Strong isochronism of polynomial differential systems with a center
Let \(\dot x= P(x,y)\), \(\dot y= Q(x,y)\) be a plane differential system \((S)\). A center \(O\) of \((S)\) is said to be strongly isochronous of order \(n\) when, in a neighbourhood \(U\) of \(O\), all trajectories starting at time \(t_0\) from the ray with polar angle \(\phi_0\) pass from the ray \(\phi= \phi_0+{2k\pi\over n}\) to the ray \(\phi= \phi_0+ {2(k+ 1)\pi\over n}\), \(k= 0,\dots, n-1\), in the same time \(T= {2\pi\over n}\).NEWLINENEWLINENEWLINEHere, some necessary and/or sufficient conditions for the strong isochronicity of a center of a polynomial system are given. Hamiltonian systems of degree 4 are considered, as well as special classes of non-Hamiltonian polynomial systems of degree 3, 4, 5. Most of the paper is concerned with strong isochronicity of order 2, which is often related to symmetry properties of the orbits to \((S)\).
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