Higher-order Melnikov functions for degenerate cubic Hamiltonians
The author considers quadratic perturbation problems of Hamiltonian systems in the plane with degenerate cubic Hamiltonians. He first uses the scheme of J. P. Francoise to compute explicitly the first four Melnikov functions \(M_1(h),\dots, M_4(h)\), then he proves the following five interesting theorems. Theorem 1. Assume \(H\) is a nongeneric cubic Hamiltonian with a center. Then the perturbed system (1) \(\dot x= H_y+ \varepsilon f(x, y)\), \(\dot y= - H_x+ \varepsilon g(x, y)\), where (2) \(H= x[y^2+ Ax^2- 3(A- 1) x+ 3(A- 2)]\), \(A\in \mathbb{R}\), is integrable and belongs to the union \(Q^H_3\cup Q^R_3\) (in the notation of H. Zoladek) provided: (i) \(M_1(h)= M_2(h)\equiv 0\), for \(H\) corresponding to \(A= 0\) in (2); (ii) \(M_1(h)+\cdots+ M_4(h)\equiv 0\), for \(H\) corresponding to \(A\not\equiv -1\) in (2). Theorem 2. In any quadratic Hamiltonian system the cyclicity of the saddle loop (or the total cyclicity of two such loops, if they exist) under quadratic perturbation is two. Theorem 3. Suppose in (1), \(H\) is the standard elliptic Hamiltonian. Then for \(\varepsilon\) small and for arbitrary quadratic perturbations \((f, g)\) the system has no more than two limit cycles in the finite part of the plane. Theorem 4. Suppose in (2) we have \(A= 0\), then (1) has at most two limit cycles simultaneously born from the center and the set of periodic orbits of \((1)_{\varepsilon= 0}\). Theorem 5. Assume the formula for the higher-order Melnikov functions is \(I(h)= k_1 J_1+ k_0 J_0+ k_{- 1} J_{- 1}= c_1+ c_2(h- h_s)|n|h- h_s|+ c_3(h- h_s)+\cdots\), where \(J_i(h)= \int_{H= h} x^i y dx\) \((i= 0, \pm 1)\), then \(c_1= c_2= c_3= 0\) implies \(k_0= k_1= k_{- 1}= 0\).
scientific article; zbMATH DE number 883516