Poincaré map under small pertubations











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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.










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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















up vote
1
down vote

favorite












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.










share|cite|improve this question




















  • 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













up vote
1
down vote

favorite









up vote
1
down vote

favorite











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.










share|cite|improve this question















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






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share|cite|improve this question













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edited Nov 25 at 19:47

























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














  • 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















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