Periodic point

of a dynamical system

A point on a trajectory of a periodic motion of a dynamical system ( or ) defined on a space , i.e. a point such that there is a number for which but for . This number is called the period of the point (sometimes, the name period is also given to all integer multiples of ).

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 and considers some class of equivalent parametrizations: If is a continuous action of the group on a topological space , a loop is considered as a circle that is topologically imbedded in ; if is a differentiable action of the group on a differentiable manifold , a loop is considered as a circle that is smoothly imbedded in .

If is a periodic point (and is a metric space), then the -limit set and the -limit set (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 is given is a complete metric space and if a point is such that , then is a fixed or a periodic point of .

References

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