A new autonomous hyperchaotic system was proposed. The chaotic system mainly relies on a nonlinear quadratic hyperbolic cosine term and a quadratic cross-product term. The basic dynamical properties of the system were studied by means of numerical and theoretical analysis. Moreover, a synchronization method for the new hyperchaotic system was established to synchronize the two non-identical chaotic systems asymptotically and globally. The nonlinear terms in the drive system were not dropped. Numerical simulations are provided to show the effectiveness and feasibility of the methods developed.