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

Abstract

We consider the singularly perturbed problem (*) $$\begin{cases} \epsilon\dot u & = Ju+\epsilon^p g(u,v,\epsilon), \\ \dot v & = f(u,v,\epsilon) \end{cases} $, $J= \begin{pmatrix} 0 &-1 \\ 1 &0 \end{pmatrix}$$, assuming that the degenerate system $\dot v=f(0,v,0)$ has an orbit $v_0(t)$ homoclinic to the hyperbolic equilibrium $v=0$. Under certain conditions on $f, g$ (that are stated in Section 2), we show that the bifurcation towards a homoclinic orbit $v(t,\epsilon)$ of (*) depends on three bifurcation functions $G_i(\alpha ,\epsilon), \, i=1,2,3, \, \alpha \in \mathbb{C} $, that are $2\pi \epsilon$-periodic in $\alpha $, for $\epsilon\neq 0$, and satisfy $|G_i(\alpha ,\epsilon)-G_i^0(\epsilon)| \leq C\epsilon^{-1} e^{-\eta _0/|\epsilon|}$, $\eta _0 >0$, where $G_i^0(\epsilon)={1\over {2\pi \epsilon}}\int_{0}^{2\pi \epsilon} G_i(\alpha ,\epsilon)\, d\alpha$. Thus we see that if $G_i^0(\epsilon)=0$ the bifurcation functions are exponentially small. This fact is then used to recover some of the results shown in [4].