Homoclinic transition to chaos in the Duffing oscillator driven by periodic piecewise linear forces. (English) Zbl 1513.70079
Summary: We have applied the Melnikov criterion to examine a homoclinic bifurcation and transition to chaos in the Duffing oscillator driven by different forms of periodic piecewise linear forces. The periodic piecewise linear forces considered are triangular, hat, trapezium, quadratic and rectangular types of forces. For all the forces, an analytical threshold condition for the homoclinic transition to chaos is derived using Melnikov method and Melnikov threshold curves are drawn in a parameter space. Using the Melnikov threshold curves, we have found a critical forcing amplitude \(f_c\) above which the system may behave chaotically. We have analyzed both analytically and numerically the homoclinic transition to chaos in the Duffing system with \(\epsilon\)-parametric force also. The predictions from Melnikov method have been further verified numerically by integrating the governing equation and finding areas of chaotic behaviour.
MSC:
| 70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |
| 70K44 | Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics |
| 70K50 | Bifurcations and instability for nonlinear problems in mechanics |
Keywords:
Duffing oscillator; Melnikov criterion; piecewise linear force; homoclinic bifurcation; chaosReferences:
| [1] | F.C. Moon and P.J. Holmes, A magnetoelastic strange attarctor, Journal of Sound and Vibration, 65, 1979, 275-296. · Zbl 0405.73082 |
| [2] | Y. Ueda and N. Akamatsu, Chaotically traditional phenomena in the forced negative-resistance oscillator, IEEE Transactions on Circuits and Systems, 28, 1981, 217-224. |
| [3] | C.S. Wang, Y.H. Kao, J.C. Huang and Y.S. Gou, Potential dependence of the bifurcation structure in generalized Duffing oscillators, Phys.Rev.E, 45, 1992, 3471-3485. |
| [4] | 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. |
| [5] | V. Ravichandran, V. Chinnathambi and S. Rajasekar, Nonlinear resonance in Duffing oscillator with fixed and integrative time-delay feedback, Pramana Journal of Physics, 78, 2012, 347-360. |
| [6] | G. Chong, W. Hai and Q. Xei, Spatial chaos of trapped Bose-Einstein condensate in onedimensional weak optical lattice potential, Chaos, 14, 2004, 217-223. |
| [7] | J. Heagy and W.L. Ditto, Dynamics of a two-frequency parametrically driven Duffing oscillator, J. of Nonlinear Science, 1, 1991, 423-455. · Zbl 0794.34028 |
| [8] | S. ValliPriyatharsini, M.V. Sethu Meenakshi, V. Chinnathambi and S. Rajasekar, Horseshoe dynamics in Duffing oscillator with fractional damping and multi-frequency excitation, Journal of Mathematical Modeling, 7(3), 2019, 263-276. · Zbl 1449.37027 |
| [9] | J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1987. · Zbl 0515.34001 |
| [10] | M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics, Integrability, Chaos and Patterns, Springer-Verlag, Berlin, 2003. · Zbl 1038.37001 |
| [11] | F.C. Moon, Chaotic Vibrations, An Introduction for Applied Scientists and Engineers, WileyInterscience, New York, 1987. · Zbl 0745.58003 |
| [12] | G. Litak, A. Siva and M. Borowiec, Homoclinic transition to chaos in the Ueda oscillator with external foring, arXiv:nlin/0610018v1[nlin.CD], 9-th Oct 2006. |
| [13] | E. Tyrkiel, On the role of chaotic saddles in generating chaotic dynamics in nonlinear driven oscillators, Int. J. Bifur. & Chaos, 15, 2005, 1215-1238. · Zbl 1089.37030 |
| [14] | L. Ravisankar, V. Ravichandran, V. Chinnathambi and S. Rajasekar, Nonlinear resonance in regular, random and small-world networks, Int. Journal of Physics, 7(1), 2014, 91-101. |
| [15] | B.K. Jones and G. Trefan, The Duffing oscillator: A precise electronic analog chaos demonstrator for the undergraduate laboratory, Am. J. Phys., 69(4), 2001, 464-469. |
| [16] | I.M. Kyprianidis, CH. Volos, I.N. Stouboulos and J. Hadjidemetriou, Dynamics of two resistively coupled Duffing-type electrical oscillators, Int. J. of Bifur. & Chaos, 16(6), 2006, 1765-1775. · Zbl 1131.34033 |
| [17] | V.K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12(3), 1963, 1-57. · Zbl 0135.31001 |
| [18] | V. Ravichandran, S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M.A.F. Sanjuan, Role of asymmetries on the chaotic dynamics of Double-well Duffing oscillator, Pramana Journal of Physics, 72, 2009, 927-937. |
| [19] | Z. Zing and J. Yang, Complex dynamics in pendulum equation with parametric and external perturbations, Int. J. of Bifur. & Chaos, 16(10), 2006, 3053-3078. · Zbl 1188.70058 |
| [20] | M.V. SethuMeenakshi, S. Athisayanathan, V. Chinnathambi and S. Rajasekar, Analytical estimates of the effect of amplitude modulated signal in nonlinearly damped Duffing-vander Pol oscillator, Chinese Journal of Physics, 55, 2017, 2208-2217. · Zbl 1543.34082 |
| [21] | M.V. Sethumeenakshi, S. Athisyanathan, V. Chinnathambi and S. Rajasekar, Effect of narrow band frequency modulated signal on horseshoe chaos in nonlinearly damped Duffing-vander Pol oscillator, Annual Review of Chaos, Theory, Bifurcations, and Dynamical Systems, 7, 2017, 41-55. |
| [22] | V. Ravichandran, V. Chinnathambi and S. Rajasekar, Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude modulated force, Physica A, 376, 2007, 223-236. |
| [23] | B. Ravindra and A.K. Mallik, Role of nonlinear dissipation in soft Duffing oscillators, Phys. Rev. E., 49, 1994, 4950-4954. |
| [24] | M. Cai and J. Yang, Bifurcation of periodic orbits and chaos in Duffing equation, Acta Math. Appl. Sin, Engl. Ser., 22, 2006, 495-�508. https://doi.org/10.1007/s10255-006-0325-4 · Zbl 1102.65130 |
| [25] | A.Y.T. Leung and Z. Liu, Suppressing chaos for some nonlinear oscillators, Int. J. of Bifur. & Chaos, 14(4), 2004, 1455-1465. · Zbl 1099.70017 |
| [26] | M.A.F. Sanjuan, The Effect of nonlinear damping on the universal escape oscillator, Int. J. Bifur. & Chaos, 9, 1999, 735-744. · Zbl 0977.34041 |
| [27] | 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. and Numeri. Simulat., 176, 2012, 2254-2269. |
| [28] | M.V. Sethu Meenakshi, S. Athisayanathan, V. Chinnathambi and S. Rajasekar, Effct of fractional damping in double-well Duffing-vander Pol oscillator driven by different sinusoidal forces, Int. J. of Nonlinear Sci. and Numerical Simulation, 27(2), 2019, 81-94 · Zbl 07048611 |
| [29] | R. Chacon, Control of homoclinic chaos by weak periodic perturbations, World Scientific, Singapore, 2005. · Zbl 1094.37016 |
| [30] | L. Ravisankar, V. Ravichandran and V. Chinnathambi, Prediction of horseshoe chaos in DVP oscillator driven by different periodic forces, Int. J. of Eng. and Sci., 1(5), 2012, 17-25. |
| [31] | Babar Ahmad, Stabilization of driven pendulum with periodic linear forces, Journal of Nonlinear Dynamics, Hindawi Publishing Corporation, 2013, Article ID: ID824701, 9 pages. http://dx.doi.org/10.1155/2013/824701, 2013. · Zbl 1407.70027 |
| [32] | S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990 · Zbl 0701.58001 |
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