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

State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, P. R. China

E-mail Address: 591098607@mail.dlut.edu.cn

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Xuegang Yuan

https://orcid.org/0000-0003-4719-364X

School of Science, Dalian Minzu University, Dalian 116600, P. R. China

E-mail Address: yxg1971@163.com

Corresponding author.

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Datian Niu

School of Science, Dalian Minzu University, Dalian 116600, P. R. China

E-mail Address: niudt@dlnu.edu.cn

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Wenzheng Zhang

School of Science, Dalian Minzu University, Dalian 116600, P. R. China

E-mail Address: ytuzwz@163.com

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, and

Hongwu Zhang

E-mail Address: zhanghw@dlut.edu.cn

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https://doi.org/10.1142/S0218127421300159Cited by:4 (Source: Crossref)

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.

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