Gromov V. Models, methods, and algorithms of bifurcation theory for nonlinear elliptic equations of von Karman type

The object of study is nonlinear phenomena in the fields of medicine, mechanics, and biology able to exhibit bifurcations. The subject of study is the mathematical models governed by the nonlinear elliptic equations of von Karman-type equations; methods and algorithms to solve the direct and inverse bifurcation problems for that class of models. The goal of study. This thesis makes the original contribution towards one of the challenging issues facing computational mathematics, which is how to develop models, methods, and algorithms of bifurcation theory for nonlinear elliptic equations of von Karman type. The methods of study are numerical and analytical modelling of nonlineas phenomena, methods of nonlinear functional analysis, numerical methods to contruct and examine branching structure for nonlinear boundary problems for partial differential equations, and numerical clustering methods as well. In the framework of novel iterative method, the solution of the nonlinear boundary problem for PDEs is constructed as a sequence of nonlinear boundary problem for ODEs; in turn, these ones are reduced to the equivalent Cauchy problems. The numerical approach is proposed to localize and analyze singular points as well trace the respective bifurcation paths. All methods are theoretically justified. In combination, the methods consitute a novel approach to obtain bifurcation structure for nonlinear boundary problems for von Karman-type equations. The approach was utilized to explore bifurcation structure for the nonlinear boundary problem for von Karman equations, defined on close and open cylindrical domains, systems of coupled von Karman equations. For all cases mentioned above, wide-ranging simulation reveals basic bifurcation structures (with possible primary, secondary, and tertiary bifurcation paths) and feasible scenarios of its destruction; it also allows analyzing non-monotonicity of bifurcation set sections, associating their maxima and minima with different bifurcation paths of the basic bifurcation structures. The novel approach to solve the problem to identify pre-bifurcation state implies that one should seek for typical sequences preceding to possible bifurcations (topological bifurcation precursors). To construct the set of precursors, we apply knowledge extraction (clustering) technique to a set of all sequences of post-bifurcation solutions observable on bifurcation paths. The approaches in question to solve direct and inverse problems of bifurcation theory are applied to analyze bifurcations of particular systems frequent in space, nano-, and biotechnologies. Methods, numerical algorithms, and program pakages may be used to design and calculate thin-walled structures in scientific laboratories and design offices.