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This is a summary relevant to the problem at hand of part 2 of Simplest Bifurcation Diagrams of Monotone Vector Fields on a Torus (C Baesens and R S MacKay, 2018). BMK equations are coupled autonomous ODEs with monotone parameters of the form
for small epsilon greater than 0. Where 
and 
Analysing the equations, for parameters inside the region 
, the ODE has two fixed points. This is dubbed the resonance region, R. Inside R, the fixed points are one saddle and one node. Consequentially, the boundaries of R form curves of generic saddle-node bifurcations.
There are exactly 4 points that have trace 0 and det 0 and correspond to 4 Bogdanov-Takens points. Connecting these points along trace-zero loops create curves of neutral saddle (trace of the saddle point is 0) and curves of centre (trace of the node is 0) as such:
The centres and neutral saddle can be detected via a Hopf bifurcation. The last type of bifurcation considered is rotational homoclinic connections: This is similar to a homoclinic bifurcation with the constraint that the connection be a homotopically non-trivial loop. There are two curves of RHC, denoted here as RHC+ and RHC-. These two curves intersect at a unique point which is a Necklace point.
The final goal is to extend the family of BMK equations to look for parameter regimes that will shift the Necklace point on the bifurcation diagram to inside the trace-zero loop. Then I will seek to analytically prove that the resulting system satisfies all the criteria. If on the other hand the numerics do not reveal a suitable parameter regime I will look for a mathematical obstruction.
Throughout the numerics, I consider the natural lift of the torus onto the plane and study the corresponding ODE that is considered to be in the phase space [0,1]x[0,1] and any path that goes outside this region will be considered as having taken a homotopically non-trivial path around.
Using PyDSTool.PyCont, I compute the curves of SNE and the trace-zero loops.
To compute RHC, I compute the saddle point then find the unstable eigenvector. Perturbing slightly in this direction, I use this point as the initial condition of the flow to estimate the unstable manifold. This flow is computed until the distance between the flow and the saddle point in the region [1,2]x[0,1] (called the (1,0)-lift of the saddle point) is minimised.
This distance is denoted the Pontryagin Energy (PE) of the branch of the unstable manifold. A zero of PE is then a RHC point. I fix 
and then apply Newton-Raphson method (NR) to find the zero. Experimentally, the plot of PE with fixed
) is differentiable almost everywhere except for the zero. As a result standard gradient descent algorithm does not work. I multiply the error term in NR by a learning rate to ensure smooth convergence (results without this scaled rate did not converge smoothly and tended to oscillate around the zero).
Once the zero is located, I increment and use the previous zero as a starting seed for the next application of gradient descent. This then continues the RHC+ curve.
To compute the RHC- curve, I apply the same methodology described above except computing the Pontryagin Energy of the stable manifold by perturbing the (1,0)-lift of the saddle point along the unstable manifold in the negative direction.
The intersection of these two curves is the N point.
In a communication with Baesens and MacKay, they suspect that adding coefficients to the sines and cosines will create a suitable parameter regime to move the Necklace point inside the trace-zero loop. Using this repo I should be able to directly numerically study such generalised BMK equations.