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

Subscription Access

Authors

  • Jamal-Odysseas Maaita Aristotle University of Thessaloniki, Thessaloniki, Greece Author

DOI:

https://doi.org/10.5890/JAND.2016.06.006

Abstract

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.

References

[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] 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] Gendelman, O.V. (2004), Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment, Nonlinear Dynamics vol.37, 115–128.

[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] 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] 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] 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.

Downloads

Article Metrics

Citations 12 Crossref
PublishedJune 2016

Usage tracking begins September 1, 2026.

History Published

Issue

Vol. 5 No. 2 (2016): Regular Issue

Section

Research Articles

License

Unless an article landing page states a Creative Commons or other article-specific license, the article is published by L&H Scientific Publishing, LLC and all rights are reserved. The article-level license and copyright notice govern access and reuse.

How to Cite

Maaita, J.-O. (2026). 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), 193-197. https://doi.org/10.5890/JAND.2016.06.006

Share

Tweet Share