# Limit cycle

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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. ).

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 , ). 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.

How to Cite This Entry:
Limit cycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit_cycle&oldid=15601
This article was adapted from an original article by L.A. Cherkas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article