Homoclinic orbits in families of hypersurfaces with hyperbolic periodic orbits
The author considers the Hamiltonian system \(\dot{X}=J\nabla H(X)\) on \(\mathbb{C}^n\) with a \(C^2\)-Hamiltonian \(H:\mathbb{C}^n\to\mathbb{R}\). Here \(J\) induces the standard symplectic structure on \(\mathbb{C}^n\). Denote \(X=(x,y)\in\mathbb{C}\times\mathbb{C}^{n-1}\). The basic hypothesis on \(H\) is that it has the form \(H(x,z)={1\over 2}\omega|x|^2+{1\over 2}\langle Az,z\rangle+W(x,z)\) with \(\sigma (JA)\cap i\mathbb{R}=\emptyset\), \(B|z|^\alpha\leq W(x,z)\leq C(x)|z|^\alpha\), \(\nabla_zW(x,z)\leq C(x)|z|^{\alpha-1}\) near \(\mathbb{C}\times\{0\}\), some \(\alpha>2\). Thus \(\mathbb{C}\times\{0\}\) is foliated by periodic orbits \(O_r(t)\) with period \(2\pi/\omega\) and energy \(\omega r^2/2\). The orbit \(O_r\) is hyperbolic in its energy shell and has \((n-1)\)-dimensional stable and unstable manifolds which may intersect along a homoclinic orbit. Set \({\mathcal{R}}=\{r>0:O_r\) has a homoclinic orbit\(\}\). The main result of the paper states that the closure \(\overline{\mathcal{R}}\) contains an interval \([m,\infty)\) provided \(W\) grows superquadratically: \(\langle \nabla W(X), X\rangle\geq \mu W(X)\) for some \(\mu>2\). The homoclinic orbits are obtained as limits of subharmonic orbits. The latter are obtained via variational methods.
scientific article; zbMATH DE number 1750351