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 the Duffing Oscillator with Distributed Time-Delayed Feedback

Journal of Applied Nonlinear Dynamics 4(4) (2015) 391--404 | DOI:10.5890/JAND.2015.11.006

C. Jeevarathinam$^{1}$; S. Rajasekar$^{1}$; M.A.F. Sanjuán$^{2}$

$^{1}$ School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamilnadu, India

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

Abstract

We analyze the vibrational resonance in the Duffing oscillator system in the presence of (i) a gamma distributed time-delayed feedback and (ii) integrative time-delayed (uniformly distributed time delays over a finite interval) feedback. Particularly, applying a theoretical procedure we obtain an expression for the response amplitude Q at the low-frequency of the driving biharmonic force. For both double-well potential and single-well potential cases we are able to identify the regions in parameter space where either (i) two resonances, (ii) a single resonance or (iii) no resonance occur. Theoretically predicted values of Q and the values of a control parameter at which resonance occurs are in good agreement with our numerical simulation. The analysis shows a strong influence of both types of time-delayed feedback on vibrational resonance.

Acknowledgments

CJ expresses his gratitude to University Grants Commission (U.G.C.), India for financial support in the form of U.G.C. meritorious fellowship. MAFS acknowledges financial support from the Spanish Ministry of Economy and Competitivity under Project No. FIS2013-40653-P.

References

1.  [1] Lakshmanan, M. and Senthilkumar, D.V. (2010), Dynamics of Nonlinear Time-Delay Systems, Springer, Berlin.
2.  [2] Knight, B.W. (1972), Dynamics of Encoding in a Population of Neurons, J. Gen. Physiol. 59, 734-766.
3.  [3] Bak, P. Tang, C. and Wisenfeld, K. (1998), Self-organized criticality, Phys. Rev. A, 38, 364-374.
4.  [4] Saxena, G. Prasad, A. and Ramaswamy, R. (2010), Dynamical effects of integrative time-delay coupling, Phys. Rev. E, 82, 017201.
5.  [5] Ravichandran, V. Chinnathambi, V. and Rajasekar, S. (2012), Nonlinear resonance in Duffing oscillator with fixed and integrative time-delayed feedbacks, Pramana J Phys. 78, 347-360.
6.  [6] Atay, F.M. (2003), Distributed Delays Facilitate Amplitude Death of Coupled Oscillators, Phys. Rev. Lett. 91, 094101.
7.  [7] Kyrychko, Y.N. Blyuss, K.B. and Schöll, E. (2011), Amplitude death in systems of coupled oscillators with distributed-delay coupling, Eur. Phys. J. B, 84, 307-315.
8.  [8] Liang, J. and Cao, J. (2004), Global asymptotic stability of bi-directional associative memory networks with distributed delays, Appl. Math. Comput. 152, 415-424.
9.  [9] Park, J.H. and Cho, H.J. (2007), A delay-dependent asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays, Chaos, Solitons and Fractals, 33, 436-442.
10.  [10] Meyer, U., Shao, J., Chakrabarty, S., Brandt, S.F., Lukshch, H. and Wessel, R. (2008), Distributed delays stabilize neural feedback systems, Biol. Cyber, 99, 79-87.
11.  [11] Liu, N. and Guan, Z.H. (2011), Chaotification for a class of cellular neural networks with distributed delays, Phys. Lett. A, 375, 463-467.
12.  [12] Wu, J., Zhan,X.S., Zhang,X.H. and Gao,H.L. (2012), Stability and Hopf Bifurcation Analysis on a Numerical Discretization of the Distributed Delay Equation, Chin. Phys. Lett. 29, 050203.
13.  [13] Kyrychko, Y.N., Blyuss, K.B. and Schöll, E. (2011), Amplitude death in systems of coupled oscillators with distributed-delay coupling, Eur. Phys. J. B, 84, 307-315.
14.  [14] Kyrychko, Y.N., Blyuss, K.B. and Schöll, E. (2013), Amplitude and phase dynamics in oscillators with distributed-delay coupling, Phil. Trans. R. Soc. A, 371, 20120466.
15.  [15] Wolkowicz, G.S.K., Xia, H. and Ruan, S. (1997), Competition in the chemostat: a distributed delay model and its global asymptotic behavior, SIAM J Appl. Math. 57, 1281-1310.
16.  [16] Gourley, S.A. and So, J.W.H. (2003), Extinction and wavefront propagation in a reaction-diffusion model of a structured population with distributed maturation delay, Proc. R. Soc. Edinburgh, 133, 527-548.
17.  [17] Faria, T. and Trofimchuk, S. (2010), Positive travelling fronts for reaction diffusion systems with distributed delay, Nonlinearity, 23, 2457-2481.
18.  [18] Blyuss, K.B. and Kyrychko, Y.N. (2010), Stability and bifurcations in an epidemic model with varying immunity period, Bull. Math. Biol. 72, 490-505.
19.  [19] Elsheikh, S.M.A.S., Patidar,K.C. and Ouifki, R. (2014), Analysis of a malaria model with a distributed delay, IMA J Appl. Maths. 79, 1139.
20.  [20] Sipahi, R., Atay, F.M. and Niculescu, S.I. (2007), Stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers, SIAM J Appl. Math. 68, 738-759.
21.  [21] Caseres,M.O. (2014), Passagetime statistics in exponential distributed time-delay models: Noisy asymptotic dynamics J. Stat. Phys. 156, 94-118.
22.  [22] Brett, T. and Galla, T. (2013), Stochastic Processes with Distributed Delays: Chemical Langevin Equation and Linear-Noise Approximation, Phys. Rev. Lett. 110, 250601.
23.  [23] Landa, P.S. and McClintock, P.V.E. (2000), Vibrational resonance, J. Phys. A: Math. Gen. 33, L433-38.
24.  [24] Baltanás, J.P., López, L., Blechman, I.I., Landa, P.S., Zaikin, A., Kurths, J. and Sanjuán, M.A.F. (2003), Experimental evidence, numerics, and theory of vibrational resonance in bistable systems, Phys. Rev. E, 67, 066119.
25.  [25] Rajasekar, S., Abirami, K. and Sanjuan, M.A.F. (2011), Novel vibrational resonance in multistable systems, Chaos, 21, 033106.
26.  [26] Yang, J.H. and Liu, X.B. (2010), Delay induces quasi-periodic vibrational resonance, J. Phys. A: Math. Theor. 43, 122001.
27.  [27] Jeevarathinam, C., Rajasekar, S. and Sanjuan, M.A.F. (2011), Theory and numerics of vibrational resonance in Duffing oscillators with time-delayed feedback, Phys. Rev. E, 83, 066205.
28.  [28] Hu, D., Yang, J. and Liu, X. (2012), Delay-induced vibrational multiresonance in FitzHugh-Nagumo system, Commun. Nonlinear Sci. Numer. Simul. 17, 1031-1035.
29.  [29] Jeevarathinam, C., Rajasekar, S. and Sanjuan, M.A.F. (2013), Effect of multiple time-delay on vibrational resonance, Chaos, 23, 013136.