Relation between the orientation preserving flow and the eigenvalues of the stable and unstable manifolds?

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Consider the 3D Dynamical system $dotX = F_mu(X)$, Say that the stable and unstable manifolds of the fixed point at $mu = 0$ intersect each other(Homoclinic tangency to a periodic orbit $C$ of period $tau_c$).

I am referring this article - Homoclinic Orbits and Mixed-Mode Oscillations
in Far-from-Equilibrium Systems by Gaspard and Wang.

Let $lambda_s,lambda_u$ denote the eigenvalues of the stable and unstable manifolds, where we have the condition that $|lambda_s|<1<|lambda_u|<frac1$.

Then I was trying to understand how to prove the following statement below -

"Since the flow preserves orientation in the phase space $BbbR^3$, either $lambda_s,lambda_u <0$ OR $lambda_s,lambda_u>0$" ?

Any reference or idea of the proof?

eigenvalues-eigenvectors manifolds dynamical-systems

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edited Jul 26 at 7:19

asked Jul 26 at 7:08

BAYMAX

2,43121021

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do stable and unstable manifolds intersect each other to periodic orbit? What does that mean?
  – user539887
  Jul 26 at 7:16
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please see the edit.
  – BAYMAX
  Jul 26 at 7:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Perhaps the authors introduced their own peculiar notation $lambda_s$ and $lambda_u$ (without knowing the context I cannot say). But as the possibility $lambda_u<0$ is admitted, $lambda_u$ cannot denote the eigenvalue corresponding to the unstable manifold.
  – user539887
  Jul 26 at 7:28
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, for the context I have given the article name!
  – BAYMAX
  Jul 26 at 7:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Perhaps $lambda_u$ and $lambda_s$ denote the eigenvalues of the linearization of the Poincaré map at the periodic orbit $C$? That would make sense.
  – user539887
  Jul 26 at 7:32
  
  

  
  
  
  

 | 
show 2 more comments

up vote
0
down vote

favorite

Consider the 3D Dynamical system $dotX = F_mu(X)$, Say that the stable and unstable manifolds of the fixed point at $mu = 0$ intersect each other(Homoclinic tangency to a periodic orbit $C$ of period $tau_c$).

I am referring this article - Homoclinic Orbits and Mixed-Mode Oscillations
in Far-from-Equilibrium Systems by Gaspard and Wang.

Let $lambda_s,lambda_u$ denote the eigenvalues of the stable and unstable manifolds, where we have the condition that $|lambda_s|<1<|lambda_u|<frac1$.

Then I was trying to understand how to prove the following statement below -

"Since the flow preserves orientation in the phase space $BbbR^3$, either $lambda_s,lambda_u <0$ OR $lambda_s,lambda_u>0$" ?

Any reference or idea of the proof?

eigenvalues-eigenvectors manifolds dynamical-systems

share|cite|improve this question

edited Jul 26 at 7:19

asked Jul 26 at 7:08

BAYMAX

2,43121021

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do stable and unstable manifolds intersect each other to periodic orbit? What does that mean?
  – user539887
  Jul 26 at 7:16
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please see the edit.
  – BAYMAX
  Jul 26 at 7:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Perhaps the authors introduced their own peculiar notation $lambda_s$ and $lambda_u$ (without knowing the context I cannot say). But as the possibility $lambda_u<0$ is admitted, $lambda_u$ cannot denote the eigenvalue corresponding to the unstable manifold.
  – user539887
  Jul 26 at 7:28
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, for the context I have given the article name!
  – BAYMAX
  Jul 26 at 7:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Perhaps $lambda_u$ and $lambda_s$ denote the eigenvalues of the linearization of the Poincaré map at the periodic orbit $C$? That would make sense.
  – user539887
  Jul 26 at 7:32
  
  

  
  
  
  

 | 
show 2 more comments

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Consider the 3D Dynamical system $dotX = F_mu(X)$, Say that the stable and unstable manifolds of the fixed point at $mu = 0$ intersect each other(Homoclinic tangency to a periodic orbit $C$ of period $tau_c$).

I am referring this article - Homoclinic Orbits and Mixed-Mode Oscillations
in Far-from-Equilibrium Systems by Gaspard and Wang.

Let $lambda_s,lambda_u$ denote the eigenvalues of the stable and unstable manifolds, where we have the condition that $|lambda_s|<1<|lambda_u|<frac1$.

Then I was trying to understand how to prove the following statement below -

"Since the flow preserves orientation in the phase space $BbbR^3$, either $lambda_s,lambda_u <0$ OR $lambda_s,lambda_u>0$" ?

Any reference or idea of the proof?

eigenvalues-eigenvectors manifolds dynamical-systems

share|cite|improve this question

edited Jul 26 at 7:19

asked Jul 26 at 7:08

BAYMAX

2,43121021

Consider the 3D Dynamical system $dotX = F_mu(X)$, Say that the stable and unstable manifolds of the fixed point at $mu = 0$ intersect each other(Homoclinic tangency to a periodic orbit $C$ of period $tau_c$).

I am referring this article - Homoclinic Orbits and Mixed-Mode Oscillations
in Far-from-Equilibrium Systems by Gaspard and Wang.

Let $lambda_s,lambda_u$ denote the eigenvalues of the stable and unstable manifolds, where we have the condition that $|lambda_s|<1<|lambda_u|<frac1$.

Then I was trying to understand how to prove the following statement below -

"Since the flow preserves orientation in the phase space $BbbR^3$, either $lambda_s,lambda_u <0$ OR $lambda_s,lambda_u>0$" ?

Any reference or idea of the proof?

eigenvalues-eigenvectors manifolds dynamical-systems

share|cite|improve this question

edited Jul 26 at 7:19

asked Jul 26 at 7:08

BAYMAX

2,43121021

share|cite|improve this question

share|cite|improve this question

edited Jul 26 at 7:19

edited Jul 26 at 7:19

edited Jul 26 at 7:19

asked Jul 26 at 7:08

BAYMAX

2,43121021

asked Jul 26 at 7:08

BAYMAX

2,43121021

asked Jul 26 at 7:08

BAYMAX

2,43121021

2,43121021

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do stable and unstable manifolds intersect each other to periodic orbit? What does that mean?
  – user539887
  Jul 26 at 7:16
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please see the edit.
  – BAYMAX
  Jul 26 at 7:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Perhaps the authors introduced their own peculiar notation $lambda_s$ and $lambda_u$ (without knowing the context I cannot say). But as the possibility $lambda_u<0$ is admitted, $lambda_u$ cannot denote the eigenvalue corresponding to the unstable manifold.
  – user539887
  Jul 26 at 7:28
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, for the context I have given the article name!
  – BAYMAX
  Jul 26 at 7:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Perhaps $lambda_u$ and $lambda_s$ denote the eigenvalues of the linearization of the Poincaré map at the periodic orbit $C$? That would make sense.
  – user539887
  Jul 26 at 7:32
  
  

  
  
  
  

 | 
show 2 more comments

• 
  
  
  

  
  

  
  
  
  
  
  

  
  How do stable and unstable manifolds intersect each other to periodic orbit? What does that mean?
  – user539887
  Jul 26 at 7:16
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please see the edit.
  – BAYMAX
  Jul 26 at 7:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Perhaps the authors introduced their own peculiar notation $lambda_s$ and $lambda_u$ (without knowing the context I cannot say). But as the possibility $lambda_u<0$ is admitted, $lambda_u$ cannot denote the eigenvalue corresponding to the unstable manifold.
  – user539887
  Jul 26 at 7:28
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks, for the context I have given the article name!
  – BAYMAX
  Jul 26 at 7:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Perhaps $lambda_u$ and $lambda_s$ denote the eigenvalues of the linearization of the Poincaré map at the periodic orbit $C$? That would make sense.
  – user539887
  Jul 26 at 7:32
  
  

  
  
  
  

How do stable and unstable manifolds intersect each other to periodic orbit? What does that mean?
– user539887
Jul 26 at 7:16

How do stable and unstable manifolds intersect each other to periodic orbit? What does that mean?
– user539887
Jul 26 at 7:16

Please see the edit.
– BAYMAX
Jul 26 at 7:22

Please see the edit.
– BAYMAX
Jul 26 at 7:22

Perhaps the authors introduced their own peculiar notation $lambda_s$ and $lambda_u$ (without knowing the context I cannot say). But as the possibility $lambda_u<0$ is admitted, $lambda_u$ cannot denote the eigenvalue corresponding to the unstable manifold.
– user539887
Jul 26 at 7:28

Perhaps the authors introduced their own peculiar notation $lambda_s$ and $lambda_u$ (without knowing the context I cannot say). But as the possibility $lambda_u<0$ is admitted, $lambda_u$ cannot denote the eigenvalue corresponding to the unstable manifold.
– user539887
Jul 26 at 7:28

Thanks, for the context I have given the article name!
– BAYMAX
Jul 26 at 7:30

Thanks, for the context I have given the article name!
– BAYMAX
Jul 26 at 7:30

Perhaps $lambda_u$ and $lambda_s$ denote the eigenvalues of the linearization of the Poincaré map at the periodic orbit $C$? That would make sense.
– user539887
Jul 26 at 7:32

Perhaps $lambda_u$ and $lambda_s$ denote the eigenvalues of the linearization of the Poincaré map at the periodic orbit $C$? That would make sense.
– user539887
Jul 26 at 7:32

 | 
show 2 more comments

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