This paper focuses on the dispersion properties and mechanism of the one-dimensional strongly nonlinear acoustic metamaterials (NAMMs) based on the homotopy method. The local bifurcation mechanism, which is different from conventional local resonance, is found. It is demonstrated that the local period-doubling bifurcation of multiple cells will induce chaotic bands in the NAMMs, which can significantly expand the bandwidth for wave suppression. The saddle-node bifurcation leads the system state jumping to the chaotic branch. Furthermore, the amplitude-dependent dispersion properties enable NAMMs to manipulate elastic waves externally. Study of broadband tunable abilities reveals that stronger nonlinearity (larger nonlinear coefficient or higher amplitude) presents a broader nonlinear band gap and larger transmission loss. Moreover, with less attached mass, a low frequency and broadband are achievable simultaneously. This research may provide useful approaches for elastic wave control.