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Prove existence of a Heteroclinic Orbit

3

1

$begingroup$

How would you go around proving that there exists a heteroclinic orbit between two equilibria ( in the problem I'm trying to solve, one is a stable node(say n) and the other is a saddle (say s))?

I started by finding the stable manifold of the stable node and the unstable manifold of the saddle point.

Next, I want to show that their intersection is not empty and so there must be a trajectory that starts at s and end at n. And therefore, a heteroclinic orbit.
But the problem is that the stable manifold only "covers" a neighborhood of n. Same for the unstable manifold.

So how do we figure out the neighborhoods I guess?
Or is there a better way to do this than using the stable and unstable manifolds?

Edit: This is the system of ODEs I'm dealing with.
$$U' = V\
V' = -fraccDV -fracmuD U (1-U)\$$
All of the parameters are positive. and $c>sqrt4mu D$.
So we have two equilibria:
A stable node: (U,V) = (0,0).
And a saddle point: (U,V) = (1,0).

Thanks

ordinary-differential-equations manifolds dynamical-systems stability-in-odes hamilton-equations

share|cite|improve this question

edited Apr 8 at 21:10

Rasha Nasri

asked Apr 8 at 19:41

Rasha NasriRasha Nasri

162

New contributor

Rasha Nasri is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Related: this and this. Would be easier to say if you write your particular system.
  $endgroup$
  – Evgeny
  Apr 8 at 20:04
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks @Evgeny. I'll take a look at those links. Also, I added the system to the question!
  $endgroup$
  – Rasha Nasri
  Apr 8 at 21:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your system is quite similar to the one that was discussed in my second answer. I guess most of this solution applies to your problem.
  $endgroup$
  – Evgeny
  Apr 8 at 23:25
  

  
  
  
  

add a comment | 

3

1

$begingroup$

How would you go around proving that there exists a heteroclinic orbit between two equilibria ( in the problem I'm trying to solve, one is a stable node(say n) and the other is a saddle (say s))?

I started by finding the stable manifold of the stable node and the unstable manifold of the saddle point.

Next, I want to show that their intersection is not empty and so there must be a trajectory that starts at s and end at n. And therefore, a heteroclinic orbit.
But the problem is that the stable manifold only "covers" a neighborhood of n. Same for the unstable manifold.

So how do we figure out the neighborhoods I guess?
Or is there a better way to do this than using the stable and unstable manifolds?

Edit: This is the system of ODEs I'm dealing with.
$$U' = V\
V' = -fraccDV -fracmuD U (1-U)\$$
All of the parameters are positive. and $c>sqrt4mu D$.
So we have two equilibria:
A stable node: (U,V) = (0,0).
And a saddle point: (U,V) = (1,0).

Thanks

ordinary-differential-equations manifolds dynamical-systems stability-in-odes hamilton-equations

share|cite|improve this question

edited Apr 8 at 21:10

Rasha Nasri

asked Apr 8 at 19:41

Rasha NasriRasha Nasri

162

New contributor

Rasha Nasri is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Related: this and this. Would be easier to say if you write your particular system.
  $endgroup$
  – Evgeny
  Apr 8 at 20:04
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks @Evgeny. I'll take a look at those links. Also, I added the system to the question!
  $endgroup$
  – Rasha Nasri
  Apr 8 at 21:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your system is quite similar to the one that was discussed in my second answer. I guess most of this solution applies to your problem.
  $endgroup$
  – Evgeny
  Apr 8 at 23:25
  

  
  
  
  

add a comment | 

$begingroup$

How would you go around proving that there exists a heteroclinic orbit between two equilibria ( in the problem I'm trying to solve, one is a stable node(say n) and the other is a saddle (say s))?

I started by finding the stable manifold of the stable node and the unstable manifold of the saddle point.

Next, I want to show that their intersection is not empty and so there must be a trajectory that starts at s and end at n. And therefore, a heteroclinic orbit.
But the problem is that the stable manifold only "covers" a neighborhood of n. Same for the unstable manifold.

So how do we figure out the neighborhoods I guess?
Or is there a better way to do this than using the stable and unstable manifolds?

Edit: This is the system of ODEs I'm dealing with.
$$U' = V\
V' = -fraccDV -fracmuD U (1-U)\$$
All of the parameters are positive. and $c>sqrt4mu D$.
So we have two equilibria:
A stable node: (U,V) = (0,0).
And a saddle point: (U,V) = (1,0).

Thanks

ordinary-differential-equations manifolds dynamical-systems stability-in-odes hamilton-equations

share|cite|improve this question

edited Apr 8 at 21:10

Rasha Nasri

asked Apr 8 at 19:41

Rasha NasriRasha Nasri

162

New contributor

Rasha Nasri is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited Apr 8 at 21:10

Rasha Nasri

asked Apr 8 at 19:41

Rasha NasriRasha Nasri

162

New contributor

Rasha Nasri is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited Apr 8 at 21:10

Rasha Nasri

asked Apr 8 at 19:41

Rasha NasriRasha Nasri

162

New contributor

Rasha Nasri is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Apr 8 at 19:41

Rasha NasriRasha Nasri

162

New contributor

Rasha Nasri is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Apr 8 at 19:41

Rasha NasriRasha Nasri

162