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