This article is about the notion of a limit set in the area of dynamical systems. For the notion of a limit in set theory, see
Set-theoretic limit .
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
ω
{\displaystyle \omega }
homoclinic or heteroclinic orbits connecting those fixed points.
Definition for iterated functions
edit
Let
X
{\displaystyle X}
metric space , and let
f
:
X
→
X
{\displaystyle f:X\rightarrow X}
continuous function . The
ω
{\displaystyle \omega }
x
∈
X
{\displaystyle x\in X}
ω
(
x
,
f
)
{\displaystyle \omega (x,f)}
cluster points of the forward orbit
{
f
n
(
x
)
}
n
∈
N
{\displaystyle \{f^{n}(x)\}_{n\in \mathbb {N} }}
iterated function
f
{\displaystyle f}
[1]
y
∈
ω
(
x
,
f
)
{\displaystyle y\in \omega (x,f)}
if and only if there is a strictly increasing sequence of natural numbers
{
n
k
}
k
∈
N
{\displaystyle \{n_{k}\}_{k\in \mathbb {N} }}
f
n
k
(
x
)
→
y
{\displaystyle f^{n_{k}}(x)\rightarrow y}
k
→
∞
{\displaystyle k\rightarrow \infty }
ω
(
x
,
f
)
=
⋂
n
∈
N
{
f
k
(
x
)
:
k
>
n
}
¯
,
{\displaystyle \omega (x,f)=\bigcap _{n\in \mathbb {N} }{\overline {\{f^{k}(x):k>n\}}},}
where
S
¯
{\displaystyle {\overline {S}}}
closure of set
S
{\displaystyle S}
recurrent points limsup ) of a sequence of sets, such that
ω
(
x
,
f
)
=
⋂
n
=
1
∞
⋃
k
=
n
∞
{
f
k
(
x
)
}
¯
.
{\displaystyle \omega (x,f)=\bigcap _{n=1}^{\infty }{\overline {\bigcup _{k=n}^{\infty }\{f^{k}(x)\}}}.}
If
f
{\displaystyle f}
homeomorphism (that is, a bicontinuous bijection), then the
α
{\displaystyle \alpha }
i.e.
α
(
x
,
f
)
=
ω
(
x
,
f
−
1
)
{\displaystyle \alpha (x,f)=\omega (x,f^{-1})}
Both sets are
f
{\displaystyle f}
X
{\displaystyle X}
compact , they are compact and nonempty.
Definition for flows
edit
Given a real dynamical system
(
T
,
X
,
φ
)
{\displaystyle (T,X,\varphi )}
flow
φ
:
R
×
X
→
X
{\displaystyle \varphi :\mathbb {R} \times X\to X}
x
{\displaystyle x}
y an
ω
{\displaystyle \omega }
limit point of
x
{\displaystyle x}
(
t
n
)
n
∈
N
{\displaystyle (t_{n})_{n\in \mathbb {N} }}
R
{\displaystyle \mathbb {R} }
lim
n
→
∞
t
n
=
∞
{\displaystyle \lim _{n\to \infty }t_{n}=\infty }
lim
n
→
∞
φ
(
t
n
,
x
)
=
y
{\displaystyle \lim _{n\to \infty }\varphi (t_{n},x)=y}
For an orbit
γ
{\displaystyle \gamma }
(
T
,
X
,
φ
)
{\displaystyle (T,X,\varphi )}
y
{\displaystyle y}
ω
{\displaystyle \omega }
limit point of
γ
{\displaystyle \gamma }
ω
{\displaystyle \omega }
limit point of some point on the orbit.
Analogously we call
y
{\displaystyle y}
α
{\displaystyle \alpha }
limit point of
x
{\displaystyle x}
(
t
n
)
n
∈
N
{\displaystyle (t_{n})_{n\in \mathbb {N} }}
R
{\displaystyle \mathbb {R} }
lim
n
→
∞
t
n
=
−
∞
{\displaystyle \lim _{n\to \infty }t_{n}=-\infty }
lim
n
→
∞
φ
(
t
n
,
x
)
=
y
{\displaystyle \lim _{n\to \infty }\varphi (t_{n},x)=y}
For an orbit
γ
{\displaystyle \gamma }
(
T
,
X
,
φ
)
{\displaystyle (T,X,\varphi )}
y
{\displaystyle y}
α
{\displaystyle \alpha }
limit point of
γ
{\displaystyle \gamma }
α
{\displaystyle \alpha }
limit point of some point on the orbit.
The set of all
ω
{\displaystyle \omega }
α
{\displaystyle \alpha }
γ
{\displaystyle \gamma }
ω
{\displaystyle \omega }
limit set (
α
{\displaystyle \alpha }
limit set ) for
γ
{\displaystyle \gamma }
lim
ω
γ
{\displaystyle \lim _{\omega }\gamma }
lim
α
γ
{\displaystyle \lim _{\alpha }\gamma }
If the
ω
{\displaystyle \omega }
α
{\displaystyle \alpha }
γ
{\displaystyle \gamma }
lim
ω
γ
∩
γ
=
∅
{\displaystyle \lim _{\omega }\gamma \cap \gamma =\varnothing }
lim
α
γ
∩
γ
=
∅
{\displaystyle \lim _{\alpha }\gamma \cap \gamma =\varnothing }
lim
ω
γ
{\displaystyle \lim _{\omega }\gamma }
lim
α
γ
{\displaystyle \lim _{\alpha }\gamma }
ω-limit cycle α-limit cycle
Alternatively the limit sets can be defined as
lim
ω
γ
:=
⋂
s
∈
R
{
φ
(
x
,
t
)
:
t
>
s
}
¯
{\displaystyle \lim _{\omega }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t>s\}}}}
and
lim
α
γ
:=
⋂
s
∈
R
{
φ
(
x
,
t
)
:
t
<
s
}
¯
.
{\displaystyle \lim _{\alpha }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t<s\}}}.}
Examples
edit
For any periodic orbit
γ
{\displaystyle \gamma }
lim
ω
γ
=
lim
α
γ
=
γ
{\displaystyle \lim _{\omega }\gamma =\lim _{\alpha }\gamma =\gamma }
For any fixed point
x
0
{\displaystyle x_{0}}
lim
ω
x
0
=
lim
α
x
0
=
x
0
{\displaystyle \lim _{\omega }x_{0}=\lim _{\alpha }x_{0}=x_{0}}
Properties
edit
lim
ω
γ
{\displaystyle \lim _{\omega }\gamma }
lim
α
γ
{\displaystyle \lim _{\alpha }\gamma }
closed if
X
{\displaystyle X}
lim
ω
γ
{\displaystyle \lim _{\omega }\gamma }
lim
α
γ
{\displaystyle \lim _{\alpha }\gamma }
nonempty , compact and connected
lim
ω
γ
{\displaystyle \lim _{\omega }\gamma }
lim
α
γ
{\displaystyle \lim _{\alpha }\gamma }
φ
{\displaystyle \varphi }
φ
(
R
×
lim
ω
γ
)
=
lim
ω
γ
{\displaystyle \varphi (\mathbb {R} \times \lim _{\omega }\gamma )=\lim _{\omega }\gamma }
φ
(
R
×
lim
α
γ
)
=
lim
α
γ
{\displaystyle \varphi (\mathbb {R} \times \lim _{\alpha }\gamma )=\lim _{\alpha }\gamma }
See also
edit
References
edit
^ Alligood, Kathleen T.; Sauer, Tim D.; Yorke, James A. (1996). Chaos, an introduction to dynamical systems . Springer.
Further reading
edit
