Multi-pulse jumping orbits and chaotic dynamics of cantilevered pipes conveying time-varying fluid

  • Li ZhangEmail author
  • Fangqi ChenEmail author
Original paper


Global bifurcations and multi-pulse chaotic motions of cantilevered pipes conveying time-varying fluid under external excitation are investigated. The method of multiple scales and Galerkin’s approach are utilized on the partial differential governing equation to yield the four-dimensional averaged equation with 1:2 internal resonance and primary parameter resonance. Based on the averaged equations, the normal form theory is adopted to derive the explicit expressions of normal form associated with a double zeroes and a pair of purely imaginary eigenvalues. Then the energy-phase method is employed to analyze the chaotic dynamics by identifying the existence of the multi-pulse Silnikov-type orbits in the perturbed phase space. The homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated in both Hamiltonian and dissipative perturbation. The diagrams indicate a gradual breakup of homoclinic tree in the system as the dissipative factor grows. Numerical simulations are performed to show the multi-pulse jumping orbits and Silnikov-type chaotic behaviors may occur. The influence of the external excitation and the flow velocity of the cantilevered pipe on the dynamics of the system is discussed simultaneously in numerical results. The global dynamics also exhibits the existence of the chaos in the sense of Smale horseshoes for the cantilevered pipe conveying time-varying fluid under external excitation.


Cantilevered pipe Multi-pulse orbits Chaotic dynamics Energy-phase method 



The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11572148, 11872201,11772148 and the Natural Science Research Project for Colleges and Universities of Anhui Province (KJ2018A0048).

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MechanicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.School of Mathematics and PhysicsAnhui University of TechnologyMa’anshanPeople’s Republic of China
  3. 3.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  4. 4.College of Mathematics and System ScienceShandong University of Science and TechnologyQingdaoPeople’s Republic of China

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