The propagation of waves in nonlinear acoustic metamaterial (NAM) is fundamentally different from that in conventional linear ones. In this article we consider two one-dimensional (1D) NAM systems featuring respectively a diatomic and a tetratomic meta unit-cell. We investigate the attenuation of waves, band structures, and bifurcations to demonstrate novel nonlinear effects, which can significantly expand the bandwidth for elastic wave suppression and cause nonlinear wave phenomena. The harmonic averaging approach, continuation algorithm, and Lyapunov exponents (LEs) are combined to study the frequency responses, nonlinear modes, bifurcations of periodic solutions, and chaos. The nonlinear resonances are studied, and the influence of damping on hyperchaotic attractors is evaluated. Moreover, a 'quantum' behavior is found between the low-energy and high-energy orbits. This work provides a theoretical base for furthering understandings and applications of NAMs.