<p>The propagation of solitary waves in fiber-reinforced hyperelastic cylindrical shells holds tremendous potential for structural health monitoring. However, solitary waves under external forces are unstable, and may break then cause chaos in severe cases. In this paper, the stability of solitary waves and chaos suppression in fiber-reinforced compressible hyperelastic cylindrical shells are investigated, and sufficient conditions for chaos generation as well as chaos suppression in cylindrical shells are provided. Under the radial periodic load and structural damping, the traveling wave equation describing the single radial symmetric motion of the cylindrical shell is obtained by using the variational principle and traveling wave method. By employing the bifurcation theory of dynamical systems, the parameter space for the appearance of peak solitary waves, valley solitary waves, and periodic waves in an undisturbed system is determined. The sufficient conditions for chaos generation are derived by the Melnikov method. It is found that the disturbed system leads to chaotic motions in the form of period-doubling bifurcation. Furthermore, a second weak periodic disturbance is applied as the non-feedback control input to suppress chaos, and the initial phase difference serves as the control parameter. According to the Melnikov function, the sufficient conditions for the second excitation amplitude and initial phase difference to suppress chaos are determined. The chaotic motions can be successfully converted to some regular motions by weak periodic perturbations. The results of theoretical analyses are compared with numerical simulation, and they are in good agreement. This paper extends the research scope of nonlinear elastic dynamics, and provides a strategy for controlling chaotic responses of hyperelastic structures.</p>