Abstract
In this work, we demonstrate numerically that two-frequency excitation is an effective method to produce chaotification over very large regions of the parameter space for the Duffing oscillator with single- and double-well potentials. It is also shown that chaos is robust in the last case. Robust chaos is characterized by the existence of a single chaotic attractor which is not altered by changes in the system parameters. It is generally required for practical applications of chaos to prevent the effects of fabrication tolerances, external influences, and aging that can destroy chaos. After showing that very large and continuous regions in the parameter space develop a chaotic dynamics under two-frequency excitation for the double-well Duffing oscillator, we demonstrate that chaos is robust over these regions. The proof is based upon the observation of the monotonic changes in the statistical properties of the chaotic attractor when the system parameters are varied and by its uniqueness, demonstrated by changing the initial conditions. The effects of a second frequency in the single-well Duffing oscillator is also investigated. While a quite significant chaotification is observed, chaos is generally not robust in this case.
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References
Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press, Boulder (2015)
Cuomo, K.M., Oppenheim, A.V.: Circuit implementation of synchronized chaos with applications to communications. Phys. Rev. Lett. 71, 65–68 (1993)
Kocarev, L.: Chaos-based cryptography: a brief overview. IEEE Circuits Syst. Mag. 1, 6–21 (2001)
Verschaffelta, G., Khoder, M., Van der Sande, G.: Random number generator based on an integrated laser with on-chip optical feedback. Chaos 27, 114310 (2017)
Dantas, W.G., Rodrigues, L.R., Ujevic, S., Gusso, A.: Using nanoresonators with robust chaos as hardware random number generators. Chaos 30, 043126 (2020)
Deane, J.H.B., Hamill, D.C.: Improvement of power supply EMC by chaos. Electron. Lett. 32, 1045 (1996)
Carroll, T.L., Rachford, F.J.: Target recognition using nonlinear dynamics. In: Leung, H. (ed.) Chaotic Signal Processing, pp. 23–48. SIAM, Philadelphia (2013)
Zeraoulia, E., Sprott, J.C.: Robust Chaos and Its Applications. World Scientific Publishing, Singapore (2012)
Zhang, H., Liu, D., Wang, Z.: Controlling Chaos. Springer, London (2009)
Kovacic, I., Brennan, M.J.: The Duffing Equation Nonlinear Oscillators and Their Behavior. Wiley, London (2011)
Gallas, J.: The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows. Int. J. Bifurc. Chaos 20, 197–211 (2010)
Banerjee, S., Yorke, J.A., Grebogi, C.: Robust chaos. Phys. Rev. Lett. 80, 3049–3052 (1998)
Kuznetsov, S.P., Seleznev, E.P.: A strange attractor of the Smale–Williams type in the chaotic dynamics of a physical system. J. Exp. Theor. Phys. 102, 355–364 (2006)
Isaeva, O.B., Kuznetsov, S.P., Sataev, I.R., Savin, D.V., Seleznev, E.P.: Hyperbolic chaos and other phenomena of complex dynamics depending on parameters in a nonautonomous system of two alternately activated oscillators. Int. J. Bifurc. Chaos 25, 1530033 (2015)
Gusso, A., Dantas, W.G., Ujevic, S.: Prediction of robust chaos in micro and nanoresonators under two-frequency excitation. Chaos 29, 033112 (2019)
Wang, Y.C., Adams, S.G., Thorp, J.S., MacDonald, N.C., Hartwell, P., Bertsch, F.: Chaos in MEMS, parameter estimation and its potential application. IEEE Trans. Circuits Syst. I(45), 1013–1020 (1998)
DeMartini, B.E., Butterfield, H.E., Moehlis, J., Turner, K.L.: Chaos for a microelectromechanical oscillator governed by the nonlinear Mathieu equation. J. Microelectromech. Syst. 16, 1314–1323 (2007)
Barceló, J., de Paúl, I., Bota, S., Segura, J., Verd J.: Chaotic signal generation in the MHz range with a monolithic CMOS-MEMS microbeam resonator. In: 2019 IEEE 32nd International Conference on Micro Electro Mechanical Systems (MEMS), pp. 1037–1040 (2019). https://doi.org/10.1109/MEMSYS.2019.8870887
Cleland, A.N.: Foundations of Nanomechanics. Springer, Berlin (2003)
Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, New York (2011)
Tamaseviciute, E., Tamasevicius, A., Mykolaitis, G., Bumeliene, S., Lindberg, E.: Analogue electrical circuit for simulation of the Duffing–Holmes equation. Nonlinear Anal. Model 13, 241–252 (2008)
Moon, F.C., Holmes, W.T.: Double Poincaré sections of a quasiperiodically forced chaotic attractor. Phys. Lett. A 111, 157–160 (1985)
Wiggins, S.: Chaos in quasiperiodically forced Duffing oscillator. Phys. Lett. B 124, 138–142 (1987)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
Ueda, Y.: Random phenomena resulting from nonlinearity in the system described by Duffing’s equation. Int. J. Non-Linear Mech. 20, 481–491 (1985)
Yang, T., Chua, L.: Secure communication via chaotic parameter modulation. IEEE Trans. Circuit. Syst. -I 43, 817–819 (1996)
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R. L. Viana was supported by the National Research Funding Agency, CNPq-Brasil, Grant 301019/2019-3.
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Gusso, A., Ujevic, S. & Viana, R.L. Strong chaotification and robust chaos in the Duffing oscillator induced by two-frequency excitation. Nonlinear Dyn 103, 1955–1967 (2021). https://doi.org/10.1007/s11071-020-06183-4
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DOI: https://doi.org/10.1007/s11071-020-06183-4