Limes amount

This article deals with limit sets of discrete or continuous dynamic systems, for the related terms of limit sets, Klein groups or general convergence groups, see Klein group # Limes set or convergence group #Limes set .

In the theory of dynamic systems , limit sets (or limit value sets ) are those points in the state space that orbits approach infinitely often (for positive or negative time).

${\displaystyle \omega }$-Limes amount (limit cycle) of the van der Pol oscillator

definition

Be a dynamic system with or . Let be a point of the state space. ${\displaystyle (T,X,\Phi )}$ ${\displaystyle T=\mathbb {Z} }$ ${\displaystyle T=\mathbb {R} }$ ${\displaystyle x\in X}$

The limit set of is ${\displaystyle \omega }$ ${\displaystyle x}$

${\displaystyle \omega (x,\Phi ):=\left\{y\in X:\exists t_{n}\rightarrow \infty ,\Phi (t_{n},x)\rightarrow y\right\}}$.

The limit set of is ${\displaystyle \alpha }$ ${\displaystyle x}$

${\displaystyle \alpha (x,\Phi ):=\left\{y\in X:\exists t_{n}\rightarrow -\infty ,\Phi (t_{n},x)\rightarrow y\right\}}$.

Alternatively, Limes sets can also be characterized as follows:

${\displaystyle \omega (x,\Phi )=\bigcap _{n\in T}{\overline {\left\{\Phi (t,x):t>n\right\}}}}$,

${\displaystyle \alpha (x,\Phi )=\bigcap _{n\in T}{\overline {\left\{\Phi (t,x):t<n\right\}}}}$.

The Limes sets are closed and invariant under . If is compact, the Limes sets are not empty. ${\displaystyle \Phi }$ ${\displaystyle X}$

Types

• fixed point
• Periodic orbit
• Limit cycle
• Strange attractor

literature

• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).