My question is to get an idea of state of art in bifurcation analysis for the following class of problems : Consider the following nonlinear coupled-ODE system over a time-interval $ [0,1]$ :

$ \dot{x}=N_1(x,y;a)\ \dot{y}=N_2(x,y;a)\ (x,y)\in\mathbb{R}^{2N}$ ,

with boundary conditions given as $ x(t=0)=x0, y(t=1)=y0$ . The parameter that we vary (to explore bifurcations) is $ a$ .

These problems often arise in optimal control for instance, where the first equation is simply the dynamics of the given control system, and the second equation is the co-state equation (governing the “adjoint” variables).

I understand that the theory of such BVPs is under-developed compared to ODE theory, so it is not surprising that a google scholar search did not help me much while looking for bifurcation theory of such BVPs.

So I am looking for any review papers/monographs/books which discuss the bifurcation structure of such systems.

For the regular old ODEs (without any symmetry), we know for example that generic bifurcations are only a few (saddle-node, hopf, transcritical). Is anything similar known for the systems above ?