Intuition on Strongly Stable Sets in a dynamical system
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I am trying to follow the beginnings of the following paper, I will post the relevant definitions after. https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/conleytype-decomposition-of-the-strong-chain-recurrent-set/44B2B4258A6F1571FE697D701D83EEAF
The setting is $phi$ is a homeomorphism on a compact metric space $X$
The authors are looking to give a stronger version of Chain Recurrent sets and attractors, and in doing so, the first thing they do is define a what it means for a subset $Bsubset X$ to be strongly stable if there exists a $rho >$, a family $(U_eta)_{eta in (0,rho)}$ of closed, nested neighborhoods of $B$ and a function $$(eta)in (0,rho)rightarrow t(eta)in (0,+infty)$$
bounded on compact subsets of $(0,rho)$ such that
$$(i) text {For any } 0<eta<lambda<rho, {xin X|d(x,U_eta)<lambda -eta }subseteq U_lambda $$
$$(ii)B=bigcap_{eta in (0,rho)}omega(U_eta) $$
$$text {For any } 0<eta<rho,cl(phi_{(t(eta),+infty)}(U_eta) subseteq U_eta$$
They then note that any such set is closed and invariant and thus equal to its own omega limit set, and that all attractors are strongly stable.
So, somehow these things are then going to give rise to a strong chain recurrent set.
I'm looking for any help/intuition understanding this property, what this function and nested neighborhood mean. It looks like property $iii$ is some kind of forward invariance after a certain point of time based on this t function, whatever that is. The first property seems to have some kind of nesting closeness property, but I'm not really following it.
Any thoughts/ideas for clarity?
dynamical-systems
asked Dec 6 '18 at 23:15
AlanAlan
8,60221636
$endgroup$
add a comment |
$begingroup$
I am trying to follow the beginnings of the following paper, I will post the relevant definitions after. https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/conleytype-decomposition-of-the-strong-chain-recurrent-set/44B2B4258A6F1571FE697D701D83EEAF
The setting is $phi$ is a homeomorphism on a compact metric space $X$
The authors are looking to give a stronger version of Chain Recurrent sets and attractors, and in doing so, the first thing they do is define a what it means for a subset $Bsubset X$ to be strongly stable if there exists a $rho >$, a family $(U_eta)_{eta in (0,rho)}$ of closed, nested neighborhoods of $B$ and a function $$(eta)in (0,rho)rightarrow t(eta)in (0,+infty)$$
bounded on compact subsets of $(0,rho)$ such that
$$(i) text {For any } 0<eta<lambda<rho, {xin X|d(x,U_eta)<lambda -eta }subseteq U_lambda $$
$$(ii)B=bigcap_{eta in (0,rho)}omega(U_eta) $$
$$text {For any } 0<eta<rho,cl(phi_{(t(eta),+infty)}(U_eta) subseteq U_eta$$
They then note that any such set is closed and invariant and thus equal to its own omega limit set, and that all attractors are strongly stable.
So, somehow these things are then going to give rise to a strong chain recurrent set.
I'm looking for any help/intuition understanding this property, what this function and nested neighborhood mean. It looks like property $iii$ is some kind of forward invariance after a certain point of time based on this t function, whatever that is. The first property seems to have some kind of nesting closeness property, but I'm not really following it.
Any thoughts/ideas for clarity?
dynamical-systems
asked Dec 6 '18 at 23:15
AlanAlan
8,60221636
$endgroup$
$begingroup$
Take on $[0,1]$ the flow of $dot{x}=f(x)$, where $f$ is a $C^1$ function such that $f(x)=0$ for $xin{1/n:ninmathbb{N}}cup{0}$, and positive otherwise. Then ${0}$ is stable, but perhaps not strongly stable. I have to admit that this is my first impression: hopefully some better expert will respond?
$endgroup$
– user539887
Dec 7 '18 at 8:29
add a comment |
$begingroup$
I am trying to follow the beginnings of the following paper, I will post the relevant definitions after. https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/conleytype-decomposition-of-the-strong-chain-recurrent-set/44B2B4258A6F1571FE697D701D83EEAF
The setting is $phi$ is a homeomorphism on a compact metric space $X$
The authors are looking to give a stronger version of Chain Recurrent sets and attractors, and in doing so, the first thing they do is define a what it means for a subset $Bsubset X$ to be strongly stable if there exists a $rho >$, a family $(U_eta)_{eta in (0,rho)}$ of closed, nested neighborhoods of $B$ and a function $$(eta)in (0,rho)rightarrow t(eta)in (0,+infty)$$
bounded on compact subsets of $(0,rho)$ such that
$$(i) text {For any } 0<eta<lambda<rho, {xin X|d(x,U_eta)<lambda -eta }subseteq U_lambda $$
$$(ii)B=bigcap_{eta in (0,rho)}omega(U_eta) $$
$$text {For any } 0<eta<rho,cl(phi_{(t(eta),+infty)}(U_eta) subseteq U_eta$$
They then note that any such set is closed and invariant and thus equal to its own omega limit set, and that all attractors are strongly stable.
So, somehow these things are then going to give rise to a strong chain recurrent set.
I'm looking for any help/intuition understanding this property, what this function and nested neighborhood mean. It looks like property $iii$ is some kind of forward invariance after a certain point of time based on this t function, whatever that is. The first property seems to have some kind of nesting closeness property, but I'm not really following it.
Any thoughts/ideas for clarity?
dynamical-systems
asked Dec 6 '18 at 23:15
AlanAlan
8,60221636
$endgroup$
I am trying to follow the beginnings of the following paper, I will post the relevant definitions after. https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/conleytype-decomposition-of-the-strong-chain-recurrent-set/44B2B4258A6F1571FE697D701D83EEAF
The setting is $phi$ is a homeomorphism on a compact metric space $X$
The authors are looking to give a stronger version of Chain Recurrent sets and attractors, and in doing so, the first thing they do is define a what it means for a subset $Bsubset X$ to be strongly stable if there exists a $rho >$, a family $(U_eta)_{eta in (0,rho)}$ of closed, nested neighborhoods of $B$ and a function $$(eta)in (0,rho)rightarrow t(eta)in (0,+infty)$$
bounded on compact subsets of $(0,rho)$ such that
$$(i) text {For any } 0<eta<lambda<rho, {xin X|d(x,U_eta)<lambda -eta }subseteq U_lambda $$
$$(ii)B=bigcap_{eta in (0,rho)}omega(U_eta) $$
$$text {For any } 0<eta<rho,cl(phi_{(t(eta),+infty)}(U_eta) subseteq U_eta$$
They then note that any such set is closed and invariant and thus equal to its own omega limit set, and that all attractors are strongly stable.
So, somehow these things are then going to give rise to a strong chain recurrent set.
I'm looking for any help/intuition understanding this property, what this function and nested neighborhood mean. It looks like property $iii$ is some kind of forward invariance after a certain point of time based on this t function, whatever that is. The first property seems to have some kind of nesting closeness property, but I'm not really following it.
Any thoughts/ideas for clarity?
dynamical-systems
dynamical-systems
asked Dec 6 '18 at 23:15
AlanAlan
8,60221636
asked Dec 6 '18 at 23:15
AlanAlan
8,60221636
asked Dec 6 '18 at 23:15
AlanAlan
8,60221636
asked Dec 6 '18 at 23:15
AlanAlan
8,60221636
asked Dec 6 '18 at 23:15
AlanAlan
8,60221636
8,60221636
$begingroup$
Take on $[0,1]$ the flow of $dot{x}=f(x)$, where $f$ is a $C^1$ function such that $f(x)=0$ for $xin{1/n:ninmathbb{N}}cup{0}$, and positive otherwise. Then ${0}$ is stable, but perhaps not strongly stable. I have to admit that this is my first impression: hopefully some better expert will respond?
$endgroup$
– user539887
Dec 7 '18 at 8:29
add a comment |
$begingroup$
Take on $[0,1]$ the flow of $dot{x}=f(x)$, where $f$ is a $C^1$ function such that $f(x)=0$ for $xin{1/n:ninmathbb{N}}cup{0}$, and positive otherwise. Then ${0}$ is stable, but perhaps not strongly stable. I have to admit that this is my first impression: hopefully some better expert will respond?
$endgroup$
– user539887
Dec 7 '18 at 8:29
$begingroup$
Take on $[0,1]$ the flow of $dot{x}=f(x)$, where $f$ is a $C^1$ function such that $f(x)=0$ for $xin{1/n:ninmathbb{N}}cup{0}$, and positive otherwise. Then ${0}$ is stable, but perhaps not strongly stable. I have to admit that this is my first impression: hopefully some better expert will respond?
$endgroup$
– user539887
Dec 7 '18 at 8:29
$begingroup$
Take on $[0,1]$ the flow of $dot{x}=f(x)$, where $f$ is a $C^1$ function such that $f(x)=0$ for $xin{1/n:ninmathbb{N}}cup{0}$, and positive otherwise. Then ${0}$ is stable, but perhaps not strongly stable. I have to admit that this is my first impression: hopefully some better expert will respond?
$endgroup$
– user539887
Dec 7 '18 at 8:29
add a comment |
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$begingroup$
Take on $[0,1]$ the flow of $dot{x}=f(x)$, where $f$ is a $C^1$ function such that $f(x)=0$ for $xin{1/n:ninmathbb{N}}cup{0}$, and positive otherwise. Then ${0}$ is stable, but perhaps not strongly stable. I have to admit that this is my first impression: hopefully some better expert will respond?
$endgroup$
– user539887
Dec 7 '18 at 8:29