Exponentially small bifurcation functions in singular systems of O.D.E (Q1916345)

Consider the singularly perturbed problem (*) \(\varepsilon \dot u = Ju + \varepsilon^p g(u,v, \varepsilon)\), \(\dot v = f(u,v, \varepsilon)\) where \(J = \left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix} \right)\). Assume that the degenerate system \(\dot v = f(0,v,0)\) has an orbit \(v_0 (t)\) homoclinic to the hyperbolic equilibrium \(v = 0\). Then, it is shown that the bifurcation towards a homoclinic orbit \(v(t, \varepsilon)\) of (*) depends on three bifurcation functions \(G_i (\alpha, \varepsilon)\), \(i = 1,2,3\), \(\alpha \in C\) that are \(2 \pi \varepsilon\)-periodic in \(\alpha\), for \(\varepsilon \neq 0\) and satisfy \(|G_i (\alpha, \varepsilon) - G^0_i (\varepsilon) |\leq c \varepsilon^{-1} \exp (- \eta_0/ |\varepsilon |) \), \(\eta_0 > 0\) and \(G^0_i (\varepsilon) = {1 \over 2 \pi \varepsilon} \int^{2 \pi \varepsilon}_0 G_i (\alpha, \varepsilon) d \alpha\). The analysis relies essentially on the Lyapunov-Schmidt reduction.