The number of limit cycles of cubic Hamiltonian system with perturbation
The following perturbed Hamiltonian system is considered: \[ \frac{dx}{dt} = y + \varepsilon \sum_{j=0}^{l} a_j x^j | y| ^{2m-1}, \quad \frac{dy}{dt} = -x - x^3, \tag{1} \] where \(0<\varepsilon<<1\), \(l=2n+1\) or \(2n+2\), \(n\) and \(m\) are arbitrary positive integers and \(a_0\), \(a_1, \dots, a_l\) are real. The main result of the work is that this perturbed Hamiltonian system has at most \(n+m\) limit cycles. The particular case \(m=1\) is studied and an upper bound of \(n\) limit cycles is given, which is sharp by a suitable choice of the parameters \(a_i\), \(i=0,1,2, \dots, l\). Moreover, the particular cases \((m,n)=(1,1)\) and \((m,n)=(1,2)\) are extensively studied and general conditions for the number and stability of existing limit cycles are described in terms of the coefficients of the perturbation. The main tool to prove these results is the following well-known result. Let us consider a perturbed Hamiltonian system of the form \[ \frac{dx}{dt} = -\frac{\partial H}{\partial y} + \varepsilon P(x,y), \quad \frac{dy}{dt} = \frac{\partial H}{\partial x} + \varepsilon Q(x,y), \] where \(0<\varepsilon<<1\), \(H\), \(P\) and \(Q\) are of class \(\mathcal{C}^1\) in \(\mathbb{R}^2\) and \(\Gamma_h\) is a closed curve defined by \(H(x,y)=h\). We assume that the closed orbit \(\Gamma_{h_1}\) surrounds \(\Gamma_{h_0}\) if \(h_1>h_0\). Let us define the Abelian integral: \[ A(h) = \int_{\Gamma_h} Q(x,y) dx - P(x,y) dy. \] Then, the following facts hold: 1) If the perturbed Hamiltonian system has a limit cycle near \(\Gamma_{h^{*}}\), then \(A(h^{*})=0\). 2) If \(A(h^*)=0\) and \(A'(h^*) \neq 0\), then the perturbed Hamiltonian system has a unique limit cycle near \(\Gamma_{h^{*}}\), which is stable (resp. unstable) when \(A'(h^*)<0\) (resp. \(A'(h^*)>0\)). We refer the reader to the following two classical references [\textit{R. Roussarie}, Bifurcation of planar vector fields and Hilbert's sixteenth problem. Progress in Mathematics 164. Basel: Birkhäuser (1998; Zbl 0898.58039)] and [\textit{Y. Ilyashenko}, Bull. Am. Math. Soc., New Ser. 39, No. 3, 301--354 (2002; Zbl 1004.34017)] and the references therein for the complete definitions, history and settings of this result. We remark that the Hamiltonian considered in system (1) is \(H(x,y)=(x^2+y^2)/2+x^4/4\) and that the closed orbits \(\Gamma_h\) exist for \(h>0\). The form of this Hamiltonian and the perturbative terms are chosen so that the computation of \(A(h)\) can be explicitly done by means of the usual integration of polynomials. The parameter change \(h=(2+\lambda)\lambda/4\) in the rank \(\lambda>0\), gives that \(A(h)=\lambda^{m+1/2} I(\lambda)\) with \(I(\lambda)\) a polynomial of degree \(n+m\) in \(\lambda\) and whose coefficients depend linearly on \(a_0\), \(a_1, \dots, a_l\). This fact, and the proof that no limit cycles come from the infinity, leads to the conclusion of the main result. Moreover, it is shown that the even degree terms of \(x\) in the perturbative part of (1) do not affect the number of limit cycles since they do not appear in the coefficients of \(I(\lambda)\). When \(m=1\), the coefficients of the polynomial \(I(\lambda)\) are analyzed from the point of view of Descartes's rule and the sharp bound of \(n\) limit cycles is attained. Finally, the polynomials \(I(\lambda)\) for the cases \((m,n)=(1,1)\) and \((m,n)=(1,2)\) are explicitly computed. These polynomials give rise to a detailed description of the bifurcation diagram in terms of the coefficients \(a_1\), \(a_3\) and \(a_5\).
scientific article; zbMATH DE number 5128066