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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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A Theorem on the Bifurcations of the Slow Invariant Manifold of a System of Two Linear Oscillators Coupled to a k-order Nonlinear Oscillator

Journal of Applied Nonlinear Dynamics 5(2) (2016) 193--197 | DOI:10.5890/JAND.2016.06.006

Jamal-Odysseas Maaita

Aristotle University of Thessaloniki, Thessaloniki, Greece

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we study a system of two linear oscillators coupled to a k-order nonlinear oscillator with a mass much smaller than the mass of the linear oscillators. We prove that the Slow Invariant Manifold of the system may bifurcate only when the order of the nonlinear oscillator is an odd number.


  1. [1]  Vakakis A.F. (2010), Relaxation Oscillations, Subharmonic Orbits and Chaos in the Dynamics of a Linear Lattice with a Local Essentially Nonlinear Attachment, Nonlinear Dynamics, 61, 443–463.
  2. [2]  Vakakis A.F., Gendelman O.V., Bergman L.A., McFarland D.M., Kerschen G., Lee Y.S. (2008), Nonlinear Target Energy Transfer in Mechanical and Structural Systems, Springer Verlag.
  3. [3]  Gendelman, O.V. (2004), Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment, Nonlinear Dynamics vol.37, 115–128.
  4. [4]  Maaita J.O., Meletlidou E., Vakakis A.F., Rothos V. (2013), The effect of Slow Flow Dynamics on the Oscillations of a singular damped system with an essentially nonlinear attachment, Journal of Applied Nonlinear Dynamics 2(4), 315-328, DOI: 10.5890/JAND.2013.11.001.
  5. [5]  Maaita J.O., Meletlidou E., Vakakis A.F., Rothos V. (2014), The dynamics of the slow flow of a singular damped nonlinear system and its Parametric Study, Journal of Applied Nonlinear Dynamics, 3(1), 37–49.
  6. [6]  Maaita J.O., Meletlidou E.(2014), The Effect of Slow Invariant Manifold and Slow Flow Dynamics on the Energy Transfer and Dissipation of a Singular Damped System with an Essential Nonlinear Attachment, Journal of Nonlinear Dynamics, 2014, Article ID 208171, 10 pages.
  7. [7]  L. I. Manevitch (1999), Complex representation of dynamics of coupled oscillators, in Mathematical Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed Systems, Kluwer Academic Publishers/ Plenum, New York, 269–300.