SIMPLEST NORMAL FORMS FOR PLANARSYSTEMS ON EQUILIBRIUM MANIFOLDS

摘要

Equilibrium manifold is a manifold that consists of equilibrium points. Planar systems with one-dimensional equilibrium manifolds are considered in this paper. First, for such planar systems, aunified equation with the horizontal axis as the equilibrium curve is formulated. Then, accordingto the corresponding linearized systems, different cases are discussed: For the nondegeneratecase, the simplest normal form of a system with simplified Bogdanov—Takens singularities isobtained; for the general first-order degenerative case, the simplest normal forms are completelycharacterized; finally, for the general higher-order degenerative case, deduction of the simplestnormal form is illustrated.