Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Periodic point
Summary
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Given a mapping f from a set X into itself,
a point x in X is called periodic point if there exists an n so that
where f_n is the nth iterate of f. The smallest positive integer n satisfying the above is called the prime period or least period of the point x. If every point in X is a periodic point with the same period n, then f is called periodic with period n (this is not to be confused with the notion of a periodic function).
If there exist distinct n and m such that
then x is called a preperiodic point. All periodic points are preperiodic.
If f is a diffeomorphism of a differentiable manifold, so that the derivative is defined, then one says that a periodic point is hyperbolic if
that it is attractive if
and it is repelling if
If the dimension of the stable manifold of a periodic point or fixed point is zero, the point is called a source; if the dimension of its unstable manifold is zero, it is called a sink; and if both the stable and unstable manifold have nonzero dimension, it is called a saddle or saddle point.
A period-one point is called a fixed point.
The logistic map
exhibits periodicity for various values of the parameter r. For r between 0 and 1, 0 is the sole periodic point, with period 1 (giving the sequence 0, 0, 0, ..., which attracts all orbits). For r between 1 and 3, the value 0 is still periodic but is not attracting, while the value is an attracting periodic point of period 1. With r greater than 3 but less than 1 + \sqrt 6, there are a pair of period-2 points which together form an attracting sequence, as well as the non-attracting period-1 points 0 and As the value of parameter r rises toward 4, there arise groups of periodic points with any positive integer for the period; for some values of r one of these repeating sequences is attracting while for others none of them are (with almost all orbits being chaotic).
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood