Discontinuity, Nonlinearity, and Complexity
Vibrational Resonance in a System with a Signum Nonlinearity
Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 4358  DOI:10.5890/DNC.2016.03.006
K. Abirami$^{1}$, S. Rajasekar$^{1}$, M.A.F. Sanjuan$^{2}$
$^{1}$ School of Physics, Bharathidasan University, Tiruchirapalli 620 024, Tamilnadu, India
$^{2}$ Departamento de F´ısica, Universidad Rey Juan Carlos, Tulipán s/n, 28933 M´ostoles, Madrid, Spain
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Abstract
We present our investigation on vibrational resonance in a system with a signum nonlinearity. We construct an exact analytical solution of the system in the presence of an external biharmonic force with two frequencies ω and Ω, Ω≥ω and use it for the computation of the response amplitude Q at the lowfrequency ω. We analyse the effect of the strength of the signum nonlinearity on vibrational resonance for the cases of the potential with a singlewell, a doublewell and a singlewell with a doublehump. An interesting feature of vibrational resonance in the system is that Q does not decay to zero for g (the amplitude of the highfrequency force) → ꝏ. We compare the features of the vibrational resonance of these two systems, since the potential of the system with the signum nonlinearity and that of the Duffing oscillator show ssimilar forms. The strength of the nonlinearity in these two systems is found to give rise distinct effects on resonance.
Acknowledgments
KA acknowledges the support from University Grants Commission (UGC), India in the form of UGCRajiv Gandhi National Fellowship. MAFS acknowledges financial support from the Spanish Ministry of Economy and Competitiveness under Project No. FIS201340653P.
References

[1]  Moon, F.C. and Holmes, P.J. (1979) A magnetoelastic strange attractor, Journal of Sound and Vibration, 65, 275296. 

[2]  Lakshmanan, M. and Rajasekar, S. (2003) Nonlinear Dynamics: Integrability, Chaos and Patterns, SpringerVerlag, New York. 

[3]  Fortuna, L., Frasca, M. and Xibilia, M.G. (2009) Chua's Circuit Implementations: Yesterday, Today and Tomorrow, World Scientific, Singapore. 

[4]  Kilic, R. (2010) A Practical Guide for Studying Chua's Circuits, World Scientific, Singapore. 

[5]  Chua, L.O., Desoer, C.A. and Kuh, E.S. (1987) Linear and Nonlinear Circuits, McGrawHill, Singapore. 

[6]  Lakshmanan, M. and Murali, K. (1996) Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific, Singapore. 

[7]  Chua, L.O. (1977) Sectionwise piecewiselinear functions: Canonical representation, properties, and applications Proccedings of IEEE, 65, 915926. 

[8]  Bernardo, M., Budd, C., Champneys, A.R. and Kowalczyk, P. (2008) Piecewisesmooth Dynamical Systems: Theory and Applications, SpringerVerlag, New York. 

[9]  De Feo, O. and Storace, M. (2007) Piecewiselinear identification of nonlinear dynamical systems in view of their circuit implementations Circuits and Systems I: Regular Papers, IEEE Transactions, 54, 15421554. 

[10]  Li, C., Sprott, J.C., Thio, W. and Zhu, H. (2014) A new piecewise linear hyperchaotic circuit, IEEE Transactions on Circuits and SystemsII, 61, 977981. 

[11]  Lowe, G.K. and Zohdy, G.K. (2010) Modeling nonlinear systems using multiple piecewise linear equations, Nonlinear Analysis: Modelling and Control, 15, 451458. 

[12]  Kominis, Y. and Bountis, T. (2010) Analytical solutions with piecewise linear dynamics, International Journal of Bifurcation and Chaos, 20, 509518. 

[13]  Grantham,W.J. and Lee, B. (1993) A chaotic limit cycle paradox, Dynamics and Control, 3, 157171. 

[14]  Sprott, J.C. (2000) Simple chaotic systems and circuits, American Journal of Physics, 68, 758763. 

[15]  Gottlieb, H.P.W. and Sprott, J.C. (2001) Simplest driven conservative chaotic oscillator, Physics Letters A, 291, 385 388. 

[16]  Danca, M.F. and Codreanu, S. (2002) On a possible approximation of discontinuous dynamical systems, Chaos, Solitons and Fractals, 13, 681691. 

[17]  Hai, D.D. (2007) On a class of quasilinear systems with signchanging nonlinearities, Journal of Mathematical Analysis and Applications, 334, 965976. 

[18]  Cortes, J. (2008) Discontinuous dynamical systems, Control Systems, IEEE, 28, 3673. 

[19]  Sun, K. and Sprott, J.C. (2010) Periodically forced chaotic system with signum nonlinearity, International Journal of Bifurcation and Chaos, 20, 14991508. 

[20]  Zhang, J., Zhang, Y., Ali, W. and Shieh, L. (2011) Linearization modeling for nonsmooth dynamical systems with approximated scalar sign function, Decision and Control and European Control Conference (CDCECC), 50th IEEE Conference, 52055210. 

[21]  Cheng, Z. and Li, C. (2015) Shooting method with signchanging nonlinearity, Nonlinear Analysis, 114, 212. 

[22]  Sun, K., Likun, A.D.D., Dong, Y., Wang, H. and Zhong, K. (2013) Multiple coexisting attractors and hysteresis in the generalized Ueda oscillator, Mathematical Problems in Engineering, 2013, 17. 

[23]  Landa, P.S. and McClintock, P.V.E. (2000) Vibrational resonance, Journal of Physics A: Mathematical and General, 33, L433L438. 

[24]  Gitterman, M. (2001) Bistable oscillator driven by two periodic fields, Journal of Physics A: Mathematical and General, 34, L355L357. 

[25]  Ullner, E., Zaikin, A., GarciaOjalvo, J., Bascones, R. and Kurths, J. (2003) Vibrational resonance and vibrational propagation in excitable systems, Physics Letters A, 312, 348354. 

[26]  Blechman, I.I. and Landa, P.S. (2004) Conjugate resonances and bifurcations in nonlinear systems under biharmonical excitation, International Journal of NonLinear Mechanics, 39, 421426. 

[27]  Chizhevsky, V.N. and Giacomelli, G. (2008) Vibrational resonance and the detection of aperiodic binary signals, Physical Review E, 77, 05112617 

[28]  Chizhevsky, V.N. (2008) Analytical study of vibrational resonance in an overdamped bistable oscillator, International Journal of Bifurcation and Chaos, 18, 17671774. 

[29]  Jeyakumari, S., Chinnathambi, V., Rajasekar, S. and Sanjuan, M.A.F. (2009) Single and multiple vibrational resonance in a quintic oscillator with monostable potentials, Physical Review E, 80, 04660817. 

[30]  Jeyakumari, S., Chinnathambi, V., Rajasekar, S. and Sanjuan, M.A.F. (2009) Analysis of vibrational resonance in a quintic oscillator, Chaos, 19, 04312818. 

[31]  Deng, B., Wang, J., Wei, X., Tsang, K.M. and Chan W.L. (2010) Vibrational resonance in neuron populations, Chaos, 20, 01311317. 

[32]  Jeevarathinam, C., Rajasekar, S. and Sanjuan, M.A.F. (2011) Theory and numerics of vibrational resonance in Duffing oscillators with timedelayed feedback, Physical Review E, 83, 066205112. 

[33]  Rajasekar, S., Abirami, K. and Sanjuan, M.A.F. (2011) Novel vibrational resonance in multistable systems, Chaos, 21, 03310617. 

[34]  Chizhevsky, V.N., Smeu, E. and Giacomelli, G. (2003) Experimental evidence of vibrational resonance in an optical system, Physical Review Letters, 91, 22060214. 

[35]  Ghosh, S. and Ray, D.S. (2015) Optical Bloch equations in a bichromatic field; vibrational resonance, The European Physical Journal B, 88, 15. 

[36]  Ghosh, S. and Ray, D.S. (2013) Nonlinear vibrational resonance, Physical Review E, 88, 04290416. 

[37]  Daza, A.,Wagemakers, A. and Sanjuan M.A.F. (2013) Strong sensitivity of the vibrational resonance induced by fractal structure, International Journal of Bifurcation and Chaos, 23, 13501291350136. 

[38]  Rajasekar, S., Used, J., Wagemakers, A. and Sanjuan, M.A.F. (2012) Vibrational resonance in biological nonlinear maps, Communication on Nonlinear Science and Numerical Simulation, 17, 34353445. 

[39]  Wang, C., Yang, K. and Qu, S. (2014) Vibrational resonance in a discrete neuronal model with time delay, International Journal of Modern Physics B, 28, 1450103112. 

[40]  Abirami, K., Rajasekar, S. and Sanjuan, M.A.F. (2013) Vibrational resonance in the Morse oscillator, Pramana Journal of Physics, 81, 127141. 