Hysteresis, Quasiperiodicity and Chaoticity in a Nonlinear Dissipative Hybrid Oscillator
Keywords:Chaos, Floquet theory, Hysteresis, Modified Rayleigh-Duffing oscillator, Whittaker method.
Hysteresis, quasi-periodicity and chaoticity in a nonlinear dissipative hybrid oscillator are studied. The modified Rayleigh-Duffing oscillator is considered. We simultaneously take into account the new nonlinear cubic, pure quadratic and hybrid dissipative terms which modify the classical Rayleigh-Duffing oscillator. The influence of each of these new parameters on the dynamics of the oscillator has been seriously studied and interesting results are obtained. It is clear that each of these new dissipation terms can be used to control the dynamics of this oscillator. Some may be used to reduce or eliminate hysteresis, amplitude jump and resonance phenomena; others may accentuate them. Similarly, these new parameters can be used to impose on the systems modeled by this oscillator, a regular, quasi-periodic or even chaotic behavior according to their field of application. Thus, one of the original results obtained is the equation of the curve delimiting the zone of instabilities of the amplitudes of harmonic oscillations. This equation thus makes it possible to know the zone of amplitude permitted or of the amplitude jump for the systems and thus to control and predict the loss or gain of energy during this jump. Finally, the second stability of the oscillations of the system is studied as well as the influence of the dissipation parameters on this stability. It should be noted that the influence of some of these parameters depends on the simultaneous presence of these parameters.
 Carrol, T. L., Communicating With Use of filtered, Synchronized, Chaotic Signals. Fundam. Theory Appl. (1995);42:105.
 Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations. Wiley, New York (1979).
 Wang, B., Barcilon, A. and Fang, Z., Stochastic Dynamics of El Nino-Southern Oscillation, Journal of Atmospheric Sciences (1996) 56:5-23.
 Francescutto, A. and Contento, G., Bifurcations in ship rolling: experimental
results and parameter identification technique. Ocean Engineering, (1999); 26: 1095-1123.
 Hayashi, C., Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York (1964).
 Rand, R. H., Ramani, D. V, Keith, W. L. and Cipolla, K. M., The quadratically damped Mathieu equation and its application to submarine dynamics. Control of Vibration and Noise: New Millennium (2000); 61: 39-50.
 Zalalutdinov, M., Parpia, J.M., Aubin, K.L., Craighead, H.G., T.Alan, Zehnder, A.T. and Rand, R.H., Hopf Bifurcation in a Disk-Shaped NEMS, Pro- ceedings of the 2003 ASME Design Engineering Technical Conferences, Biennial Conference on Mechanical Vibrations and Noise, Chicago, IL, Sept. 2-6, (2003); 56: 5-23.
 Zalalutdinov, M., Olkhovets, A., Zehnder, A., Ilic, B., Czaplewski, D. and Craighead, H. G. Optically pumped parametric amplification for micro-mechanical systems, Applied Physics Letters (2008); 78:3142-3144.
 Soliman, M. S. and Thompson, J. M. T., The effect of damping on the steady state and basin bifurcation patterns of a nonlinear mechanical oscillator. Int. J. Bifurcation and Chaos 1992); 2: 81-92.
 Wirkus, S., Rand, R. H. and Ruina, A., How to pump a swing, The College Mathematics Journal (1998) 29:266-275.
 Ostrikov, K. and Xu, S., Plasma-aided Nanofabrication: from Plasma Sources to Nano assembly, John Wiley Sons, Weinheim (2007).
 Yamapi, R., Aziz-Alaoui, M. A., Vibration analysis and bifurcations in the self-sustained electromechanical system with multiple functions, Communications in Nonlinear Science and Numerical Simulation (2007) 12:1534-1549.
 Miwadinou, C.H., Monwanou, A. V. and Chabi Orou, J. B., Active Control of the Parametric Resonance in the Modified Rayleigh-Duffing Oscillator, African Review of Physics (2O14); 9: 227-235.
 Miwadinou, C. H., Monwanou, A. V. and Chabi Orou, J. B., Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh-Duffing Oscillator. Int. J. Bifurcation and Chaos (2015); 2: 1550024.
 Miwadinou, C. H., Hinvi, A. L, Monwanou, A. V. and Chabi Orou, J. B., Nonlinear dynamics of a f6ô€€€ modified Duffing oscillator: resonant oscillations and transition to chaos. Nonlinear Dyn. (2017); 88:97-113.
 Miwadinou, C. H., Hinvi, A. L, Monwanou, A. V., Koukpemedji, A. A., Ainamon, C. and Chabi Orou, J. B., Melnikov Chaos in a Modified Rayleighâ€“Duffing Oscillator with f6ô€€€ Potential. Int. J. Bifurcation and Chaos (2016); 5: 1650085.
 Pandey, M., Rand, R. and Zehnder, A., Perturbation Analysis of Entrainment in a Micromechanical Limit Cycle Oscillator, Communications in Nonlinear Science and Numerical Simulation (2006).
 Morrison, T. M., Three Problems in Nonlinear Dynamics With 2:1 Parametric Exitation. Ph.D. Cornell University (2006).
 Wang, B. and Fang, Z., Chaotic Oscillations of Tropical Climate: A Dynamic System Theory for ENSO. Journal of Atmospheric Sciences (1996); 53:2786-2802.
 K.W. Holappa, J.M. Falzarano Application of extended state space to nonlinear ship rolling, Ocean Engineering (1999); 26: 227-240.
 Enjieu, K. H. G., Nana, N. B. R. , Orou, C. J. B. and Talla, P. K., Nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator. Phys. Plasmas, (2008);15:1-13.
 Enjieu, K. H. G. and Nana, N. B. R., Nonlinear dynamics of plasma oscillations modeled by an anharmonic oscillator. Commun. Nonlinear Sci. Numer. Simulat., (2012);17:1779-1794.
 Nayfeh, A. H., Introduction to perturbation techniques, John Wiley and Sons, New York (1981).