Question

A smooth vector field on the phase plane is known to have exactly three closed orbits. Two of the cycles, say C1 and C2 , lie inside the third cycle C3 . However, C1 does not lie inside C2 , nor vice-versa. a) Sketch the arrangement of the three cycles. b) Show that there must be at least one fixed point in the region bounded by C1 ,C2 , C3 .

Answer 1

a) attached below

b) Ic3 = Ic2 + Ic1 + index of a fixed point in ( C°₁ U C°₂ )^c n c₃

Explanation:

A) sketch of the arrangement of the cycles

attached below

b) prove that at least one fixed point in the attached region is bounded

Given that : Ic3 = Ic2 = Ic1 = 1

hence : Ic3 = Ic2 + Ic1 + index of a fixed point in ( C°₁ U C°₂ )^c n c₃

This is a prove that at least one fixed point is bounded by C1 , C2 , C3