attracting set dynamical systems
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There are some definitions about attracting set, but one of them is confusing for me. First, definitions:
$$
begin{array}{l}
1) textrm{flow:} enspace phi_t({bf x}) \
2) textrm{for a subset} enspace M: enspace phi_t(M) = bigcuplimits_{{bf x} in M} phi_t({bf x}) \
3) M enspace textrm{is a trapping region if it is compact and } phi_t(M) subset int(M), t>0 \
4) A enspace textrm{is an attracting set if there exist a trapping region
} M enspace textrm{such that} enspace A subset M enspace textrm{and}\
A=bigcap_{t>0} phi_t(M)
end{array}
$$
Suppose you have a system in $mathbb{R}^2$ with a fixed point $P$ that is a sink. Consider a small neighborhood $U$ of $P$, with two points of it, namely $(x_0,y_0) ne (x_1,y_1)$, that belong to different orbits $Gamma_0$ and $Gamma_1$ respectively. By uniqueness $Gamma_0 cap Gamma_1 = emptyset$.
If the limit $t to infty$ is allowed in the previous definition then $Gamma_0 cap Gamma_1 = {P}$, which is the attracting set, as expected.
Coud you help me to understand this definition? Thank you
differential-equations
asked Nov 18 at 23:32
Joe
11
|
show 3 more comments
up vote
0
down vote
favorite
There are some definitions about attracting set, but one of them is confusing for me. First, definitions:
$$
begin{array}{l}
1) textrm{flow:} enspace phi_t({bf x}) \
2) textrm{for a subset} enspace M: enspace phi_t(M) = bigcuplimits_{{bf x} in M} phi_t({bf x}) \
3) M enspace textrm{is a trapping region if it is compact and } phi_t(M) subset int(M), t>0 \
4) A enspace textrm{is an attracting set if there exist a trapping region
} M enspace textrm{such that} enspace A subset M enspace textrm{and}\
A=bigcap_{t>0} phi_t(M)
end{array}
$$
Suppose you have a system in $mathbb{R}^2$ with a fixed point $P$ that is a sink. Consider a small neighborhood $U$ of $P$, with two points of it, namely $(x_0,y_0) ne (x_1,y_1)$, that belong to different orbits $Gamma_0$ and $Gamma_1$ respectively. By uniqueness $Gamma_0 cap Gamma_1 = emptyset$.
If the limit $t to infty$ is allowed in the previous definition then $Gamma_0 cap Gamma_1 = {P}$, which is the attracting set, as expected.
Coud you help me to understand this definition? Thank you
differential-equations
asked Nov 18 at 23:32
Joe
11
You have to define all the terms you use. What is $phi (t,M)$?
– Kavi Rama Murthy
Nov 18 at 23:35
Thank you Kavi, I have included some definitions
– Joe
Nov 19 at 0:18
So which is the definition that you are confused about?
– carsandpulsars
Nov 19 at 0:23
I have problems with number 4). I think that the intersection is empty, and below 4) I gave an example.
– Joe
Nov 19 at 0:30
So I think the confusion here is that $phi_t(M)$ will be another set (i.e other region in $mathbb{R}^n$. We aren't intersecting individual orbits. So essentially if you were to think of all solutions starting at some point in $M$ then evolving the system forward by $t$, then $phi_t(M)$ is where all those points have now ended up after some time $t$.
– carsandpulsars
Nov 19 at 1:37
|
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
There are some definitions about attracting set, but one of them is confusing for me. First, definitions:
$$
begin{array}{l}
1) textrm{flow:} enspace phi_t({bf x}) \
2) textrm{for a subset} enspace M: enspace phi_t(M) = bigcuplimits_{{bf x} in M} phi_t({bf x}) \
3) M enspace textrm{is a trapping region if it is compact and } phi_t(M) subset int(M), t>0 \
4) A enspace textrm{is an attracting set if there exist a trapping region
} M enspace textrm{such that} enspace A subset M enspace textrm{and}\
A=bigcap_{t>0} phi_t(M)
end{array}
$$
Suppose you have a system in $mathbb{R}^2$ with a fixed point $P$ that is a sink. Consider a small neighborhood $U$ of $P$, with two points of it, namely $(x_0,y_0) ne (x_1,y_1)$, that belong to different orbits $Gamma_0$ and $Gamma_1$ respectively. By uniqueness $Gamma_0 cap Gamma_1 = emptyset$.
If the limit $t to infty$ is allowed in the previous definition then $Gamma_0 cap Gamma_1 = {P}$, which is the attracting set, as expected.
Coud you help me to understand this definition? Thank you
differential-equations
asked Nov 18 at 23:32
Joe
11
There are some definitions about attracting set, but one of them is confusing for me. First, definitions:
$$
begin{array}{l}
1) textrm{flow:} enspace phi_t({bf x}) \
2) textrm{for a subset} enspace M: enspace phi_t(M) = bigcuplimits_{{bf x} in M} phi_t({bf x}) \
3) M enspace textrm{is a trapping region if it is compact and } phi_t(M) subset int(M), t>0 \
4) A enspace textrm{is an attracting set if there exist a trapping region
} M enspace textrm{such that} enspace A subset M enspace textrm{and}\
A=bigcap_{t>0} phi_t(M)
end{array}
$$
Suppose you have a system in $mathbb{R}^2$ with a fixed point $P$ that is a sink. Consider a small neighborhood $U$ of $P$, with two points of it, namely $(x_0,y_0) ne (x_1,y_1)$, that belong to different orbits $Gamma_0$ and $Gamma_1$ respectively. By uniqueness $Gamma_0 cap Gamma_1 = emptyset$.
If the limit $t to infty$ is allowed in the previous definition then $Gamma_0 cap Gamma_1 = {P}$, which is the attracting set, as expected.
Coud you help me to understand this definition? Thank you
differential-equations
differential-equations
asked Nov 18 at 23:32
Joe
11
asked Nov 18 at 23:32
Joe
11
edited Nov 19 at 0:17
asked Nov 18 at 23:32
Joe
11
asked Nov 18 at 23:32
Joe
11
asked Nov 18 at 23:32
Joe
11
11
You have to define all the terms you use. What is $phi (t,M)$?
– Kavi Rama Murthy
Nov 18 at 23:35
Thank you Kavi, I have included some definitions
– Joe
Nov 19 at 0:18
So which is the definition that you are confused about?
– carsandpulsars
Nov 19 at 0:23
I have problems with number 4). I think that the intersection is empty, and below 4) I gave an example.
– Joe
Nov 19 at 0:30
So I think the confusion here is that $phi_t(M)$ will be another set (i.e other region in $mathbb{R}^n$. We aren't intersecting individual orbits. So essentially if you were to think of all solutions starting at some point in $M$ then evolving the system forward by $t$, then $phi_t(M)$ is where all those points have now ended up after some time $t$.
– carsandpulsars
Nov 19 at 1:37
|
show 3 more comments
You have to define all the terms you use. What is $phi (t,M)$?
– Kavi Rama Murthy
Nov 18 at 23:35
Thank you Kavi, I have included some definitions
– Joe
Nov 19 at 0:18
So which is the definition that you are confused about?
– carsandpulsars
Nov 19 at 0:23
I have problems with number 4). I think that the intersection is empty, and below 4) I gave an example.
– Joe
Nov 19 at 0:30
So I think the confusion here is that $phi_t(M)$ will be another set (i.e other region in $mathbb{R}^n$. We aren't intersecting individual orbits. So essentially if you were to think of all solutions starting at some point in $M$ then evolving the system forward by $t$, then $phi_t(M)$ is where all those points have now ended up after some time $t$.
– carsandpulsars
Nov 19 at 1:37
You have to define all the terms you use. What is $phi (t,M)$?
– Kavi Rama Murthy
Nov 18 at 23:35
You have to define all the terms you use. What is $phi (t,M)$?
– Kavi Rama Murthy
Nov 18 at 23:35
Thank you Kavi, I have included some definitions
– Joe
Nov 19 at 0:18
Thank you Kavi, I have included some definitions
– Joe
Nov 19 at 0:18
So which is the definition that you are confused about?
– carsandpulsars
Nov 19 at 0:23
So which is the definition that you are confused about?
– carsandpulsars
Nov 19 at 0:23
I have problems with number 4). I think that the intersection is empty, and below 4) I gave an example.
– Joe
Nov 19 at 0:30
I have problems with number 4). I think that the intersection is empty, and below 4) I gave an example.
– Joe
Nov 19 at 0:30
So I think the confusion here is that $phi_t(M)$ will be another set (i.e other region in $mathbb{R}^n$. We aren't intersecting individual orbits. So essentially if you were to think of all solutions starting at some point in $M$ then evolving the system forward by $t$, then $phi_t(M)$ is where all those points have now ended up after some time $t$.
– carsandpulsars
Nov 19 at 1:37
So I think the confusion here is that $phi_t(M)$ will be another set (i.e other region in $mathbb{R}^n$. We aren't intersecting individual orbits. So essentially if you were to think of all solutions starting at some point in $M$ then evolving the system forward by $t$, then $phi_t(M)$ is where all those points have now ended up after some time $t$.
– carsandpulsars
Nov 19 at 1:37
|
show 3 more comments
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You have to define all the terms you use. What is $phi (t,M)$?
– Kavi Rama Murthy
Nov 18 at 23:35
Thank you Kavi, I have included some definitions
– Joe
Nov 19 at 0:18
So which is the definition that you are confused about?
– carsandpulsars
Nov 19 at 0:23
I have problems with number 4). I think that the intersection is empty, and below 4) I gave an example.
– Joe
Nov 19 at 0:30
So I think the confusion here is that $phi_t(M)$ will be another set (i.e other region in $mathbb{R}^n$. We aren't intersecting individual orbits. So essentially if you were to think of all solutions starting at some point in $M$ then evolving the system forward by $t$, then $phi_t(M)$ is where all those points have now ended up after some time $t$.
– carsandpulsars
Nov 19 at 1:37