Classificação de centros Hamiltonianos polinomiais biquadrados, e isocronicidade trivial versus formas canônicas para aplicações polinomiais no plano de Jacobiano unitário

Abstract

This work, we classify a non- degenerate center at the origin of a planar Hamiltonian system associated to a function of the form $H(x,y)=A(x)+B(x)y^2 + C(x)y^4$, where $A$, $B$ and $C$ are polynomials. After seing a relation between trivial isochronous centers and the Jacobian Conjecture on the plane, we study polynomial maps $f: \mathbb{R}^2 \longrightarrow \mathbb{R}^{2}$, with $f(0,0)=(0,0)$ and Jacobian determinant constant and equal to $1$, and we present sufficient conditions to its injectivity. At last, as a consequence of the study, we characterize the trivial isochronous centers of planar polynomial Hamiltonian system associated to polynomial function of degrees $10$, $12$, $14$ and $22$.