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