Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop (Q423682)

The authors begin with the piecewise linear Hamiltonian system NEWLINE\[NEWLINE\dot x = H_y,\;\dot y = -H_xNEWLINE\]NEWLINE for NEWLINE\[NEWLINE\displaystyle H(x,y) = \begin{cases}\frac12((x-1)^2 + y^2),&x \geq 0,\\ -\frac12(x^2 + y^2),&x < 0.\end{cases}NEWLINE\]NEWLINE They perturb it to NEWLINE\[NEWLINE\dot x = H_y + \epsilon p(x,y),\;\dot y = -H_x + \epsilon q(x,y),NEWLINE\]NEWLINE where \(\epsilon\) is near zero and each of \(p\) and \(q\) is a degree \(n\) or less polynomial in \(x\) and \(y\) whose coefficients can change when crossing the \(y\)-axis. Using a first-order Melnikov function, they derive lower bounds on the maximum number of limit cycles that can appear in Hopf and homoclinic loop bifurcation as \(\epsilon\) is varied from zero. For \(n \leq 4\), they give a precise upper bound on the maximum number of zeros of the first order Melnikov function on a segment covering the period annulus; they give upper and lower bounds when \(n \geq 5\) and make a conjecture as to the precise value in this latter situation.