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Objectives The emergence of modern high-strength materials leads to the creation of thin-walled structures in various fields of technology. To obtain the necessary information about their behavior under load, one should analyze all the characteristic features encountered at all stages of their loading - at the initial (initial) stage of their operation, taking into account one or more types of nonlinearities, find possible critical states and, depending on the type of stability loss, study the nature of the initial stage of postcritical deformation. Based on an algorithm combining approximate analytical and numerical methods, the article solves the model problem — studying the behavior of a thin-walled spherical shell under load.Method. The study is based on solving the nonlinear problem of determining the stress-strain state at the initial - axisymmetric stage of work; critical (bifurcation) load values; analysis of the nature of post-bifurcation behavior. The work uses a variant of the general theory of stability and postcritical behavior of structures previously developed by V.T. Coiter.Result. The solution of such a general problem associated with discontinuous phenomena is carried out on the basis of mathematical ideas formulated in the theory of branching solutions of nonlinear equations. The values of the coefficients characterizing the initial stage of the post-bifurcation behavior of the shells and, from a practical point of view, the relations between the critical and limiting values of the loads are obtained. It is shown that depending on the area of the shell surface part loaded by the distributed load, the nature of the initial stage of postcritical deformation changes not only quantitatively, but also qualitatively.Conclusion. The most effective in solving problems associated with discontinuous phenomena are combinations of approximate analytical ones - catastrophe theory and numerical methods that do not require complex, timeconsuming and significant amounts of computation. Analysis of the initial stage of the postbifurcation behavior of structures allows us to assess the degree of danger of reaching a critical state, which is achieved by taking into account the values of the corresponding reliability coefficients in the calculations.