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Homoclinic orbits in a shallow arch subjected to periodic excitation

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Abstract

Homoclinic orbits in a shallow arch subjected to periodic excitation are investigated in the presence of 1:1 internal resonance and external resonance. The method of multiple scales is used to obtain a set of near-integrable systems. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect equilibria in a resonance band of the slow manifold. These orbits arise from singular homoclinic orbits and are composed of alternating slow and fast pieces. The result obtained imply the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the class of shallow arch systems.

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References

  1. Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Moscow Math. 12, 1–57 (1963)

    Google Scholar 

  2. Kovačič, G., Wiggins, S.: Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation. Physica D 57, 185–225 (1992)

  3. Haller, G., Wiggins, S.: N-pulse homoclinic orbits in perturbations of resonant hamiltonian systems. Arch. Ration. Mech. Anal. 130, 25–101 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Feng, Z.C., Wiggins, S.: On the existence of chaos in a class of two-degree-of-freedom damped parametrically forced mechanical systems with broken O(2) symmetry. Z. Angew. Math. Phys. 44, 201–248 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Feng, Z.C., Liew, K.M.: Global bifurcations in parametrically excited systems with zero-to-one internal resonance. Nonlinear Dyn. 21, 249–263 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Samaranayake, G., Samaranayake, S., Bajaj, A.K.: Nonresonant and resonant chaotic dynamics in externally excited cyclic systems. Acta Mech. 150, 139–160 (2001)

    Article  MATH  Google Scholar 

  7. Zhang, W., Wang, F.X., Yao, M.H.: Global bifurcations and chaotic dynamics in nonlinear nonplanar oscillations of a parametrically excited cantilever beam. Nonlinear Dyn. 40, 251–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Yu, W.Q., Chen, F.Q.: Global bifurcations of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation. Nonlinear Dyn. 59, 129–141 (2010)

    Article  MATH  Google Scholar 

  9. Haller, G., Wiggins, S.: Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forces nonlinear Schrödinger equation. Physica D 85, 311–347 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. McDonald, R.J., Namachchivaya, N.S.: Pipes conveying pulsating fluid near a 0:1 resonance: global bifurcations. J. Fluid Struct. 21, 665–687 (2005)

    Article  Google Scholar 

  11. Yao, M.H., Zhang, W., Zu, J.W.: Multi-pulse chaotic dynamics in non-planar motion of parametrically excited viscoelastic moving belt. J. Sound Vib. 331, 2624–2653 (2012)

    Article  Google Scholar 

  12. Yu, W.Q., Chen, F.Q.: Multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam. Arch. Appl. Mech. 83, 647–660 (2013)

    Article  MATH  Google Scholar 

  13. Kaper, T.J., Kovačič, G.: Multi-bump orbits homoclinic to resonance bands. Trans. Amer. Math. Soc. 348, 3835–3887 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Camassa, R., Kovačič, G., Tin, S.K.: A Melnikov method for homoclinic orbits with many pulses. Arch. Ration. Mech. Anal. 143, 105–193 (1998)

  15. Zhang, W., Zhang, J.H., Yao, M.H.: The extended Melnikov method for non-autonomous nonlinear dynamical systems and application to multi-pulse chaotic dynamics of a buckled thin plate. Nonlinear Anal. Real World Appl. 11, 1442–1457 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, W., Zhang, J.H., Yao, M.H., Yao, Z.G.: Multi-pulse chaotic dynamics of a non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Acta Mech. 211, 23–47 (2010)

    Article  MATH  Google Scholar 

  17. Zhang, W., Hao, W.L.: Multi-pulse chaotic dynamics of six-dimensional non-autonomous nonlinear system for a composite laminated piezoelectric rectangular plate. Nonlinear Dyn. 73, 1005–1033 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bi, Q., Dai, H.H.: Analysis of non-linear dynamics and bifurcation of a shallow arch subjected to periodic excitation with internal resonance. J. Sound vib. 233, 557–571 (2000)

    Article  Google Scholar 

Download references

Acknowledgments

The authors greatly appreciate the anonymous reviews for their insightful comments and suggestions for further improving the quality of this work. This research was supported by the National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11326132) the National Natural Science Foundation of China (No. 11202095, 11172125, 61203128) and the National Research Foundation for the Doctoral Program of Higher Education of China (20133218110025).

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Authors and Affiliations

  1. School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, People’s Republic of China

    Weiqin Yu

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Fangqi Chen

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  1. Weiqin Yu
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Correspondence to Weiqin Yu.

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Yu, W., Chen, F. Homoclinic orbits in a shallow arch subjected to periodic excitation. Nonlinear Dyn 78, 713–727 (2014). https://doi.org/10.1007/s11071-014-1471-5

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  • DOI: https://doi.org/10.1007/s11071-014-1471-5

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