An algebraic analysis for bifurcation problems
Hisato Fujisaka, Chikara Sato · Electronics and Communications in Japan (Part III Fundamental Electronic Science) · 1991
Abstract An algorithm for computing the codimensions of tangent space for bifurcation equations using the Gröbner basis is presented. When a bifurcation equation with perturbation added becomes equivalent to the original bifurcation problem, the perturbation term belongs to the tangent space; this contains an ideal of a certain equation set on real functions. Therefore, perturbation terms that prevent the equivalence relationship belong to the codimensional space of the tangent space. On the other hand, the residue system of ideals on polynomial rings is found as a set of terms that are not multiples of the highest order of the Gröbner basis. In this paper an algorithm will compute the codimension algebraically by associating the codimensional space of a tangent space with a residue system. Thus, a symbolic manipulation description is adequate for this algorithm. An established algorithm can be similarly applied not only to the case in which a bifurcation equation can be expressed by one variable and one parameter but also to the case in which it is expressed by simultaneous equations of multivariables and one parameter. As an example of computation an equation expressing bifurcation such as Pitch‐fork and Hilltop, etc., was used. The results were verified by illustrating the solution curve when perturbation terms belonging to the codimensional space are added.