The effect of slow flow dynamics on the oscillations of a singular damped system with an essentially nonlinear attachment

J. O. Maaita, E. Meletlidou, Alexander F Vakakis, V. Rothos

Research output: Contribution to journalArticle


We study a three degree of freedom autonomous system with damping, composed of two linear coupled oscillators with an essentially nonlinear lightweight attachment. In particular, we are interested in strongly nonlinear interactions between the linear oscillators and the essentially nonlinear attachment. First, we reduce our system to a non-autonomous second order nonlinear damped oscillator. Then, we introduce a slow-fast partition of the dynamics and average out the main frequency components in order to obtain a reduced system that is studied through the Slow Invariant Manifold (SIM) approach. Depending on the parameters of the system we find different interesting nonlinear dynamical phenomena. With the help of the SIM approach we can study how the parameters of the original problem influence the asymptotic behavior of the orbits of the system. This is accomplished with the application of Tikhonov's theorem. We classify the different cases of the dynamics according to the values of the parameters and the theoretically predicted asymptotic behavior of the orbits. Interesting phenomena are reported such as orbit capture, relaxation oscillations and complex structure of the basins of attraction.

Original languageEnglish (US)
Pages (from-to)315-328
Number of pages14
JournalJournal of Applied Nonlinear Dynamics
Issue number4
StatePublished - Jan 1 2013



  • Relaxation oscillations
  • Singular perturbation analysis
  • Slow Invariant Manifold
  • Tikhonov's theorem

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Mechanical Engineering

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