Limit cycles for a mechanical system coming from the perturbation of a four-dimensional linear center (Q854723)

It is shown that the system \[ \dot x= u,\quad \dot u= -x+\varepsilon f(x,y),\quad\dot y= v,\quad\dot v= -y+\varepsilon g(x,y)\tag{1} \] with \(f\) and \(g\) arbitrary polynomials of degree 3 in the variables \(x\) and \(y\), has at most 2 limit cycles bifurcating from the periodic orbits of the system \[ \dot x= u,\quad\dot u=-x,\quad\dot y= v,\quad\dot v=-y, \] up to first-order expansion of the displacement function of (1) with respect to the small parameter \(\varepsilon\). Moreover, there are systems (1) having exactly 2 limit cycles.