Bifurcation and chaos of a micro-void centered at the sphere composed of the transversely isotropic incompressible Gent–Thomas materials

• Published in: International journal of dynamics and control 2024-08, Vol.12 (8), p.2629-2647
• Main Authors: Ma, Minfu, Zhao, Zhentao, Zhang, Wenzheng, Niu, Datian, Yuan, Xuegang
• Format: Article
• Language: English
• Subjects: Bifurcations; Complexity; Control; Control and Systems Theory; Damping; Deformation effects; Differential equations; Dynamic response; Dynamical Systems; Engineering; Equilibrium; Nonlinear dynamics; Nonlinear response; Ordinary differential equations; Parameters; Perturbation; Qualitative analysis; Saddle points; Steering; Vibration
• ISSN: 2195-268X; 2195-2698
• DOI: 10.1007/s40435-024-01396-6

The phenomena of bifurcation and chaos are examined for a class of second-order nonlinear non-autonomous ordinary differential equations, which are formulated by the nonlinear dynamic response of a micro-void at the center of the sphere subjected to periodically perturbed loads and structural damping, and the sphere is composed of the radial transversely isotropic incompressible Gent–Thomas materials. Firstly, based on the variational principle, the mathematical model describing the problem is established with the assumption of spherical symmetric deformation. Then, the solution is derived by the first integral and so on, through qualitative analysis of the solution, some meaningful conclusions are obtained: (1) For constant loads, the influences of relevant parameters on the number of equilibrium points are discussed. Moreover, the secondary steering bifurcation of equilibrium curves and the effects of structural damping on the qualitative properties of equilibrium points are analyzed in detail. (2) For periodic loads, the quasi-periodic and chaotic motions of the micro-void are discussed, and the influences of perturbation parameters on the chaotic motions are analyzed. Particularly, when there is structural damping, periodic and quasi-periodic motions near the center are discussed, the chaos threshold near the saddle point is obtained by the Melnikov method. In addition, the bifurcation characteristics of micro-void are analyzed by bifurcation diagrams. The results show that with the increase in perturbation parameters, the motions of the micro-void present a process from periodic to chaotic and then to periodic motion alternately.