autonomous system
of ordinary differential equations that is an
-
or
-limit
set (cf.
Limit set of a trajectory)
of at least one other trajectory of this system. A limit cycle is called
orbit stable,
or
stable,
if for any
there is a
such that all trajectories starting in a
-neighbourhood
of it for
do not leave its
-neighbourhood
for
(cf.
Orbit stability).
A limit cycle corresponds to a periodic solution
of the system, differing from a constant. In order for
a periodic solution to correspond to a stable limit cycle
it is sufficient that the moduli of all its
multipliers,
except one, be less than one (cf.
Characteristic exponent;
Andronov–Witt theorem).
From the physical point of view, a limit cycle corresponds to periodic behaviour, or an
auto-oscillation,
of the system (cf.
[2]).
Suppose that an autonomous system
defined in a region
,
where
is a differentiable manifold, e.g.
,
has a closed trajectory
.
Draw the hyperplane
intersecting
transversally at a point
.
Then every trajectory of the system starting for
at a point
,
with
a sufficiently small neighbourhood of
,
intersects
again, at a point
,
as
increases. The diffeomorphism
has fixed point
and is called the
Poincaré return map.
Its properties determine the behaviour of trajectories
of the system in a neighbourhood of
.
A limit cycle, as distinct from an arbitrary closed trajectory,
always determines a Poincaré return map that is not the identity. If
is a saddle point of the diffeomorphism
,
then the limit cycle
is said to be
of saddle type.
A system having a limit cycle of saddle type can have
homoclinic curves,
i.e. trajectories for which the limit cycle is both the
-
and the
-limit
set.
In the case of a two-dimensional system
(*)
one takes a straight line for
and considers the function
,
,
which is called the
Poincaré return function.
The multiplicity of the zero
of
is called the
multiplicity of the limit cycle.
A limit cycle of even multiplicity is called
semi-stable.
The limit cycles, together with the rest points and the separatrices (cf.
Separatrix),
determine the qualitative picture of the behaviour of the other trajectories (cf.
Poincaré–Bendixson theory,
as well as
[3],
[4]).
In the case of an analytic function
the limit cycles belong to one of the following three types:
1) stable; 2) unstable, i.e. stable if the direction of
is reversed; or 3) semi-stable. E.g., the system
where
,
,
,
has for
(
)
and
odd a stable (unstable) limit cycle of multiplicity
,
while for
even it has a limit cycle of multiplicity
.
In all cases the limit cycle is the circle
,
i.e. the trajectory of the solution
If the system
(*)
is given on a simply-connected domain
,
then a limit cycle encircles at least one rest point of the system.
In order to find limit cycles of second-order systems one uses
methods based on the following fact: If a vector field
is directed inwards (outwards) an annular domain
and if
does not contain rest points, then there is at least one stable (unstable) limit cycle in
.
The choice of
is based on physical considerations or results from analytic or numerical computations.