Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems (Q1956537)

The paper considers a singularly perturbed system of the form \[ \begin{cases} \dot x=\varepsilon f(x,y,\varepsilon),\\ \dot y=g(x,y,\varepsilon), \end{cases} \tag{1} \] where \(x\in\mathbb R^m\), \(y\) belongs to an open subset \(\Omega\subset\mathbb R^n\), \(\varepsilon\) is a small parameter and \(f\), \(g\) are \(C^r\)-functions bounded with their derivatives, \(r\geq 2\). It is assumed that the following conditions hold: {\parindent8mm \begin{itemize}\item[(i)] for any \(x\in\mathbb R^m\), the equation \(g(x,y,0)=0\) has a \(C^r\) solution \(y=v(x)\) such that \(v(x)\) and its derivatives are bounded on \(\mathbb R^m\); \item[(ii)] the infima over \(x\in\mathbb R^m\) of the moduli of the real parts of the eigenvalues of the Jacobian matrix \(g_y(x,v(x),\varepsilon)\) are separated from zero; \item[(iii)] the equation \(\dot y=g(\xi_0,y,0)\) has a solution \(y_0(t)\) homoclinic to the fixed point \(v(\xi_0)\); \item[(iv)] \(\dot y_0(t)\) is the unique bounded solution of the linear variational system \(\dot y=g_y(\xi_0,y_0(t),0)y\) up to a scalar multiple; \item[(v)] the Cauchy problem (with \(\varepsilon=0\)) \[ \dot x=f(x,v(x),0),\qquad x(0)=\xi_0 \] has the hyperbolic periodic solution \(u_0(t)\). \end{itemize}} The conditions imply the existence of a global center manifold \(y=v(x,\varepsilon)\) and of a hyperbolic periodic orbit \(q(t,\varepsilon)=(u_0(\varepsilon t,\varepsilon), v(u_0(\varepsilon t,\varepsilon),\varepsilon))\) for system (1). The paper presents an old and a new result concerning the existence of solutions of (1) that are homoclinic to \(q(t,\varepsilon)\) and their transversality. Some examples in dimension greater than three of Sil'nikov periodic-to-periodic homoclinic orbits are given.