Analytical and Numerical Results on Global Dynamics of the Generalized Boussinesq Equation with Cubic Nonlinearity and External Excitation

Mathematical Problems in Engineering: Theory, Methods and Applications

article

Mingyuan Li¹  Wei Zhang¹  Qiliang Wu⁵ 

[1] Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, Beijing University of Technology;College of Mechanical Engineering, Beijing University of Technology;Big Data Applied Research and Collaborative Innovation Center, Inner Mongolia University of Finance and Economics;School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics;School of Artificial Intelligence, Tiangong University;Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, Tiangong University

DOI  :  10.1155/2021/6629095

来源: Hindawi Publishing Corporation

【 摘 要 】

This paper analytically and numerically presents global dynamics of the generalized Boussinesq equation (GBE) with cubic nonlinearity and harmonic excitation. The effect of the damping coefficient on the dynamical responses of the generalized Boussinesq equation is clearly revealed. Using the reductive perturbation method, an equivalent wave equation is then derived from the complex nonlinear equation of the GBE. The persistent homoclinic orbit for the perturbed equation is located through the first and second measurements, and the breaking of the homoclinic structure will generate chaos in a Smale horseshoe sense for the GBE. Numerical examples are used to test the validity of the theoretical prediction. Both theoretical prediction and numerical simulations demonstrate the homoclinic chaos for the GBE.

【 授权许可】

Unknown