Effect of Fractional Damping in Double-Well Duffing–Vander Pol Oscillator Driven by Different Sinusoidal Forces

References

[1] S. J. Elliot, M. G. Tehrani and R. S. Langley, Nonlinear damping and quasi-linear modelling, Phil. Trans. R. Soc. A 373 (2015), 20140402.10.1098/rsta.2014.0402Suche in Google Scholar

[2] B. Ravindra and A. K. Mallik, Stability analysis of a nonlinearly damped Duffing oscillator, J. Sound. Vib. 171 (1994), 708–716.10.1006/jsvi.1994.1153Suche in Google Scholar

[3] B. Ravindra and A. K. Mallik, Role of nonlinear dissipation in soft Duffing oscillators, Phys. Rev. E. 49 (1994), 4950–4954.10.1103/PhysRevE.49.4950Suche in Google Scholar

[4] A. Sharma, V. Patidar, G. Purokit and K. K. Sud, Effects on the bifurcation and chaos in forced Duffing oscillator due to nonlinear damping, Commun. in Nonlinear Sci. Numeri. Simulat. 176 (2012), 2254–2269.10.1016/j.cnsns.2011.10.032Suche in Google Scholar

[5] M. S. Soliman and J. M. T. Thompson, The effect of damping on the steady state and basin bifurcation patterns of a nonlinear mechanical oscillator, Int. J. Bifur. Chaos. 2 (1992), 81–91.10.1142/S0218127492000082Suche in Google Scholar

[6] J. P. Baltanas, J. L. Trueba and M. A. F. Sanjuan, Energy dissipation in a nonlinearly damped Duffing oscillators, Physica D. 159 (2001), 22–34.10.1016/S0167-2789(01)00329-3Suche in Google Scholar

[7] J. Padovan and J. T. Sawicki, Nonlinear vibration of fractional damped systems, Nonlinear Dyn. 16 (1198), 321–336.10.1023/A:1008289024058Suche in Google Scholar

[8] R. E. Mickons, Fractional Vander Pol equations, J. Sound. Vib. 259 (2003), 457–460.10.1006/jsvi.2002.5170Suche in Google Scholar

[9] R. E. Mickens, K. O. Oyedeji and S. A. Rucker, Analysis of the simple harmonic oscillator with fractional damping, J. Sound Vib. 268 (2003), 839–842.10.1016/S0022-460X(03)00371-7Suche in Google Scholar

[10] Z. M. Ge and C. Y. Ou, Chaos in a fractional order modified Duffing system, Chaos, Solitons & Fractals. 34 (2007), 262–291.10.1016/j.chaos.2005.11.059Suche in Google Scholar

[11] K. Konishi, Generating chaotic behaviours in an oscillator driven by periodic forces, Phys. Lett. A 320 (2003), 200–206.10.1016/j.physleta.2003.11.024Suche in Google Scholar

[12] Z. M. Ze and W. Y. Leu, Anticontrol of chaos of two degrees of freedom loud speaker system and synchronization of different order systems, Chaos, Solitons & Fractals. 20 (2004), 503–521.10.1016/j.chaos.2003.07.001Suche in Google Scholar

[13] Y. C. Lai, Z. Liu, A. Nachman and L. Zhu, Suppression of jamming inexcitable systems by aperiodic stochastic resonance, Int. J. Bifur. Chaos. 14 (2004), 3519–3539.10.1142/S0218127404011454Suche in Google Scholar

[14] R. Chacon, Control of Homoclinic chaos by weak periodic perturbations, World Scientific, Singapore, 2005.10.1142/5001Suche in Google Scholar

[15] V. Ravichandran, V. Chinnathambi and S. Rajasekar, Effect of rectified and modulated sine forces on chaos in Duffing oscillator, Indian J. Phys. 83 (11) (2009), 1593–1603.10.1007/s12648-009-0143-7Suche in Google Scholar

[16] J. Guckenheimar and P. Holmes, The nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer, New York, 1983.10.1007/978-1-4612-1140-2Suche in Google Scholar

[17] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Springer, New York, 1990.10.1007/978-1-4757-4067-7Suche in Google Scholar

[18] J. L. Trueba, J. Rams and M. A. F. Sanjuan, Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators, Int. J. Bifur. Chaos. 109 (2000), 2257–2267.10.1142/S0218127400001419Suche in Google Scholar

[19] M. Borowice, G. Litak and A. Syta, Vibration of the Duffing oscillator; Effect of fractional damping, Shock Vib. 14 (2007), 29–36.10.1155/2007/276515Suche in Google Scholar

[20] L. J. Sheu, H. K. Chen, J. H. Chen and L. M. Tam, Chaotic dynamics of the fractionally damped Duffing equation, Chaos, Solitons Fractals. 31 (2007), 1203–1212.10.1016/j.chaos.2005.11.066Suche in Google Scholar

[21] M. A. F. Sanjuan, The effect of nonlinear damping on the universal escape oscillator, Int. J. Bifur. & Chaos. 9 (1999), 735–744.10.1142/S0218127499000523Suche in Google Scholar

[22] D. G. Silva and P. S. Varoto, Effects of variations in nonlinear damping coefficients in the parametric vibration of a cantilever beam with a lumped mass, Hindawi Publishing Corporation, Mathematical Problems in Engineering, Vol. 2008, Article ID 185351, 19 pages, doi:10.1155/2008/185351.Suche in Google Scholar

[23] Q. Liu, Y. Xu, C. Xu and J. Kurths, The sliding mode control for an airfoil system driven by harmonic and colored Gaussian noise excitations, Appl. Math. Modell. 64. doi:10.1016/j.apm.2018.07.032.Suche in Google Scholar

[24] Y. Xu, Y. Li and W. Jia, Response of Duffing oscillator with fractional damping and random phase, Nonlinear Dyn. 74 (2013), 745–753.10.1007/s11071-013-1002-9Suche in Google Scholar

[25] A. Li, Li. Ma, D. Keene, J. Klingel, M. Payene and X.J. Wang, Forced oscillations with linear and nonlinear damping, Am. J. Phys. 84 (2016), 32–40.10.1119/1.4935358Suche in Google Scholar

[26] Y. Xu, R. Gu, H. Zhang and D. Li, Chaos in diffusionless Lorenz system with a fractional order and its control, Int. J. Bifur. & Chaos. 22 (14) (2012), 1250087.10.1142/S0218127412500885Suche in Google Scholar

[27] Q. Liu, Y. Xu and J. Kurths, Active vibration suppression of a novel airfoil model with fractional order viscoelastic constitutive relationaship, J. Sound Vib. 432 (2018), 50–64.10.1016/j.jsv.2018.06.022Suche in Google Scholar

[28] Y. Xu, H. Wang, Di. Liu and H. Huang, Sliding mode control of a class of fractional chaotic systems in the presence of parameter perturbation, J. Vib. Control. 21 (3) (2015), 435–448.10.1177/1077546313486283Suche in Google Scholar

[29] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D. 16 (1985), 285–317.10.1016/0167-2789(85)90011-9Suche in Google Scholar