Phenomena of Bifurcation and Chaos in the Dynamically Loaded Hyperelastic Spherical Membrane Based on a Noninteger Power-Law Constitutive Model

Abstract

The phenomena of bifurcation and chaos are studied for a class of second order nonlinear nonautonomous ordinary differential equations, which may be formulated by the nonlinear radially symmetric motion of the dynamically loaded hyperelastic spherical membrane composed of the Rivlin–Saunders material model with a noninteger power-law exponent. Firstly, based on the variational principle, the governing equation describing the problem is obtained with the spherically symmetric deformation assumption. Then, the dynamic characteristics of the system are qualitatively analyzed in detail in terms of different values of material parameters. Particularly, for a given constant load, the parameter spaces describing the bifurcation behaviors of equilibrium curves are established and the characteristics of equilibrium points are presented; for a periodically perturbed load, the quasi-periodic and chaotic behaviors are discussed for the systems with two and three equilibrium points, respectively.