SIMPLEST NORMAL FORMS FOR PLANAR SYSTEMS ON EQUILIBRIUM MANIFOLDS

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, a unified equation with the horizontal axis as the equilibrium curve is formulated. Then, according to the corresponding linearized systems, different cases are discussed: For the nondegenerate case, the simplest normal form of a system with simplified Bogdanov–Takens singularities is obtained; for the general first-order degenerative case, the simplest normal forms are completely characterized; finally, for the general higher-order degenerative case, deduction of the simplest normal form is illustrated.