The following is taken from the book Concepts and applications of Finite element methods by Cook, Malkus and Plesha.

 For bifurcation buckling, we have the following eigenvalue problem:

(K+Lambda Ksigma)delta d= 0 

K =stiffness matrix corresponding to base state, Lambda is an eigenvalue, Kisgma is the stress stiffening matrix correspondng to base state.
 
My understanding is K and K sigma is determined at a base state under the application of any particular loading . If Geometric Non linearities are included K and K sigma will based on the deformed state under the action of the base state load.For a linear analysis K and K sigma is based on the undeformed structure.
 
Also, no additional load ( beyond the base state  load) is needed for the eigenvalue analysis.

 
Using the above concept. I tried to do an eigenvalue bifurcation analysis using ABAQUS. The requirement in ABAQUS is to apply the loads in two steps:  A base state load ( P) and an additional load Q. Isn't it different from the concept presented in the above mentioned book? That is why a second loading has to be applied when the change from one equilibrium state to the other is happening w/o any additional load.

Kindly advise.

Regards