Proof of two conjectures for perturbed piecewise linear Hamiltonian systems (Q6641236)

Consider the planar piecewise Hamiltonian vector fields \(X\) and \(Y\) given by the Hamiltonian functions\N\[\NX: \left\{\begin{array}{ll} x-\frac{1}{2}x^2+\frac{1}{2}y^2 &\text{ if } x\geqslant 0, \\\N\frac{1}{2}x^2+\frac{1}{2}y^2 &\text{ if } x<0, \end{array}\right.\N\]\N\[\NY: \left\{\begin{array}{ll} x-\frac{1}{2}x^2+\frac{1}{2}y^2 &\text{ if } x\geqslant 0, \\\N-x-\frac{1}{2}x^2+\frac{1}{2}y^2 &\text{ if } x<0. \end{array}\right.\N\]\NWe observe that \(X\) has a saddle point \(r=(1,0)\) with a homoclinic loop and \(Y\) has two saddles \(r^\pm=(\pm1,0)\) forming a polycycle.\N\NGiven \(n\in\mathbb{N}\) let \(Z_X(n)\) and \(Z_Y(n)\) denote the maximum number of limit cycles bifurcating from the period annuluses of these polycycles under piecewise polynomial perturbations of maximum degree \(n\).\N\NThe authors of [Zbl 1264.34073] and [Zbl 1342.34051] proved that\N\[\NZ_X(n)=n+\left[\frac{n+1}{2}\right], \quad Z_Y(n)=n,\N\]\Nfor \(n\in\{1,2,3,4\}\) and\N\[\Nn+\left[\frac{n+1}{2}\right]\leqslant Z_X(n)\leqslant 2n+\left[\frac{n+1}{2}\right], \quad n\leqslant Z_Y(n)\leqslant n+\left[\frac{n+1}{2}\right],\N\]\Nfor \(n\geqslant 5\). They also conjectured that \(Z_X(n)\) and \(Z_Y(n)\) are equal to their lower bounds for every \(n\geqslant5\).\N\NIn this paper the authors proves that these conjectures are true. That is, they prove that\N\[\NZ_X(n)=n+\left[\frac{n+1}{2}\right], \quad Z_Y(n)=n,\N\]\Nfor every \(n\in\mathbb{N}\).\N\NThe proof follows by considering the generating functions of the Melnikov function of first order and proving that these generating functions form an \textit{extended complete Chebyshev system}.