Computation of bifurcation manifolds of linearly independent homoclinic orbits (Q947147)

The paper considers the system \[ \dot x=f_0(x)+g(x,t), \] where \(x\in\mathbb R^n\), \(f_0\) satisfies the hypotheses: (H1) \(f_0\) is \(C^3\), (H2) \(f_0(0)=0\) and the eigenvalues of the derivative \(Df_0(0)\) lie off the imaginary axis, (H3) the autonomous equation \[ \dot x=f_0(x) \] has a solution \(\gamma(t)\) homoclinic to 0, and \(g\), regarded as a functional parameter, is such that (A) \(g\in C^3\), \(g(0,t)\equiv 0\) and the norm \(| g| _{C^3}\) is small. Let \(\mathcal A\) be a subspace of \(C^3\) bounded functions \(C_b^3(\mathbb R^n\times\mathbb R,\mathbb R^n)\), where (A) is satisfied. In [J. Differ. Equations 240, No. 1, 38--57 (2007; Zbl 1138.34024)], the authors gave conditions on \(g\) for various situations of co-existence of homoclinic orbits. Those conditions actually determine some bifurcation manifolds \(\Gamma_k\) (\(k=1,\dots ,d\)) in \(\mathcal A\). Since the manifolds \(\Gamma_k\) (\(k=1,\dots ,d\)) describe conditions on parameters for degenerate homoclinic bifurcations, it is of interest to give expressions for \(\Gamma_k\) or approximationes of them. It should be noted that it is not easy to compute \(\Gamma_k\) because of their infinite dimension. In the paper, an algorithm to compute approximately \(\Gamma_k\) is given and their first order approximationes are obtained. The algorithm is illustrated with the example.