Abstract and Applied Analysis

Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

S. P. Chen and Y. H. Qian

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 879564, 11 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425048220

Digital Object Identifier
doi:10.1155/2014/879564

Mathematical Reviews number (MathSciNet)
MR3256264

Zentralblatt MATH identifier
07023244

Citation

Chen, S. P.; Qian, Y. H. Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 879564, 11 pages. doi:10.1155/2014/879564. https://projecteuclid.org/euclid.aaa/1425048220


Export citation

References

  • J. Murdock, Normal Forms and Unfoldings for Local Dynamical Systems, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003.
  • Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2004.
  • A. Rincon, F. Angulo, and G. Olivar, “Control of an anaerobic digester through normal form of fold bifurcation,” Journal of Process Control, vol. 19, no. 8, pp. 1355–1367, 2009.
  • H. Poincaré, Sur les propriétés des fonctions définies par des equations aux différences partielles thèse inaugural, Gauthier-Villars, Paris, France, 1879.
  • L. Y. Chen, N. Goldenfeld, and Y. Oono, “Renormalization group theory for global asymptotic analysis,” Physical Review Letters, vol. 73, no. 10, pp. 1311–1315, 1994.
  • R. E. L. DeVille, A. Harkin, M. Holzer, K. Josic, and T. J. Kaper, “Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations,” Physica D: Nonlinear Phenomena, vol. 237, no. 8, pp. 1029–1052, 2008.
  • H. Chiba, “Simplified renormalization group method for ordinary differential equations,” Journal of Differential Equations, vol. 246, no. 5, pp. 1991–2019, 2009.
  • M. Holzer, Renormalization group methods for singularly perturbed systems, normal forms and stability of traveling waves in a reaction-diffusion-mechanics systems [Ph.D. thesis], Boston University, 2010.
  • E. Stróżyna and H. Żoł\kadek, “The analytic and formal normal form for the nilpotent singularity,” Journal of Differential Equations, vol. 179, no. 2, pp. 479–537, 2002.
  • E. Strózyna and H. Zoladek, “Orbital formal normal forms for general Bogdanov-Takens singularity,” Journal of Differential Equations, vol. 193, no. 1, pp. 239–259, 2003.
  • H. R. Dullin and J. D. Meiss, “Nilpotent normal form for divergence-free vector fields and volume-preserving maps,” Physica D: Nonlinear Phenomena, vol. 237, no. 2, pp. 156–166, 2008.
  • J. Murdock, “On the structure of nilpotent normal form modules,” Journal of Differential Equations, vol. 180, no. 1, pp. 198–237, 2002.
  • M. Benderesky and R. Churchill, “A spectral sequence approach to normal forms,” in Recent Developments in Algebraic Topology, vol. 407 of Contemporary Mathematics, pp. 27–81, 2006.
  • M. Bendersky and R. C. Churchill, “Normal forms in a cyclically graded Lie algebra,” Journal of Symbolic Computation, vol. 41, no. 6, pp. 633–662, 2006.
  • J. A. Sanders, “Normal form theory and spectral sequences,” Journal of Differential Equations, vol. 192, no. 2, pp. 536–552, 2003.
  • Y. A. Kuznetsov, “Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 11, pp. 3535–3546, 2005.
  • W. Zhang, F. X. Wang, and J. W. Zu, “Computation of normal forms for high dimensional non-linear systems and application to non-planar non-linear oscillations of a cantilever beamčommentComment on ref. [17?]: We deleted reference [34] in the original manuscript, which was a repetition of [17?]. Please check.,” Journal of Sound and Vibration, vol. 278, no. 4-5, pp. 949–974, 2004.
  • W. Zhang, Y. Chen, and D. Cao, “Computation of normal forms for eight-dimensional nonlinear dynamical system and application to a viscoelastic moving belt,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 35–58, 2006.
  • Q. C. Zhang and A. Y. T. Leung, “Normal form of double Hopf bifurcation in forced oscillators,” Journal of Sound and Vibration, vol. 231, no. 4, pp. 1057–1069, 2000.
  • P. Yu and G. Chen, “The simplest parametrized normal forms of HOPf and generalized HOPf bifurcations,” Nonlinear Dynamics, vol. 50, no. 1-2, pp. 297–313, 2007.
  • P. Yu and A. Y. T. Leung, “Normal forms of vector fields with perturbation parameters and their application,” Chaos, Solitons & Fractals, vol. 34, no. 2, pp. 564–579, 2007.
  • M. Gazor and P. Yu, “Infinite order parametric normal form of Hopf singularity,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 18, no. 11, pp. 3393–3408, 2008.
  • G. Chen and J. D. Dora, “Further reductions of normal forms for dynamical systems,” Journal of Differential Equations, vol. 166, no. 1, pp. 79–106, 2000.
  • G. R. Itovich and J. L. Moiola, “Non-resonant double Hopf bifurcations: the complex case,” Journal of Sound and Vibration, vol. 322, no. 1-2, pp. 358–380, 2009.
  • C. Zhang, H. Yin, and H. Zheng, “Simple bifurcation of coupled advertising oscillators with delay,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1840–1844, 2011.
  • W. Jiang and Y. Yuan, “Bogdanov-Takens singularity in van der Pol's oscillator with delayed feedback,” Physica D: Nonlinear Phenomena, vol. 227, no. 2, pp. 149–161, 2007.
  • V. Gattulli, F. Di Fabio, and A. Luongo, “One to one resonant double Hopf bifurcation in aeroelastic oscillators with tuned mass dampers,” Journal of Sound and Vibration, vol. 262, no. 2, pp. 201–217, 2003.
  • Y. Li, W. Jiang, and H. Wang, “Double Hopf bifurcation and quasi-periodic attractors in delay-coupled limit cycle oscillators,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 1114–1126, 2012.
  • P. Buono and J. Belair, “Restrictions and unfolding of double Hopf bifurcation in functional differential equations,” Journal of Differential Equations, vol. 189, no. 1, pp. 234–266, 2003.
  • F. Takens, “Singularities of vector fields,” Publications Mathématiques de l'IHÉS, vol. 43, pp. 47–100, 1974.
  • W. Zhang and M. H. Yao, “Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt,” Chaos, Solitons and Fractals, vol. 28, no. 1, pp. 42–66, 2006.
  • L. H. Chen, W. Zhang, and Y. Q. Liu, “Modeling of nonlinear oscillations for viscoelastic moving belt using generalized Hamilton's principle,” Journal of Vibration and Acoustics, Transactions of the ASME, vol. 129, no. 1, pp. 128–132, 2007.
  • C. Z. Song, Studies on non-linear dynamics of a multi-degree-of freedom viscoelastic drive belt system [M.S. thesis], Beijing University of Technology, 2005. \endinput