Proving that three closed orbits must contain a fixed point

[infinitylord]

A smooth vector field on the phase plane is known to have exactly three closed orbit. Two of the cycles, C₁ and C₂ lie inside the third cycle C₃. However C₁ does not lie inside C₂, nor vice-versa.

What is the configuration of the orbits?
Show that there must be at least one fixed point bounded by C₁, C₂, and C₃.

I'm having trouble with this problem. I know that any closed orbit must enclose a number of fixed points such that the total index is 1 (also known as winding number). Therefore:

I₁ = I₂ = I₃ = 1

where I_k is the index of C_k.

I = (2π)^-1∫dθ

I don't exactly see how there can be a fixed point that is bounded by all *three* of the fixed points. Since C₁ and C₂ are not contained with in each other, the only way a fixed points could be bounded by both of them would be if they overlapped somewhat like a Venn Diagram, but I didn't think it was permissible for orbits to cross each other like that (?).

 

[mfb]

I don't exactly see how there can be a fixed point that is bounded by all *three* of the fixed points.

I don't see that either. Maybe it is three separate problems?

Show that there is at least one fixed point bounded by C₁.
Show that there is at least one fixed point bounded by C₂.
Show that there is at least one fixed point bounded by C₃.

That would be trivial, but what about this?

Show that there is at least one fixed point bounded by C₃ but not bounded by C₁ or C₂.