Wave propagation in nonlinear metamaterial multi-atomic chains based on homotopy method

This paper studies the dispersion properties and wave propagation in the tetratomic nonlinear acoustic metamaterial chain based on the homotopy analysis method (HAM). We perform a comparison between HAM and Perturbation approach, harmonic balance method (HBM) and equivalent method. Results indicate that HAM can filter the unstable multiple periodic solutions fined by HBM and be more accurate. The succinct equivalent formulas can estimate the bandgaps. There is a limit of the dispersion solution when the nonlinearity tends to infinity. Analyses demonstrate that the energy dispersion in spectrum replaces the linear energy localization because of the hyperchaos that is induced by period-doubling bifurcations. The hyper-chaotic phenomena are demonstrated with frequency spectra, bifurcation diagram and Lyapunov Exponents. This paper further proves the chaotic bands can significantly expand the bandwidth for wave suppression. Enhancing the nonlinearity will vary the behavior of nonlinear bandgaps from independent state to coupling state and then experience a transition. Approaches to manipulate bands are elucidated. The strong nonlinearity is beneficial to expand the total width about 6 times. Moreover, lightweight, low-frequency and broadband characteristics are compatible so can be achieved simultaneously for nonlinear acoustic metamaterial.