Pseudoholomorphic curves and multiplicity of homoclinic orbits (Q1895995)

The authors consider a Hamiltonian system of the form \[ \dot x = J(x) H'(t;x) \tag{1} \] on a compact, smooth Riemannian manifold \(M\). The function \(H : \mathbb{R} \times TM \to \mathbb{R}\) is smooth, 1-periodic in time and admits a point \(q_0 \in M\) such that \[ H(t; q_0, 0) = 0,\quad H'(t;q_0, 0) = 0, \] \[ H(t; q_0, p) \geq 0,\quad H(t;q,0) < 0 \] for all \(t\in \mathbb{R}\), \(p \in T_{q_0} M\) and \(q \in M\) with \(q \neq q_0\). In this setting, two solutions \(x_1,x_2 : \mathbb{R} \to TM\) of (1) are said to be geometrically equivalent, if there exists \(n \in \mathbb{Z}\) such that \(x_2(t) = x_1(t - n)\). Under suitable assumptions, the authors show that (1) admits infinitely many solutions \(x : \mathbb{R} \to TM\), homoclinic to \((q_0, 0)\), and these solutions are all geometrically distinct. Then they consider the autonomous case. In this setting, two solutions \(x_1, x_2: \mathbb{R} \to TM\) of (1) are said to be geometrically equivalent, if there exists \(\tau \in \mathbb{R}\) such that \(x_2(t) = x_1(t - \tau)\). In this case, the authors show that (1) admits at least \(g(M)\) solutions \(x : \mathbb{R} \to TM\), homoclinic to \((q_0, 0)\) and geometrically distinct, where \(g(M)\) is the minimal number of generators of the fundamental group \(\pi_1(M)\). About the existence of heteroclinic solutions of (1) on the \(n\)-torus, previous results have been established by \textit{P. L. Felmer}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, No. 5, 477-497 (1991; Zbl 0749.58021).