Centers with degenerate infinity and their commutators.
This paper is devoted to study the isochronicity of the origin of planar polynomial systems of the form \[ \dot x= P(x,y)=-y+P_2(x,y)+\cdots+ P_n(x,y), \] \[ \dot y= Q(x,y)=x+Q_2(x,y)+\cdots+ Q_n(x,y), \] where \(P_i\) and \(Q_i\) are homogeneous polynomials of degree \(i\) and \(xQ_n(x,y)\equiv yP_n(x,y).\) This last identity forces the infinity of the Poincaré compactification of the vector field to be full of critical points and, for this reason, the above systems are called systems with degenerate infinity. The authors derive necessary conditions for a system with degenerate infinity \(X=(P,Q)\) to admit a transversal polynomial commutator \(Y,\) i.e., a polynomial vector field \(Y\) such that is transversal to \(X\) in a punctured neighborhood of the origin and verifies \([X,Y]=0.\) Recall that if such an \(Y\) exists then the origin of \(X\) is an isochronous center. The results obtained are applied to constant angular speed systems (also called rigid systems), which are systems \(X\) such that \(xQ_i(x,y)\equiv yP_i(x,y)\) for \(i=2,\ldots,n,\) giving that \(Y\) has to be of the form \(Y(x,y)= (xK(x,y),yK(x,y))\) for some polynomial \(K\) of degree at most \(n-1.\) Finally, the case \(X=(-y+P_s(x,y)+xH(x,y), x+Q_s(x,y)+yH(x,y))\) where \(H\) is a polynomial of degree at most \(n-1\) and having no monomials of degree lower than \(s,\) is studied. The paper ends with several comments, examples and open problems. For instance, the system \(\dot x=-y+xy+xy^3,\) \(\dot y=x+y^2+y^4,\) that does not admit polynomial commutators, has an isochronous center. Is it true that all centers for systems with constant angular speed are either reversible or admit a polynomial commutator?
scientific article; zbMATH DE number 1903475
scientific article