Extension of the solution of nonlinear equations in the neighborhood of a bifurcation point

Using the method of continuous extension with respect to a parameter we develop a method of constructing the load trajectory of a structure having both limit points and bifurcation points. The method is applicable for the systems of nonlinear algebraic equations that describe the family of extremals that minimize the value of the total potential strain energy of the structure, and makes it possible to find all the branches of the load trajectory emanating from a bifurcation point and extend the solution along any of them. The method is based on the fact that the eigenvectors of the augmented Jacobian of the system of equations in the extended space of variables that correspond to zero eigenvalues on the main branch of the load trajectory are bifurcation vectors and form the active subspace of solutions of the equations of the extension. Meanwhile the other eigenvectors form the passive subspace that contains the extension vector with respect to the main branch of the load. As a result the entire process of computing the extension vector of the solution at any point of the load trajectory reduces to determining the eigenvectors of the augmented Jacobian of the original system of nonlinear algebraic equations, identifying them according as they belong to the active or passive subspace, and forming the extension vector of the solution using them and analytic relations Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 35–46.