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Bifurcation and chaos of an airfoil with cubic nonlinearity in incompressible flow

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Abstract

Using a combination of analytical and numerical methods, the paper studies bifurcations and chaotic motions of a two-dimensional airfoil with cubic nonlinearity in incompressible flow. One type of critical points (characterized by a negative eigenvalue, a simple zero eigenvalue and a pair of purely imaginary eigenvalues) for the bifurcation response equations is considered. With the aid of the normal form theory, the explicit expressions of the critical bifurcation lines leading to incipient and secondary bifurcations are obtained. The stability of the bifurcation solutions is also investigated. By using the undetermined coefficient method, the homoclinic orbit is found, and the uniform convergence of the homoclinic orbit series expansion is proved. It analytically demonstrates that there exists a homoclinic orbit joining the initial equilibrium point to itself, therefore Smale horseshoe chaos occurs for this system via Si’lnikov criterion. The system evolves into chaotic motion through period-doubling bifurcation, and is periodic again as the dimensionless airflow speed increases. Numerical simulations are also given, which confirm the analytical results.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    FangQi Chen & LiangQiang Zhou

  2. Department of Mechanics, Tianjin University, Tianjin, 300072, China

    YuShu Chen

  3. Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin, 300072, China

    LiangQiang Zhou & YuShu Chen

Authors
  1. FangQi Chen
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  2. LiangQiang Zhou
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  3. YuShu Chen
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Correspondence to FangQi Chen or LiangQiang Zhou.

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Chen, F., Zhou, L. & Chen, Y. Bifurcation and chaos of an airfoil with cubic nonlinearity in incompressible flow. Sci. China Technol. Sci. 54, 1954–1965 (2011). https://doi.org/10.1007/s11431-011-4456-3

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  • DOI: https://doi.org/10.1007/s11431-011-4456-3

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