A good classical book is:

The Hopf Bifurcation and Its Applications, 19. kötet

Jerrold Eldon Marsden, Marjorie MacCracken

Meet Lajos Lóczi in GesearchGate:

https://www.researchgate.net/profile/Lajos_Loczi/citations/

As you probably already know, there are many books covering general field of low dimensional local bifurcations (for example: Strogatz, S.H., 1994. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering. Perseus Books, Reading.). However, completely analytic solutions are usually limited to simple systems. Good approximations could be found for wide family of stoichiometric network systems in Clarke, B.L., 1980. Stability of Complex Reaction Networks, in: Prigogine, I., Rice, S.A. (Eds.), Advances in Chemical Physics. John Wiley & Sons, Inc., pp. 1–215.

This depends on the system, of course. I think you would like to study a dynamical regime (say, equilibrium) depending on the system parameters. Suppose you know it in the sence that you can express it analytitcally. Then you can try the following way: first, froze all the system parameters and consider the perturbation of the regime under consideration; second, assume the perturbation to be small and go to the linear system for perturbations; third, seek for linear eigenmodes proportional to exp(kt), where k is complex eigenvalue and t stands for time, finally find the critical values of the system parameters just equating real part of k to zero. Generically, the moving of the parameters through the critical values is accompanied by tje bifurcations. To know what would have happened just have a look to the books mentioned in the answers above.

Bifurcation points can be characterized and computed as solutions of so-called appended nonlinear systems of equations. See the book by Willy Govaerts

Numerical Methods for Bifurcations of Dynamical Equilibria

published by SIAM in 2000.

Complementary to the book proposed by A. Griewank is the one by R. Seydel, Practical bifurcations and stability analysis, published by Springer in 2010.

If you want to remain in the analytical realm, you may want to have a look at normal form reduction, which essentially reduces your system to a canonical form that reflects the signature of the bifurcation the system experiences.