Coexistence of Multiple Attractors, Hysteresis, and Vibrational Resonance in the Classical Morse Oscillator Driven by an Amplitude Modulated Signal
DOI:
https://doi.org/10.15407/ujpe62.01.0051Keywords:
classical Morse oscillator, coexistence of multiple attractors, hysteresis, vibrational resonance, amplitude modulated signalAbstract
We consider the classical Morse oscillator driven by an amplitude modulated signal with two widely different frequencies w and Ω, where Ω ≫ w. The dynamics of such oscillator is numerically studied for a specific set of parameters. We show the occurrence of coexistence of several period-T orbits, bifurcations of them, and hysteresis and vibrational resonance phenomena. We characterize the periodic and chaotic orbits, hysteresis, and vibrational resonance with the use of the bifurcation diagram and response amplitude.
References
P.M. Morse. Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. E 34, 57 (1929).
https://doi.org/10.1103/PhysRev.34.57
G. Herzberg. Spectra of Diatomic Molecules (Van Nostrand, 1950).
M.E. Goggin, P.W. Milonni. Driven Morse oscillator: Classical chaos, quantum theory, and photodissociation. Phys. Rev. A 37, 796 (1988).
https://doi.org/10.1103/PhysRevA.37.796
J.R. Ackerhalft, P.W. Milonni. Chaos and incoherence in a model of multiple-photon excitation of molecular vibrations. Phys. Rev. A 34, 1211 (1986).
https://doi.org/10.1103/PhysRevA.34.1211
P.S. Dardi, S.K. Gray. Classical and quantum mechanical studies of hydrogen fluoride in an intense laser field. J. Chem. Phys. 77 (3), 1345 (1982).
https://doi.org/10.1063/1.443957
S.K. Gray. Classical aspects of laser excitation of a Morse oscillator. J. Chem. Phys. 75 (1), 67 (1983).
https://doi.org/10.1016/0301-0104(83)85008-3
K.M. Christoffel, J.M. Bowman. Classical trajectory studies of multiphoton and overtone absorption of hydrogen fluoride. J. Phys. Chem. 85 (15), 2159 (1981).
https://doi.org/10.1021/j150615a004
D. Beigie, S. Wiggins. Dynamics associated with a quasiperiodically forced Morse oscillator: Application to molecular dissociation. Phys. Rev. A 45 (7), 4803 (1992).
https://doi.org/10.1103/PhysRevA.45.4803
K. Abirami, S. Rajasekar, M.A.F. Sanjuan. Vibrational resonance in the Morse oscillator. Pramana J. of Phys. 81 (1), 127 (2013).
https://doi.org/10.1007/s12043-013-0546-z
S. Behnia, A. Akhshani, M. Panahi, R. Asadi. Controlling chaos in damped and driven Morse oscillator via slavemaster feedback. Acta Physica Polon. A 123, 7 (2013).
https://doi.org/10.12693/APhysPolA.123.7
Z. Jing, J. Deng, J. Yang. Bifurcations of periodic orbits and chaos in damped and driven Morse oscillator. Chaos, Solitons and Fractals 35, 486 (2008).
https://doi.org/10.1016/j.chaos.2006.05.038
T. Kapitaniak, Yu. Maistrenko. Multiple choice bifurcations as a source of unpredictability in dynamical systems. Phys. Rev. E 58 (4), 5161 (1998).
https://doi.org/10.1103/PhysRevE.58.5161
A. Ray, D. Ghosh, A.R. Chowdhury. Topological study of multiple coexisting attractors in a nonlinear system. J. Phys. A: Math. Theor. 42 (38), 385102 (2009).
https://doi.org/10.1088/1751-8113/42/38/385102
M. Dutta, H.F. Nusse, E. Ott, J.A. Yorke, G.H. Yuan. Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems. Phys. Rev. Lett. 83 (21), 4281 (1999) .
https://doi.org/10.1103/PhysRevLett.83.4281
K. Sun, A. Di-li Duo Li-Kun, Y. Dong, H. Wang, K. Zhong. Multiple coexisting attractors and hysteresis in the generalized Ueda oscillator. Mathematical Problems in Engineering, Article ID 256092 (2013) .
https://doi.org/10.1155/2013/256092
V.M. Gandhimathi, S. Rajasekar, J. Kurths. Vibrational and stochastic resonances in two coupled overdamped anharmonic oscillators. Phys. Letts. A 360, 279 (2006).
https://doi.org/10.1016/j.physleta.2006.08.051
K. Abirami, S. Rajasekar, M.A.F. Sanjuan. Vibrational and ghost-vibrational resonances in a modified Chua's circuit model equation. Int. J. Bifurcation Chaos 24, 1430031 (2014).
https://doi.org/10.1142/S0218127414300316
O. de Feo, G. Mario Maggio. Bifurcations in the colpitts oscillator: From theory to practice. Int. J. Bifurc. Chaos 13, 2917 (2003).
https://doi.org/10.1142/S0218127403008338
V.P. Lukomsky, I.S. Gandzha. Cascades of subharmonic stationary states in strongly non-linear driven planar systems. J. Sound and Vibration 275 (1), 351 (2004).
https://doi.org/10.1016/j.jsv.2003.06.029
L. Gammaitoni, P. Hanggi, P. Jung, F. Marchesoni. Stochastic resonance. Rev. Mod. Phys. 70, 223 (1998).
https://doi.org/10.1103/RevModPhys.70.223
P.S. Landa, P.V.E. McClintock. Vibrational resonance. J. Phys. A: Math. Gen. 33, L433 (2000) ].
https://doi.org/10.1088/0305-4470/33/45/103
S. Jeyakumari, V. Chinnathambi, S. Rajasekar, M.A.F. Sanjuan. Single and multiple vibrational resonance in a quintic oscillator with monostable potentials. Phys. Rev. E 80, 046608 (2009).
https://doi.org/10.1103/PhysRevE.80.046608
S. Rajasekar, K. Abirami, M.A.F. Sanjuan. Novel vibrational resonance in multistable systems. Chaos 21, 033106 (2011) .
https://doi.org/10.1063/1.3610213
E. Ullner, A. Zaikin, J. Garc’ıa-Ojalvo, R. B’ascones, J. Kurths. Vibrational resonance and vibrational propagation in excitable systems. Phys. Lett. A 312, 348 (2003).
https://doi.org/10.1016/S0375-9601(03)00681-9
A.A. Zaikin, L. Lopez, J.P. Baltanas, J. Kurths, M.A.F. Sanjuan. Phys. Rev. E 66, 011106 (2002).
https://doi.org/10.1103/PhysRevE.66.011106
S. Rajasekar, J. Used, A. Wagemakers, M.A.F. Sanjuan. Vibrational resonance in biological nonlinear maps. Commun. Nonlinear Sci. Numer. Simulat. 17, 3435 (2012) .
https://doi.org/10.1016/j.cnsns.2011.12.014
C. Jeevarathinam, S. Rajasekar, M.A.F. Sanjuan. Theory and numerics of vibrational resonance in Duffing oscillators with time-delayed feedback. Phys. Rev. E 83, 066205 (2011).
https://doi.org/10.1103/PhysRevE.83.066205
J.H. Yang, M.A.F. Sanjuan, H.G. Liu. Vibrational subharmonic and superharmonic resonances. Commun. Nonlinear Sci. Numer. Simulat. 30, 362 (2016) .
https://doi.org/10.1016/j.cnsns.2015.07.002
V. Ravichandran, V. Chinnathambi, S. Rajasekar. Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude-modulated force. Physica A 376, 223 (2007).
https://doi.org/10.1016/j.physa.2006.11.003
J.H. Yang, X.B. Liu. Controlling vibrational resonance in a delayed multistable system driven by an amplitude-modulated signal. Phys. Scr. 82, 025006 (2010) 06].
V.M. Gandhimathi, S. Rajasekar. Vibrational and stochastic resonances in two coupled overdamped anharmonic oscillators driven by an amplitude modulated force. Phys. Scr. 76, 693 (2007) .
https://doi.org/10.1088/0031-8949/76/6/019
M.J. Feigenbaum. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25 (1978).
https://doi.org/10.1007/BF01020332
M.J. Feigenbaum. Universal behavior in nonlinear systems. Los Alamos Science 1, 4 (1980).
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