Infinitely many homoclinic orbits for a class of superquadratic Hamiltonian systems

In this paper, we prove the existence of infinitely many homoclinic orbits for the first order Hamiltonian systems $J\dot{x}-M(t)x+R^{\mathrm{\prime }}(t,x)=0$, by the minimax methods in critical point theory, when $R(t,y)$ satisfies the superquadratic condition $\frac{R(t,x)}{\mathrm{\parallel }x{\mathring{\mathrm{\mid }}}^2}⟶\pm \mathrm{\infty }$ as $\mathrm{\parallel }x\mathrm{\parallel }⟶\mathrm{\infty }$, uniformly in $t$, and need not satisfy the global Ambrosetti-Rabinowitz condition