On the limit values of Mel'nikov functions on periodic orbits (Q2459757)

The paper considers the two-dimensional nonautonomous system \[ \dot x=H'_y(x,y)+\varepsilon f(t,x,y),\qquad \dot x=-H'_x(x,y)+\varepsilon g(t,x,y), \leqno (1_\varepsilon) \] where \(\varepsilon >0\) is a small parameter, \(H(x,y)\) and \(f(t,x,y)\), \(g(t,x,y)\) are infinitely differentiable with respect to \((x,y)\in G\) and \((t,x,y)\in\mathbb R\times G\), \(G\subset\mathbb R^2\). The perturbations \(f\) and \(g\) are assumed to be periodic in \(t\). The main assumptions are: 1. The limit Hamiltonian system \((1_0)\) has a closed contour \(\Gamma\) consisting of two saddles and two separatrices; 2. In the domain bounded by \(\Gamma\) system \((1_0)\) has a one-parameter family of cycles \[ L(\delta)=\{(x_{*}(t,\delta),y_{*}(t,\delta))\}, \] where \(0<\delta<\delta_{*}\) is the distance between \(L(\delta)\) and \(\Gamma\), whose periods \(T(\delta)>0\) are such that \(T'(\delta)<0\) and \(T(\delta)\to +\infty\) as \(\delta\to +0\). 3. One has \(h'(\delta )\neq 0\), where \(h(\delta)=H(x_{*}(t,\delta),y_{*}(t,\delta))\). Under these and some other additional conditions it is shown that the number of exponentially stable periodic (with respect to \(t\)) solutions of system \((1_\varepsilon)\) infinitely grows as \(\varepsilon\to 0\).