Poincaré map under small pertubations
Let $gamma$ be a periodic orbit of a vector field $X$. Through a point $x_{0} in gamma$ we consider a section $Sigma$ transversal to the field $X$ and V a small neighbourhood which contains $x_{0}$.
Consider $P_{X}: V subset Sigma rightarrow Sigma$ the Poincaré map, which to each point $x in V$ associates $P(x)$, the first point where the orbit of $x$ returns to intersect $Sigma$.
If $Y$ is a vector field sufficiently near to vector field $X$, my questions are:
1) Is $Sigma$ transversal to $Y$?
2) Is true that the orbit of $Y$ through each point of $V$ still returns to intersect $Sigma$?
In the first question if $Sigma$ were not transversal to $Y$ (in a point p), then $Y(p) in T_{p}Sigma$, since $Y$ is close to the $X$ we have $X(p) in T_{p}Sigma$, which contradicts transversality.
(Is my argument correct? Can I take the topology over $TSigma$?)
Is the second question related to the structural stability? I don't know if the statement is true.
I'm self studying Dynamical Systems, so I'm having some trouble to make some stuffs precise.
Thanks in advance.
differential-equations dynamical-systems stability-theory
asked Nov 25 at 14:39
BBVM
17112
add a comment |
Let $gamma$ be a periodic orbit of a vector field $X$. Through a point $x_{0} in gamma$ we consider a section $Sigma$ transversal to the field $X$ and V a small neighbourhood which contains $x_{0}$.
Consider $P_{X}: V subset Sigma rightarrow Sigma$ the Poincaré map, which to each point $x in V$ associates $P(x)$, the first point where the orbit of $x$ returns to intersect $Sigma$.
If $Y$ is a vector field sufficiently near to vector field $X$, my questions are:
1) Is $Sigma$ transversal to $Y$?
2) Is true that the orbit of $Y$ through each point of $V$ still returns to intersect $Sigma$?
In the first question if $Sigma$ were not transversal to $Y$ (in a point p), then $Y(p) in T_{p}Sigma$, since $Y$ is close to the $X$ we have $X(p) in T_{p}Sigma$, which contradicts transversality.
(Is my argument correct? Can I take the topology over $TSigma$?)
Is the second question related to the structural stability? I don't know if the statement is true.
I'm self studying Dynamical Systems, so I'm having some trouble to make some stuffs precise.
Thanks in advance.
differential-equations dynamical-systems stability-theory
asked Nov 25 at 14:39
BBVM
17112
2
Regarding the first question: the flow of a $C^1$ perturbed field is still transversal, although perhaps on a proper subset of the original section. A proof can be the following: if you have a Riemannian metric on the manifold, transversality means that the field on $Sigma$ is not orthogonal to the normal, which is an open property in the $C^1$ topology. The second question has nothing to do with structural stability, either. I think that the best approach would be through the flow box theorem: the orbit of the perturbed flow gets close to the section, so it will be pushed across it.
– user539887
Nov 26 at 18:55
@user539887 Thanks for helping me with this. Where I can find about the flow box theorem? I know the Tubular flow theorem, is the same thing?
– BBVM
Nov 26 at 23:56
I think that's the same. Other names: straightening out theorem, rectification theorem. You can find it, e.g., in Chicone Ordinary Differential Equations with Applications.
– user539887
Nov 27 at 8:11
add a comment |
Let $gamma$ be a periodic orbit of a vector field $X$. Through a point $x_{0} in gamma$ we consider a section $Sigma$ transversal to the field $X$ and V a small neighbourhood which contains $x_{0}$.
Consider $P_{X}: V subset Sigma rightarrow Sigma$ the Poincaré map, which to each point $x in V$ associates $P(x)$, the first point where the orbit of $x$ returns to intersect $Sigma$.
If $Y$ is a vector field sufficiently near to vector field $X$, my questions are:
1) Is $Sigma$ transversal to $Y$?
2) Is true that the orbit of $Y$ through each point of $V$ still returns to intersect $Sigma$?
In the first question if $Sigma$ were not transversal to $Y$ (in a point p), then $Y(p) in T_{p}Sigma$, since $Y$ is close to the $X$ we have $X(p) in T_{p}Sigma$, which contradicts transversality.
(Is my argument correct? Can I take the topology over $TSigma$?)
Is the second question related to the structural stability? I don't know if the statement is true.
I'm self studying Dynamical Systems, so I'm having some trouble to make some stuffs precise.
Thanks in advance.
differential-equations dynamical-systems stability-theory
asked Nov 25 at 14:39
BBVM
17112
Let $gamma$ be a periodic orbit of a vector field $X$. Through a point $x_{0} in gamma$ we consider a section $Sigma$ transversal to the field $X$ and V a small neighbourhood which contains $x_{0}$.
Consider $P_{X}: V subset Sigma rightarrow Sigma$ the Poincaré map, which to each point $x in V$ associates $P(x)$, the first point where the orbit of $x$ returns to intersect $Sigma$.
If $Y$ is a vector field sufficiently near to vector field $X$, my questions are:
1) Is $Sigma$ transversal to $Y$?
2) Is true that the orbit of $Y$ through each point of $V$ still returns to intersect $Sigma$?
In the first question if $Sigma$ were not transversal to $Y$ (in a point p), then $Y(p) in T_{p}Sigma$, since $Y$ is close to the $X$ we have $X(p) in T_{p}Sigma$, which contradicts transversality.
(Is my argument correct? Can I take the topology over $TSigma$?)
Is the second question related to the structural stability? I don't know if the statement is true.
I'm self studying Dynamical Systems, so I'm having some trouble to make some stuffs precise.
Thanks in advance.
differential-equations dynamical-systems stability-theory
differential-equations dynamical-systems stability-theory
asked Nov 25 at 14:39
BBVM
17112
asked Nov 25 at 14:39
BBVM
17112
edited Nov 25 at 19:47
asked Nov 25 at 14:39
BBVM
17112
asked Nov 25 at 14:39
BBVM
17112
asked Nov 25 at 14:39
BBVM
17112
17112
2
Regarding the first question: the flow of a $C^1$ perturbed field is still transversal, although perhaps on a proper subset of the original section. A proof can be the following: if you have a Riemannian metric on the manifold, transversality means that the field on $Sigma$ is not orthogonal to the normal, which is an open property in the $C^1$ topology. The second question has nothing to do with structural stability, either. I think that the best approach would be through the flow box theorem: the orbit of the perturbed flow gets close to the section, so it will be pushed across it.
– user539887
Nov 26 at 18:55
@user539887 Thanks for helping me with this. Where I can find about the flow box theorem? I know the Tubular flow theorem, is the same thing?
– BBVM
Nov 26 at 23:56
I think that's the same. Other names: straightening out theorem, rectification theorem. You can find it, e.g., in Chicone Ordinary Differential Equations with Applications.
– user539887
Nov 27 at 8:11
add a comment |
2
Regarding the first question: the flow of a $C^1$ perturbed field is still transversal, although perhaps on a proper subset of the original section. A proof can be the following: if you have a Riemannian metric on the manifold, transversality means that the field on $Sigma$ is not orthogonal to the normal, which is an open property in the $C^1$ topology. The second question has nothing to do with structural stability, either. I think that the best approach would be through the flow box theorem: the orbit of the perturbed flow gets close to the section, so it will be pushed across it.
– user539887
Nov 26 at 18:55
@user539887 Thanks for helping me with this. Where I can find about the flow box theorem? I know the Tubular flow theorem, is the same thing?
– BBVM
Nov 26 at 23:56
I think that's the same. Other names: straightening out theorem, rectification theorem. You can find it, e.g., in Chicone Ordinary Differential Equations with Applications.
– user539887
Nov 27 at 8:11
2
2
Regarding the first question: the flow of a $C^1$ perturbed field is still transversal, although perhaps on a proper subset of the original section. A proof can be the following: if you have a Riemannian metric on the manifold, transversality means that the field on $Sigma$ is not orthogonal to the normal, which is an open property in the $C^1$ topology. The second question has nothing to do with structural stability, either. I think that the best approach would be through the flow box theorem: the orbit of the perturbed flow gets close to the section, so it will be pushed across it.
– user539887
Nov 26 at 18:55
Regarding the first question: the flow of a $C^1$ perturbed field is still transversal, although perhaps on a proper subset of the original section. A proof can be the following: if you have a Riemannian metric on the manifold, transversality means that the field on $Sigma$ is not orthogonal to the normal, which is an open property in the $C^1$ topology. The second question has nothing to do with structural stability, either. I think that the best approach would be through the flow box theorem: the orbit of the perturbed flow gets close to the section, so it will be pushed across it.
– user539887
Nov 26 at 18:55
@user539887 Thanks for helping me with this. Where I can find about the flow box theorem? I know the Tubular flow theorem, is the same thing?
– BBVM
Nov 26 at 23:56
@user539887 Thanks for helping me with this. Where I can find about the flow box theorem? I know the Tubular flow theorem, is the same thing?
– BBVM
Nov 26 at 23:56
I think that's the same. Other names: straightening out theorem, rectification theorem. You can find it, e.g., in Chicone Ordinary Differential Equations with Applications.
– user539887
Nov 27 at 8:11
I think that's the same. Other names: straightening out theorem, rectification theorem. You can find it, e.g., in Chicone Ordinary Differential Equations with Applications.
– user539887
Nov 27 at 8:11
add a comment |
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2
Regarding the first question: the flow of a $C^1$ perturbed field is still transversal, although perhaps on a proper subset of the original section. A proof can be the following: if you have a Riemannian metric on the manifold, transversality means that the field on $Sigma$ is not orthogonal to the normal, which is an open property in the $C^1$ topology. The second question has nothing to do with structural stability, either. I think that the best approach would be through the flow box theorem: the orbit of the perturbed flow gets close to the section, so it will be pushed across it.
– user539887
Nov 26 at 18:55
@user539887 Thanks for helping me with this. Where I can find about the flow box theorem? I know the Tubular flow theorem, is the same thing?
– BBVM
Nov 26 at 23:56
I think that's the same. Other names: straightening out theorem, rectification theorem. You can find it, e.g., in Chicone Ordinary Differential Equations with Applications.
– user539887
Nov 27 at 8:11