In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A system that has reached its limiting set is said to be at equilibrium.
In general, limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all nonempty, compact
\omega
Let
X
f:X → X
\omega
x\inX
\omega(x,f)
| n(x)\} | |
| \{f | |
| n\inN |
f
y\in\omega(x,f)
\{nk\}k\in
| nk | |
| f |
(x) → y
k → infty
\omega(x,f)=capn\in\overline{\{fk(x):k>n\}},
where
\overline{S}
S
\omega(x,f)=
| infty | |
| cap | |
| n=1 |
| infty | |
| \overline{cup | |
| k=n |
\{fk(x)\}}.
If
f
\alpha
\alpha(x,f)=\omega(x,f-1)
Both sets are
f
X
(T,X,\varphi)
\varphi:R x X\toX
x
\omega
x
(tn)n
R
\limntn=infty
\limn\varphi(tn,x)=y
\gamma
(T,X,\varphi)
y
\omega
\gamma
\omega
Analogously we call
y
\alpha
x
(tn)n
R
\limntn=-infty
\limn\varphi(tn,x)=y
\gamma
(T,X,\varphi)
y
\alpha
\gamma
\alpha
The set of all
\omega
\alpha
\gamma
\omega
\alpha
\gamma
\lim\omega\gamma
\lim\alpha\gamma
If the
\omega
\alpha
\gamma
\lim\omega\gamma\cap\gamma=\varnothing
\lim\alpha\gamma\cap\gamma=\varnothing
\lim\omega\gamma
\lim\alpha\gamma
Alternatively the limit sets can be defined as
\lim\omega\gamma:=caps\in\overline{\{\varphi(x,t):t>s\}}
\lim\alpha\gamma:=caps\in\overline{\{\varphi(x,t):t<s\}}.
\gamma
\lim\omega\gamma=\lim\alpha\gamma=\gamma
x0
\lim\omegax0=\lim\alphax0=x0
\lim\omega\gamma
\lim\alpha\gamma
X
\lim\omega\gamma
\lim\alpha\gamma
\lim\omega\gamma
\lim\alpha\gamma
\varphi
\varphi(R x \lim\omega\gamma)=\lim\omega\gamma
\varphi(R x \lim\alpha\gamma)=\lim\alpha\gamma
. Gerald. Gerald Teschl. Ordinary Differential Equations and Dynamical Systems. American Mathematical Society. Providence. 2012. 978-0-8218-8328-0.