Multiple homoclinic solutions for singular differential equations
Motivated by \textit{J. Gruendler} [Trans. Am. Math. Soc. 350, No.~9, 3797--3814 (1998; Zbl 0902.34038)], singularly perturbed systems \[ \dot x=f(x)+g(x,\epsilon,\epsilon t) \] are considered. Amongst others, the authors assume that the unperturbed system \[ \dot x=f(x) \] has a degenerate homoclinic orbit \(\gamma\) to a hyperbolic equilibrium, i.e., the tangent spaces of the corresponding stable and unstable manifolds have (along \(\gamma\)) a \(d\)-dimensional intersection, \(d>1\). The aim of the paper is to study nearby homoclinic orbits. In the bifurcation analysis, both \(\epsilon\) and the function \(g\) are considered as parameters. The main result states for \(\epsilon\not=0\) the existence of \(d\) bifurcation manifolds in the \(g\)-space. The authors employ a functional analytic approach together with the Liapunov-Schmidt method to derive the bifurcation equations. The spirit of the result and of the procedure resembles pretty much the ones of \textit{C. Zhu} and \textit{W. Zhang} [J. Differ. Equations 240, No.~1, 38--57 (2007; Zbl 1138.34024)].
scientific article; zbMATH DE number 5718434