Showing papers on "Bifurcation diagram published in 1971"

On matrices depending on parameters

Abstract: 1. Классические результаты (Аполлоний, Декарт, Ньютон, Харнак). Шестнадцатая проблема Гильберта.

Bifurcation theory of nonlinear boundary value problems

Abstract: The theory of bifurcation of solutions to two-point boundary value problems is developed for a system of nonlinear first order ordinary differential equations in which the bifurcation parameter is allowed to appear nonlinearly An iteration method is used to establish necessary and sufficient conditions for bifurcation and to construct a unique bifurcated branch in a neighborhood of a bifurcation point which is a simple eigenvalue of the linearized problem The problem of bifurcation at a degenerate eigenvalue of the linearized problem is reduced to that of solving a system of algebraic equations Cases with no bifurcation and with multiple bifurcation at a degenerate eigenvalue are considered The iteration method employed is shown to generate approximate solutions which contain those obtained by formal perturbation theory Thus the formal perturbation solutions are rigorously justified A theory of continuation of a solution branch out of the neighborhood of its bifurcation point is presented Several generalizations and extensions of the theory to other types of problems, such as systems of partial differential equations, are described The theory is applied to the problem of the axisymmetric buckling of thin spherical shells Results are obtained which confirm recent numerical computations

Bifurcation theorems and limit cycles in nonlinear systems—I.

Abstract: Two bifurcation theorems are established concerning the qualitative change in the integral curves of the hard-excitation type nonlinear systems at a point of bifurcation (or a branch point) where different regions meet. Two classes of this type (Type B) are considered. These exhibit limit cycles which do not contract to the origin, unlike the soft-excitation type nonlinear systems (Type A) reported by Jonnada and Weygandt. A Type B, Class 1, system is exemplified by the well-known van der Pol equation. A third-order example (B-1 system) is also given. Type B, Class 2 systems exhibit an unstable limit cycle. Second- and third-order examples are given for B-2 systems.