# Periodic point

*of a dynamical system*

A point on a trajectory of a periodic motion of a dynamical system $ f ^ { t } $( $ t \in \mathbf R $ or $ t \in \mathbf Z $) defined on a space $ S $, i.e. a point $ x \in S $ such that there is a number $ T > 0 $ for which $ f ^ { T } x = x $ but $ f ^ { t } x \neq x $ for $ t \in ( 0, T) $. This number $ T $ is called the period of the point $ x $( sometimes, the name period is also given to all integer multiples of $ T $).

The trajectory of a periodic point is called a closed trajectory or a loop. When the latter terms are used, one frequently abandons a concrete parametrization of the set of points on the trajectory with parameter $ t $ and considers some class of equivalent parametrizations: If $ f ^ { t } $ is a continuous action of the group $ \mathbf R $ on a topological space $ S $, a loop is considered as a circle that is topologically imbedded in $ S $; if $ f ^ { t } $ is a differentiable action of the group $ \mathbf R $ on a differentiable manifold $ S $, a loop is considered as a circle that is smoothly imbedded in $ S $.

If $ x $ is a periodic point (and $ S $ is a metric space), then the $ \alpha $- limit set $ A _ {x} $ and the $ \omega $- limit set $ \Omega _ {x} $( cf. Limit set of a trajectory) coincide with its trajectory (as point sets). This property, to a certain extent, distinguishes a periodic point among all points that are not fixed, i.e. if the space in which the dynamical system $ f ^ { t } $ is given is a complete metric space and if a point $ x $ is such that $ \Omega _ {x} = \{ f ^ { t } x \} _ {t \in \mathbf R } $, then $ x $ is a fixed or a periodic point of $ f ^ { t } $.

#### References

[1] | V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502 |

#### Comments

In arbitrary dynamical systems (where the phase space is not necessarily metric) the periodic points are characterized as follows (both for actions of $ \mathbf R $ and of $ \mathbf Z $): A point is periodic if and only if its trajectory is a compact set consisting of more than one point. The question whether a given dynamical system has periodic points has been much studied. For dynamical systems on $ 2 $- manifolds, see e.g. [a4], [a6] and also Limit cycle; Poincaré–Bendixson theory and Kneser theorem. For Hamiltonian systems (cf. Hamiltonian system) see e.g. [a5], and for Hilbert's 16th problem (i.e., what is the number of limit cycles of a polynomial vector field in the plane?) see [a2]. Well-known is the Seifert conjecture. Every $ C ^ \infty $- dynamical system on $ S ^ {3} $ has a periodic trajectory; see e.g. [a3]. For a connection between (the existence of) periodic trajectories and certain topological invariants (cf. also Singular point, index of a), see e.g. [a1].

#### References

[a1] | C. Conley, E. Zehnder, "Morse type index theory for flows and periodic solutions for Hamiltonian equations" Comm. Pure Appl. Math. , 37 (1984) pp. 207–253 MR0733717 Zbl 0559.58019 |

[a2] | N.G. Lloyd, "Limit cycles of polynomial systems - some recent developments" T. Bedford (ed.) J. Swift (ed.) , New directions in dynamical systems , Cambridge Univ. Press (1988) pp. 192–234 MR0953973 Zbl 0646.34040 |

[a3] | L. Markus, "Lectures in differentiable dynamics" , Amer. Math. Soc. (1980) pp. Appendix II MR0309152 Zbl 0214.50701 |

[a4] | D.A. Neumann, "Existence of periodic orbits on 2-manifolds" J. Differential Eq. , 27 (1987) pp. 313–319 MR0482857 Zbl 0337.34041 |

[a5] | P.H. Rabinowitz (ed.) A. Ambrosetti (ed.) I. Ekeland (ed.) E.J. Zehnder (ed.) , Periodic solutions of Hamiltonian systems and related topics , Proc. NATO Adv. Res. Workshop, 1986 , Reidel (1987) MR0920604 Zbl 0621.00013 |

[a6] | R.J. Sacker, G.R. Sell, "On the existence of periodic solutions on 2-manifolds" J. Differential Eq. , 11 (1972) pp. 449–463 MR0298706 Zbl 0242.34042 |

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Periodic point.

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