ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Vibrational Resonance in a Duffing System with a Generalized Delayed Feedback

Journal of Applied Nonlinear Dynamics 2(4) (2013) 397--408 | DOI:10.5890/JAND.2013.11.006

J.H. Yang$^{1}$; Miguel A.F. Sanjuán$^{2}$; C.J.Wang$^{3}$; H. Zhu$^{1}$

$^{1}$ School of Mechanical and Electrical Engineering, China University of Mining and Technology, Xuzhou 221116, P.R. China

$^{2}$ Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain

$^{3}$ Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721016, P.R. China

Abstract

We investigate the vibrational resonance in the Duffing system with different kinds of delayed feedback. Our approach is to consider the delayed feedback as a generalized delayed feedback in a fractional-order differential version. For three special cases, the generalized delayed feedback corresponds to displacement delayed feedback, velocity delayed feedback, and acceleration delayed feedback respectively. At first, based on the vibrational mechanism, the approximate solution of the system is obtained. Then, we give conditions for all resonance patterns. The the oretical predictions are verified by numerical simulations. Furthermore, the theoretical results are in good agreement with the numerical simulations. Since the delayed feedback is in a generalized form, our results can be regarded as universal for the vibrational resonance in nonlinear systems with different kinds of delayed feedback.

Acknowledgments

We acknowledge financial support from the China Postdoctoral Science Foundation (Grant No.2012M510192), Qihang Plan of China University of Mining and Technology (CUMT), Scientific Research Foundation for Talents Introduced into CUMT (Grant No. 2011RC13), the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Spanish Ministry of Science and Innovation (Grant No. FIS2009-09898).

References

1.  [1] Landa, P.S. and McClintock, P.V. (2000), Vibrational resonance, Journal of Physics A: Mathematical and General, 33, L433-L438.
2.  [2] Zaikin, A.A., Lopez, L., Baltanás, J.P., Kurths, J., and sánjuan, M.A.F. (2002), Vibrational resonance in a noise-induced structure, Physical Review E, 66, 011106.
3.  [3] Casado-Pascual, J. and Baltanás, J.P. (2004), Effects of additive noise on vibrational resonance in a bistable system, Physical Review E, 69, 046108.
4.  [4] Chizhevsky, V.N. and Giacomelli, G. (2005), Improvement of signal-to-noise ratio in a bistable optical system: Comparison between vibrational and stochastic resonance, Physical Review A, 71, 011801.
5.  [5] Deng, B., Wang, J., Wei, X., Tsang, K.M., and Chan, W.L. (2010), Vibrational resonance in neuron populations. Chaos: An Interdisciplinary Journal of Nonlinear Science, 20, 013113.
6.  [6] Yu, H., Wang, J., Sun, J., and Yu, H. (2012), Effects of hybrid synapses on the vibrational resonance in small-world neuronal networks, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22, 033105.
7.  [7] Qin, Y.M., Wang, J., Men, C., Deng, B., and Wei, X.L. (2011), Vibrational resonance in feedforward network, Chaos: An Interdisciplinary Journal of Nonlinear Science, 21, 023133.
8.  [8] Gandhimathi, V.M., Rajasekar, S., and Kurths, J. (2006), Vibrational and stochastic resonances in two coupled overdamped anharmonic oscillators, Physics Letters A, 360, 279-286.
9.  [9] Yao, C. and Zhan, M. (2010), Signal transmission by vibrational resonance in one-way coupled bistable systems, Physical Review E, 81, 061129.
10.  [10] Yang, J.H. and Liu, X.B. (2011), Delay-improved signal propagation in globally coupled bistable systems, Physica Scripta, 83, 065008.
11.  [11] Ullner, E., Zaikin, A., Garcia-Ojalvo, J., Báscones, R., and Kurths, J. (2003), Vibrational resonance and vibrational propagation in excitable systems, Physics Letters A, 312, 348-354.
12.  [12] Yang, J.H. and Liu, X.B. (2010), Delay induces quasi-periodic vibrational resonance, Journal of physics. A, Mathematical and theoretical, 43, 122001.
13.  [13] Yang, J.H. and Liu, X.B. (2010), Controlling vibrational resonance in a multistable system by time delay, Chaos: An Interdisciplinary Journal of Nonlinear Science, 20, 033124.
14.  [14] Yang, J.H. and Liu, X.B. (2010), Controlling vibrational resonance in a delayed multistable system driven by an amplitude-modulated signal, Physica Scripta, 82, 025006.
15.  [15] Jeevarathinam, C., Rajasekar, S., and Sanjuán, M.A.F. (2011), Theory and numerics of vibrational resonance in Duffing oscillators with time-delayed feedback, Physical Review E, 83, 066205.
16.  [16] Jeevarathinam, C., Rajasekar, S. and Sanjuán, M.A.F. (2013), Effect of multiple time-delay on vibrational resonance, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23, 013136.
17.  [17] Hu, D., Yang, J., and Liu, X. (2012), Delay-induced vibrational multiresonance in FitzHugh-Nagumo system, Communications in Nonlinear Science and Numerical Simulation, 17, 1031-1035.
18.  [18] Daza, A., Wagemakers, A., Rajasekar, S., and Sanjuán, M.A.F. (2013), Vibrational resonance in a timedelayed genetic toggle switch, Communications in Nonlinear Science and Numerical Simulation, 18, 411-416.
19.  [19] Gosak, M., Perc, M., and Kralj, S. (2012), The impact of static disorder on vibrational resonance in a ferroelectric liquid crystal, Molecular Crystals and Liquid Crystals, 553, 13-20.
20.  [20] Wickenbrock, A., Holz, P.C., Wahab, N.A., Phoonthong, P., Cubero, D., and Renzoni, F. (2012), Vibrational mechanics in an optical lattice: controlling transport via potential renormalization, Physical review letters, 108, 020603.
21.  [21] Jeevarathinam, C., Rajasekar, S., and Sanjuán, M.A.F. (2013), Vibrational resonance in groundwaterdependent plant ecosystems, Ecological Complexity, http://dx.doi.org/10.1016/j.ecocom.2013.02.003.
22.  [22] Yang, J.H. and Zhu, H. (2012), Vibrational resonance in Duffing systems with fractional-order damping, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22, 013112.
23.  [23] Ortigueira, M. and Coito, F. (2012), On the Usefulness of Riemann-Liouville and Caputo Derivatives in Describing Fractional Shift-invariant Linear Systems, Journal of Applied Nonlinear Dynamics, 1, 113- 124.
24.  [24] Mainardi, F. (2012), An historical perspective on fractional calculus in linear viscoelasticity, Fractional Calculus and Applied Analysis, 15, 712-717.
25.  [25] Jesus, I.S. and Tenreiro Machado J.A. (2012), Application of Integer and Fractional Models in Electrochemical Systems, Mathematical Problems in Engineering, 2012, 248175.
26.  [26] Ruszewski, A. and Sobolewski, A. (2012), Comparative studies of control systems with fractional controllers, Przeglad Elektrotechniczny, 88, 204-208.
27.  [27] Sheng, H., Chen, Y.Q. and Qiu, T.S. (2012), Fractional Processes and Fractional-Order Signal Processing, Springer: London.
28.  [28] Mainardi, F. (2012), Fractional calculus: some basic problems in continuum and statistical mechanics, arXiv preprint arXiv:1201.0863.
29.  [29] Yang, J.H. and Zhu, H. (2013), Bifurcation and resonance induced by fractional-order damping and time delay feedback in a Duffing system, Communications in Nonlinear Science and Numerical Simulation, 18, 1316-1326.
30.  [30] Wang, Z.H. and Zheng, Y.G. (2009), The optimal form of the fractional-order difference feedbacks in enhancing the stability of a sdof vibration system, Journal of Sound and Vibration, 326, 476-488.
31.  [31] Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., and Feliu, V. (2010), Fractional-order Systems and Controls: Fundamentals and Applications, Springer, London.
32.  [32] Blekhman, I.I. (2000), Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications, World Scientific Publishing Company, Singapore.
33.  [33] Gitterman, M. (2001), Bistable oscillator driven by two periodic fields, Journal of Physics A: Mathematical and General, 34, L355-L357.
34.  [34] Borromeo, M. and Marchesoni, F. (2006), Vibrational ratchets, Physical Review E, 73, 016142.
35.  [35] Borromeo, M. and Marchesoni, F. (2007), Artificial sieves for quasimassless particles, Physical Review Letters, 99, 150605.
36.  [36] Borromeo, M. and Marchesoni, F. (2007), Mobility oscillations in high-frequency modulated devices, Europhysics Letters, 72, 362.
37.  [37] Blekhman, I.I. and Landa, P.S. (2004), Conjugate resonances and bifurcations in nonlinear systems under biharmonical excitation, International Journal of Non-Linear Mechanics, 39, 421-426.
38.  [38] Rajasekar, S., Jeyakumari, S., Chinnathambi, V., and Sanjuán, M. A. F. (2010), Role of depth and location of minima of a double-well potential on vibrational resonance, Journal of Physics A: Mathematical and Theoretical, 43, 465101.