Center problem for systems with two monomial nonlinearities
The authors study the center problem for a planar real system which is written in the complex form as \[ \dot z=i z + A z^k \bar z^\ell+B z ^m \bar z^n \] with \(k+\ell \leq m+n, \;(k,\ell)\neq (m,n), \;A, B\in \mathbb{C}\). In the equation \(z(t)\) is \(u(t)+i v(t)\), so the phase plane is the real plane \((u,v)\). In the first theorem of the paper five series of conditions are given, the fulfillment of which yields the existence of a center at the origin of the coordinates. Then, it is proved that under some restrictions the conditions of the theorem are not only sufficient, but also necessary center conditions.
scientific article; zbMATH DE number 6537328
scientific article