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 おめが
{\displaystyle \omega }
-limit sets that contain at most finitely many fixed points as a fixed point, a periodic orbit, or a union of fixed points and homoclinic or heteroclinic orbits connecting those fixed points.
Definition for iterated functions
edit
Let
X
{\displaystyle X}
be a metric space , and let
f
:
X
→
X
{\displaystyle f:X\rightarrow X}
be a continuous function . The
ω おめが
{\displaystyle \omega }
-limit set of
x
∈
X
{\displaystyle x\in X}
, denoted by
ω おめが
(
x
,
f
)
{\displaystyle \omega (x,f)}
, is the set of cluster points of the forward orbit
{
f
n
(
x
)
}
n
∈
N
{\displaystyle \{f^{n}(x)\}_{n\in \mathbb {N} }}
of the iterated function
f
{\displaystyle f}
.[ 1] Hence,
y
∈
ω おめが
(
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} }}
such that
f
n
k
(
x
)
→
y
{\displaystyle f^{n_{k}}(x)\rightarrow y}
as
k
→
∞
{\displaystyle k\rightarrow \infty }
. Another way to express this is
ω おめが
(
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}}}
denotes the closure of set
S
{\displaystyle S}
. The points in the limit set are non-wandering (but may not be recurrent points ). This may also be formulated as the outer limit (limsup ) of a sequence of sets, such that
ω おめが
(
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}
is a homeomorphism (that is, a bicontinuous bijection), then the
α あるふぁ
{\displaystyle \alpha }
-limit set is defined in a similar fashion, but for the backward orbit; i.e.
α あるふぁ
(
x
,
f
)
=
ω おめが
(
x
,
f
−
1
)
{\displaystyle \alpha (x,f)=\omega (x,f^{-1})}
.
Both sets are
f
{\displaystyle f}
-invariant, and if
X
{\displaystyle X}
is compact , they are compact and nonempty.
Definition for flows
edit
Given a real dynamical system
(
T
,
X
,
φ ふぁい
)
{\displaystyle (T,X,\varphi )}
with flow
φ ふぁい
:
R
×
X
→
X
{\displaystyle \varphi :\mathbb {R} \times X\to X}
, a point
x
{\displaystyle x}
, we call a point y an
ω おめが
{\displaystyle \omega }
-limit point of
x
{\displaystyle x}
if there exists a sequence
(
t
n
)
n
∈
N
{\displaystyle (t_{n})_{n\in \mathbb {N} }}
in
R
{\displaystyle \mathbb {R} }
so that
lim
n
→
∞
t
n
=
∞
{\displaystyle \lim _{n\to \infty }t_{n}=\infty }
lim
n
→
∞
φ ふぁい
(
t
n
,
x
)
=
y
{\displaystyle \lim _{n\to \infty }\varphi (t_{n},x)=y}
.
For an orbit
γ がんま
{\displaystyle \gamma }
of
(
T
,
X
,
φ ふぁい
)
{\displaystyle (T,X,\varphi )}
, we say that
y
{\displaystyle y}
is an
ω おめが
{\displaystyle \omega }
-limit point of
γ がんま
{\displaystyle \gamma }
, if it is an
ω おめが
{\displaystyle \omega }
-limit point of some point on the orbit.
Analogously we call
y
{\displaystyle y}
an
α あるふぁ
{\displaystyle \alpha }
-limit point of
x
{\displaystyle x}
if there exists a sequence
(
t
n
)
n
∈
N
{\displaystyle (t_{n})_{n\in \mathbb {N} }}
in
R
{\displaystyle \mathbb {R} }
so that
lim
n
→
∞
t
n
=
−
∞
{\displaystyle \lim _{n\to \infty }t_{n}=-\infty }
lim
n
→
∞
φ ふぁい
(
t
n
,
x
)
=
y
{\displaystyle \lim _{n\to \infty }\varphi (t_{n},x)=y}
.
For an orbit
γ がんま
{\displaystyle \gamma }
of
(
T
,
X
,
φ ふぁい
)
{\displaystyle (T,X,\varphi )}
, we say that
y
{\displaystyle y}
is an
α あるふぁ
{\displaystyle \alpha }
-limit point of
γ がんま
{\displaystyle \gamma }
, if it is an
α あるふぁ
{\displaystyle \alpha }
-limit point of some point on the orbit.
The set of all
ω おめが
{\displaystyle \omega }
-limit points (
α あるふぁ
{\displaystyle \alpha }
-limit points) for a given orbit
γ がんま
{\displaystyle \gamma }
is called
ω おめが
{\displaystyle \omega }
-limit set (
α あるふぁ
{\displaystyle \alpha }
-limit set ) for
γ がんま
{\displaystyle \gamma }
and denoted
lim
ω おめが
γ がんま
{\displaystyle \lim _{\omega }\gamma }
(
lim
α あるふぁ
γ がんま
{\displaystyle \lim _{\alpha }\gamma }
).
If the
ω おめが
{\displaystyle \omega }
-limit set (
α あるふぁ
{\displaystyle \alpha }
-limit set) is disjoint from the orbit
γ がんま
{\displaystyle \gamma }
, that is
lim
ω おめが
γ がんま
∩
γ がんま
=
∅
{\displaystyle \lim _{\omega }\gamma \cap \gamma =\varnothing }
(
lim
α あるふぁ
γ がんま
∩
γ がんま
=
∅
{\displaystyle \lim _{\alpha }\gamma \cap \gamma =\varnothing }
), we call
lim
ω おめが
γ がんま
{\displaystyle \lim _{\omega }\gamma }
(
lim
α あるふぁ
γ がんま
{\displaystyle \lim _{\alpha }\gamma }
) a ω おめが -limit cycle (α あるふぁ -limit cycle ).
Alternatively the limit sets can be defined as
lim
ω おめが
γ がんま
:=
⋂
s
∈
R
{
φ ふぁい
(
x
,
t
)
:
t
>
s
}
¯
{\displaystyle \lim _{\omega }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t>s\}}}}
and
lim
α あるふぁ
γ がんま
:=
⋂
s
∈
R
{
φ ふぁい
(
x
,
t
)
:
t
<
s
}
¯
.
{\displaystyle \lim _{\alpha }\gamma :=\bigcap _{s\in \mathbb {R} }{\overline {\{\varphi (x,t):t<s\}}}.}
For any periodic orbit
γ がんま
{\displaystyle \gamma }
of a dynamical system,
lim
ω おめが
γ がんま
=
lim
α あるふぁ
γ がんま
=
γ がんま
{\displaystyle \lim _{\omega }\gamma =\lim _{\alpha }\gamma =\gamma }
For any fixed point
x
0
{\displaystyle x_{0}}
of a dynamical system,
lim
ω おめが
x
0
=
lim
α あるふぁ
x
0
=
x
0
{\displaystyle \lim _{\omega }x_{0}=\lim _{\alpha }x_{0}=x_{0}}
lim
ω おめが
γ がんま
{\displaystyle \lim _{\omega }\gamma }
and
lim
α あるふぁ
γ がんま
{\displaystyle \lim _{\alpha }\gamma }
are closed
if
X
{\displaystyle X}
is compact then
lim
ω おめが
γ がんま
{\displaystyle \lim _{\omega }\gamma }
and
lim
α あるふぁ
γ がんま
{\displaystyle \lim _{\alpha }\gamma }
are nonempty , compact and connected
lim
ω おめが
γ がんま
{\displaystyle \lim _{\omega }\gamma }
and
lim
α あるふぁ
γ がんま
{\displaystyle \lim _{\alpha }\gamma }
are
φ ふぁい
{\displaystyle \varphi }
-invariant, that is
φ ふぁい
(
R
×
lim
ω おめが
γ がんま
)
=
lim
ω おめが
γ がんま
{\displaystyle \varphi (\mathbb {R} \times \lim _{\omega }\gamma )=\lim _{\omega }\gamma }
and
φ ふぁい
(
R
×
lim
α あるふぁ
γ がんま
)
=
lim
α あるふぁ
γ がんま
{\displaystyle \varphi (\mathbb {R} \times \lim _{\alpha }\gamma )=\lim _{\alpha }\gamma }
^ Alligood, Kathleen T.; Sauer, Tim D.; Yorke, James A. (1996). Chaos, an introduction to dynamical systems . Springer.