Center problem for systems with two monomial nonlinearities. (English) Zbl 1388.34024
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.
Reviewer: Valery Romanovski (Maribor)
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
Keywords:
nondegenerate center; Poincaré-Lyapunov constants; Darboux center; reversible center; holomorphic center; persistent centerReferences:
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