Everything about Periodic Point totally explained
In
mathematics, in the study of
iterated functions and
dynamical systems, a
periodic point of a
function is a point which returns to itself after a certain number of function iterations or a certain amount of time.
Iterated functions
Given an
endomorphism f on a
set X »
a point
x in
X is called
periodic point if there exists an
n so that
»
where
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 function with period
n.
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
»
and that it's
attractive if
»
and it's
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's called a
sink; and if both the stable and unstable manifold have nonzero dimension, it's called a
saddle or
saddle point.
Examples
Dynamical system
Given a
real global dynamical system (
R,
X, Φ) with
X the
phase space and Φ the
evolution function,
»
a point
x in
X is called
periodic with
period t if there exists a
t ≥ 0 so that
»
The smallest positive
t with this property is called
prime period of the point
x.
Properties
Given a periodic point x with period t, then for all s in R
Given a periodic point x then all points on the orbit through x are periodic with the same prime period.Further Information
Get more info on 'Periodic Point'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://periodic_point.totallyexplained.com">Periodic point Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
