The theory of bifurcation of solutions to two-point boundaryvalue problems is developed for a system of nonlinear first orderordinary differential equations in which the bifurcation parameter isallowed to appear nonlinearly. An iteration method is used toestablish necessary and sufficient conditions for bifurcation and toconstruct a unique bifurcated branch in a neighborhood of a bifurcationpoint which is a simple eigenvalue of the linearized problem. Theproblem of bifurcation at a degenerate eigenvalue of the linearizedproblem is reduced to that of solving a system of algebraic equations.Cases with no bifurcation and with multiple bifurcation at adegenerate eigenvalue are considered.
The iteration method employed is shown to generateapproximate solutions which contain those obtained by formalperturbation theory. Thus the formal perturbation solutions arerigorously justified. A theory of continuation of a solution branchout of the neighborhood of its bifurcation point is presented. Severalgeneralizations and extensions of the theory to other types ofproblems, such as systems of partial differential equations, aredescribed.
The theory is applied to the problem of the axisymmetricbuckling of thin spherical shells. Results are obtained whichconfirm recent numerical computations.
