- #1
MeissnerEffect
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Homework Statement
I'm doing a course in nonlinear dynamics using Strogatz. One of the exercises in the book is "Using index theory show that the system has no closed orbits"
Homework Equations
[itex] \dot{x} = x(4-y-x^{2}) , \dot{y} = y(x-1) [/itex]
The Attempt at a Solution
Turns out there are 4 fixed points and we know that the sum of the indices of the fp within the closed orbit must equal 1.
No problem, except for the fp (1,3) which is a stable spiral(hence index of 1), which could in theory have a closed orbit around it.
Strogatz argues there can't be a closed orbit around it due to the saddle at (2,0) which has an unstable manifold in the direction of the spiral and the cycle can't cross the manifold as it would intersect another trajectory.
But as I understand things at the moment, the unstable manifold is only valid in the region close to the saddle, very clearly demonstrated by homoclinic orbits where the trajectory runs away on the unstable manifold but loops around and returns on the stable manifold.
I plotted the system in mathematica(see attachment) and it's clear that there indeed are no closed orbits, but Strogatz' argument doesn't convince me that what I drew in on the plane isn't possible.