I am dealing with a boundary value ODE problem. The (nonlinear) ODE system smoothly depends on a real parameter q. For q=1 the ODE system simplifies and I can derive analytically a complete set of valid solutions of the boundary value problem. Under which conditions are solutions for q = 1 smoothly mapped into solutions for q >1? How can one predict "singularity points" in q where the correspondence breaks down?
If the solution is non-degenerate, it can be exteded locally. For a global extension, you may use continuation methods based on a priori estimates of the solutions.
take a look at our paper "A Simple Method for Tracking Turning Points in Parameter Space" which draws on some ideas in the homotopy literature and parameter continuation literature. The examples in the above paper all have turning points. But similar ideas can be used to also find other types of bifurcation points ( e.g. Hopf, pitchfork, etc) along the solution curve.
I would suggest the Painleve analysis of the equation. It may happen the resonances depend on the parameter q. If they do then you can obtain an information about essential singularities. If the resonances are all integers independent of q then you are dealing with a possibly integrable equation for all q.
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