ON THE MINIMAX BIFURCATION FORMULA

This talk will discuss the so-called minimax bifurcation formula, which allows one to find and analyze saddle-node bifurcations in nonlinear systems. This approach allows one to directly determine bifurcation points, evaluate their stability, and provides a computationally efficient framework. An abstract version of the formula, applicable to a wide class of equations, is developed. The method is demonstrated using the example of determining a saddle-node bifurcation point for positive solutions of a nonlinear elliptic system of nonvariational form. Using the minimax formula, explicit estimates of bifurcation values ​​in perturbed systems are obtained, enabling stability analysis. Moreover, this approach turns out to be an effective tool for studying the convergence of bifurcations in finite element approximations to bifurcations of the original partial differential equations.