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How to solve a boundary value problem for a piecewise linear differential equation?

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I have two a set of two second order differential equations to which I want to find a periodic solution. The differential equations are defined as
α and β are known apriori. I have tried to use a pseudo arclength continuation method but that does not seem to be working. My next approach is to use the bvp4c or bvp5c solvers provided by MATLAB. My doubt is - how do I implement the switching condition in the bvp solvers? Does the odeevents provision work?
If you have any other approach that may be useful for solving the problem, please let me know!
(Note: My region of interest is obviously values of )

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