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Journal of Applied Nonlinear Dynamics 3(1) (2014) 37--49 | DOI:10.5890/JAND.2014.03.004

J.O. Maaita$^{1}$; E. Meletlidou$^{1}$; A.F. Vakakis$^{2}$; V. Rothos$^{3}$

$^{1}$ Physics Department, Aristotle University of Thessaloniki, Greece

$^{2}$ Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, United States

$^{3}$ Department of Mathematics, Physics and Computational Sciences, Faculty of Technology, Aristotle University of Thessaloniki, Thessaloniki, Greece

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Abstract

We study the dynamical behavior of the slow flow of a three degree of freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment and compare the behavior of the initial system to the Slow Invariant Manifold (SIM). The dynamics of the slow flow can be simple, making regular oscillations in the region of the stable branches of the SIM, having relaxation oscillations or chaotic behavior.The initial system oscillates in the region of the SIM, verifying that the SIM plays an essential role for the dynamics of the initial system.

References

1. [1]	Gendelman, O.V., Starosvetsky, Y., and Feldman, M., (2008), Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: Description of response regimes, Nonlinear Dynamics, 51, 31-46 .

2. [2]	Gendelman, O.V., Vakakis, A.F., Bergman, L.A., and McFarland, D.M., (2010), Asymptotic analysis of passive suppression mechanisms for aeroelastic instabilities in a rigid Wing in subsonic flow, SIAM Journal on Applied Mathematics, 70 (5), 1655-1677.

3. [3]	Gendelman, O., Manevitch, L.I., Vakakis, A.F., and M'Closkey, R., (2001), Energy pumping in nonlinear mechanical oscillators: Part I, Dynamics of the underlying Hamiltonian systems,Journal of Applied Mechanics, 68, 34-41 .

4. [4]	Sapsis, T., Vakakis, A.F., Gendelman, O.V., Bergman, L.A., and Kerschen, G., Quinn D.D., (2009), Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II, analytical study, Journal of Sound and Vibration , 325, 297-320.

5. [5]	Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., and Lee, Y.S.,(2008), Nonlinear Target Energy Transfer in Mechanical and Structural Systems, Springer Verlag.

6. [6]	Fenichel, N., (1979), Geometric singular perturbations theory for ordinary differential equations, Journal Different Equation, 31, 53-98 .

7. [7]	Guckenheimer, J., Hoffman, K., and Weckesser, W.,(2005), Bifurcations of relaxation oscillations near folded saddles, International Journal of Bifurcation and Chaos, 15, 3411-3421.

8. [8]	Guckenheimer, J., Wechselberger, M., and Young, L.S.,(2006), Chaotic attractors of relaxation oscillators, Nonlinearity, 19, 701-720.

9. [9]	Guckenheimer, J. and Holmes, P.,(1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag.

10. [10]	Jones, K.R.T.C. (1995), Geometric Singular Perturbation Theory, Dynamical Systems Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 13-22, 1994, 44-118, Springer.

11. [11]	Neishtadt, A.I., (1987), On the change in the adiabatic invariant on crossing a separatix in systems with two degrees of freedom, PMM USSR, 51(5), 586-592.

12. [12]	Tikhonov, A.N.,(1952), Systems of differential equations containing a small parameter multiplying the derivative, Matematicheskii Sbornik, 31, 575-586 .

13. [13]	Verhulst, F. (2007), Singular perturbation methods for slow-fast dynamics, Nonlinear Dynamics, 50, 747-753.

14. [14]	Maaita, J.O., Meletlidou, E., Vakakis, A.F., and 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.

15. [15]	Manevitch, L.I. (1999), Complex representation of dynamics of coupled oscillators, in Mathematical Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed Systems, 269-300, Kluwer Academic Publishers/Plenum, New York .

16. [16]	Skokos, Ch. (2010), The Lyapunov Characteristic Exponents and their computation, Lecture Notes in Physics, 790, 63-135.