Classification of dynamical systems near a cosymmetric equilibrium

A local classification is developed in a neighborhood of a cosymmetric equilibrium for differential equations with invertible cosymmetry and a vector parameter, under the assumption that the kernel of the linearization matrix at the cosymmetric equilibrium is two-dimensional and that the entire stability spectrum, except for the double zero eigenvalue, is stable. Equations with such properties are of codimension one among even-dimensional systems with a cosymmetric equilibrium. In all cases, such a system admits a straightenable family of non-cosymmetric equilibria near the cosymmetric one. The classification is based on the following properties: the type of the cosymmetric equilibrium (node, focus, saddle); the relative position of the cosymmetric equilibrium and the family (including the case where the cosymmetric equilibrium belongs to the family); the number of boundary equilibria of the family separating its stable and unstable regions ($\leqslant 3$); the number of intersections of each separatrix of the cosymmetric saddle equilibrium with the family ($\leqslant 3$). Each property is determined by polynomial conditions, and the classification therefore reduces to identifying sets of conditions with a non-empty intersection. The defining polynomial conditions and corresponding phase portraits are presented for each identified class. The existence of each nonempty class is established by a scalable example for non-obvious cases, while the emptiness of the remaining classes is established separately. This work continues the studies of L. G. Kurakin and V. I. Yudovich [1, 2], where analogous results were obtained in the neighborhood of a non-cosymmetric equilibrium.