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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Nonlinear Behavior of a Micro-resonator with Electrostatic Force on Both Sides that is Described by a Duffing Type Oscillator

Discontinuity, Nonlinearity, and Complexity 11(4) (2022) 735--749 | DOI:10.5890/DNC.2022.12.011

L. Laskaridis, J.O. Maaita, E. Meletlidou

Physics Department, Aristotle University of Thessaloniki, Thessaloniki, Greece

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A Duffing type oscillator simulates a micromechanical resonator with electrostatic force on both sides. The system, concerning the amplitude of the external excitation and the damping parameter, has rich dynamics that contain regular (periodic and semi- periodic) and chaotic oscillations. Melnikov's function proves the existence of homoclinic chaos.


  1. [1]  Lyshevski, S.E. (2001), Nano and Microelectromechanical Systems, CRC Press LLC.
  2. [2]  Khanna, V.K. (2016), Integrated Nanoelectronics Nanoscale CMOS, Post-CMOS and Allied Nanotechnologies, Springer.
  3. [3]  Schwab, K.C., Henriksen, E.A., Worlock, J.M., and Roukes, M.L.(2000), Measurement of the quantum of thermal conductance, Nature, 404, 974-977.
  4. [4]  Lifshitz, R. and Cross, M.C. (2008), Nonlinear Dynamics of Nanomechanical and Micromechanical Resonators, Wiley.
  5. [5]  Laurent Duraffourg Julien Arcamone (2015), Nanoelectromechanical Systems, ISTE Ltd.
  6. [6]  Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N., and Fathi, M. (2014), Nano-resonator frequency response based on strain gradient theory, Journal of Physics D: Applied Physics, 47(36), 365303.
  7. [7]  Mestrom, R.M.C., Fey, R.H.B., van Beek, J.T.M., Phan, K.L., and Nijmeijer, H. (2007), Modelling the Dynamics of a MEMS Resonator: Simulations and Experiments, Elsevier B.V.
  8. [8]  Huang, X.M.H., Zorman, C.A., Mehregany, M., and Roukes, M.L. (2003), Nanodevice motion at microwave frequencies, Nature, 421-496, 2003.
  9. [9]  Cleland, A.N. and Geller, M.R. (2004),Superconducting qubit storage and entanglement with nanomechanical resonators, Physical Review Letters, 93, 070501.
  10. [10]  Peng, H.B., Chang, C.W., Aloni, S., Yuzvinsky, T.D., and Zettl, A. (2006), Ultra high frequency nanotube resonators, Physical Review Letters, 97, 087203.
  11. [11]  Rhoads, J.F., Shaw, S.W., and Turner, K.L. (2008), Nonlinear dyamics and its applications in micro and nanoresonators, ASME Dynamic Systems and Control Conference.
  12. [12]  Requa, M.V. and Turner, K.L. (2007), Precise frequency estimation in a microelectromechanical parametric resonator, Appl. Phys. Lett., 90(17), 173508.
  13. [13]  Miandoab, E.M., Yousefi-Koma, A., Pishkenari, H.N., and Tajadodianfar, F. (2015), Study of nonlinear dynamics and chaos in MEMS/NEMS resonators, Communications in Nonlinear Science and Numerical Simulation.
  14. [14]  Miandoab, E.M., Pishkenari, H.N., Yousefi-Koma, A., and Hoorzad, H. (2014), Polysilicon nano-beam model based on modified couple stress and Eringen's nonlocal elasticity theories, Physica E: Low- dimensional Systems and Nanostructures, 63, 223-228.
  15. [15]  Tilmons, H.A. and Legtenberg, R. (1994), Electrostatically driven vacuum - encapsulated polysilicon resonators: Part II: Theory and performance, Sensors and Actuators A. Physical, 45(1), 67-84.
  16. [16]  Haghighi, H.S. and Markazi, A.H. (2009), Chaos Prediction and Control in MEMS Resonators, Elsevier B.V.
  17. [17]  Zhang, W.M., Tabata, O., Tsuchiya, T., and Meng, G. (2011), Noise-induced cChaos in the Electrostatically Actuated MEMS Resonators, Elsevier B.V.
  18. [18]  Song, Z.K. and Sun, K.B. (2013), Nonlinear and Chaos Control of a Micro-electro-mechanical System by using Second-order Fast Terminal Sliding Mode Control, Elsevier B.V.
  19. [19]  Tusset, A.M., Balthazar, J.M., Rocha, R.T., Ribeiro, M.A., and Lenz, W.B. (2020), On suppression of chaotic motion of a nonlinear MEMS oscillator, Nonlinear Dynamics, 99(1), 537-557.
  20. [20]  Sabarathinam, S. and Thamilmaran, K. (2017), Implementation of analog circuit and study of chaotic dynamics in a generalized Duffing-type MEMS resonator, Nonlinear Dynamics, 87(4), 2345-2356.
  21. [21]  Maaita, J.O., Kyprianidis, I.M., Volos, C.K., and Meletlidou, E. (2013), The study of a nonlinear duffing-type oscillator driven by two voltage sources, Journal of Engineering Science and Technology Review, 6(4), 74-80.
  22. [22]  Guo, Y., Luo, A.C.J., Reyes, Z., Reyes, A., and Goonesekere, R. (2019), On experimental periodic motions in a Duffing oscillatory circuit, J. Vibr. Test. Syst. Dyn, 3, 55-69.
  23. [23]  Xu, Y. and Luo, A.C.J. (2020),Independent period-2 motions to chaos in a van der Pol-Duffing oscillator, International Journal of Bifurcation and Chaos, 30(15), 2030045.
  24. [24]  Rajamani, S. and Rajasekar, S. (2017), Variation of response amplitude in parametrically driven single Duffing oscillator and unidirectionally coupled Duffing oscillators, Journal of Applied Nonlinear Dynamics, 6(1), 121-129.
  25. [25]  Abirami, K., Rajasekar, S., and Sanjuan, M.A.F. (2016), Vibrational resonance in a system with a signum nonlinearity, Discontinuity, Nonlinearity, and Complexity, 5(1), 43-58.
  26. [26]  Skokos, Ch. (2010), The Lyapunov characteristic exponents and their computation. Dynamics of Small Solar System Bodies and Exoplanets, Springer, Berlin, Heidelber, 63-135.
  27. [27]  Lichtenberg, A.J. and Lieberman, M.A., (2013), Regular and Stochastic Motions, 38, Springer Science and Business Media.
  28. [28]  Stephen, W. (2013), Global Bifurcations and Chaos: Analytical Methods, Springer Science and Business Media.
  29. [29]  Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer.