Dynamic Systems: Poincaré-Bendixson Theorem finite # of equilibria

[Master1022]

Homework Statement:: Can someone explain the finite number of equilibria outcome of the Poincaré-Bendixson Theorem?
Relevant Equations:: Poincaré-Bendixson Theorem

[Mentor Note -- General question moved from the schoolwork forums to the technical math forums]

Hi,

I was reading notes in dynamical systems and have the following question about the Poincaré-Bendixson theorem.

Context: The Poincaré-Bendixson Theorem states:
"Let M be a positively invariant region of a vector field, containing only a finite number of equilibria. Let $x \in M$ and consider $\omega(\mathbf{x})$. Then one of the following possibilities holds:
(i) $\omega(\mathbf{x})$ is an equilibria
(ii) $\omega(\mathbf{x})$ is a closed orbit
(iii) $\omega(\mathbf{x})$ consists of a finite number of equilibria $x_1 ^{*}, x_2 ^{*}, ..., x_n ^{*}$ and orbits $\gamma$ with $\alpha (\gamma) = \mathbf{x}_i ^{*}$ and $\omega (\gamma) = \mathbf{x}_j ^{*}$
"

'Positively invariant' basically means that once a trajectory enters a region, it won't escape (a non-mathematical explanation, but it helps me to visualize what is going on)

Question: What is meant by possibility (iii) about the finite number of equilibria and the $\omega$ and $\alpha$ limits (/cycles)? I cannot visualize what is going on and would appreciate any help (or sketch if possible).

I understand what is meant by the first two possibilities, but not the third.

Attempt:
Breaking down the 'sentence':
- I don't understand how the $\omega$ set can contain multiple equilibria, without it being a cycle.
- I don't understand what is meant by the latter half at all

Many thanks in advance.

 

[S.G. Janssens]

What (iii) is trying to capture, are homoclinic and heteroclinic orbits. Have you seen those before? The lecture notes or book probably discuss those.

In the homoclinic case, the unstable and stable manifolds of a single equilibrium coincide. In the heteroclinic case, the unstable manifold of one equilibrium coincides with the stable manifold of another equilibrium. At first, this is easiest to see geometrically, without attempting to write down a vector field explicitly.

(Note that the Poincaré-Bendixson theorem is valid for vector fields on the plane. Its classification is not exhaustive in higher dimensions.)

 

[Master1022]

Thanks for the reply @S.G. Janssens !

What (iii) is trying to capture, are homoclinic and heteroclinic orbits. Have you seen those before?

Oh okay - yes I have come across those terms

In the homoclinic case, the unstable and stable manifolds of a single equilibrium coincide.

Agreed

In the heteroclinic case, the unstable manifold of one equilibrium coincides with the stable manifold of another equilibrium.

I thought heteroclitic connected two unstable manifolds of different equilibria?However, I am still struggling to understand what point (iii) means pictorially? The heteroclinic and homoclinic orbits start/end at equilibrium rather than circling around them, so I can't seem to make the connection between what point (iii) of the theorem is saying.

 

[S.G. Janssens]

I thought heteroclitic connected two unstable manifolds of different equilibria?

No, that is not the case.

However, I am still struggling to understand what point (iii) means pictorially? The heteroclinic and homoclinic orbits start/end at equilibrium rather than circling around them, so I can't seem to make the connection between what point (iii) of the theorem is saying.

To get an idea,

1. Draw two saddle equilibria in the plane.
2. For each saddle, draw the stable and the unstable eigenvectors. This gives you an $X$-shape at each saddle, with the respective equilibrium in the center of the $X$.
3. Draw an orbit emanating from the first saddle, tangential to its unstable eigenvector, connecting to the second saddle, tangential to its stable eigenvector.
4. Do the same with the roles of the two saddles interchanged.

This gives you a heteroclinic cycle in the plane with two equilibria involved and two connecting orbits. (If you do not manage, there are a lot of pictures of this situation available in the textbooks and online.) Next, consider one of the connecting orbits and convince yourself that its $\alpha$-limit set is precisely one of the two equilibria, and its $\omega$-limit set is the other equilibrium.

 

[Master1022]

Thank you for responding once again.

No, that is not the case.

Okay - I agree.

To get an idea,

1. Draw two saddle equilibria in the plane.
2. For each saddle, draw the stable and the unstable eigenvectors. This gives you an $X$-shape at each saddle, with the respective equilibrium in the center of the $X$.
3. Draw an orbit emanating from the first saddle, tangential to its unstable eigenvector, connecting to the second saddle, tangential to its stable eigenvector.
4. Do the same with the roles of the two saddles interchanged.

This gives you a heteroclinic cycle in the plane with two equilibria involved and two connecting orbits. (If you do not manage, there are a lot of pictures of this situation available in the textbooks and online.) Next, consider one of the connecting orbits and convince yourself that its $\alpha$-limit set is precisely one of the two equilibria, and its $\omega$-limit set is the other equilibrium.

Thanks. I did make a sketch and can convince myself of the $\alpha$ and $\omega$ limits. I know I keep asking, but perhaps I should rephrase it to my current misunderstanding: what does the "orbits $\gamma$ with $\alpha (\gamma) = \mathbf{x}_i ^{*}$ and $\omega (\gamma) = \mathbf{x}_j ^{*}$" mean in from the theorem (point 3)?

Context: The Poincaré-Bendixson Theorem states:
"Let M be a positively invariant region of a vector field, containing only a finite number of equilibria. Let $x \in M$ and consider $\omega(\mathbf{x})$. Then one of the following possibilities holds:
(i) $\omega(\mathbf{x})$ is an equilibria
(ii) $\omega(\mathbf{x})$ is a closed orbit
(iii) $\omega(\mathbf{x})$ consists of a finite number of equilibria $x_1 ^{*}, x_2 ^{*}, ..., x_n ^{*}$ and orbits $\gamma$ with $\alpha (\gamma) = \mathbf{x}_i ^{*}$ and $\omega (\gamma) = \mathbf{x}_j ^{*}$
"

Does the 'orbits $\gamma$' mean that the positively invariant region encapsulates this heteroclitic (for example) orbit that we have sketched above?