Nonlinear Resonance in a Position-Dependent Mass-Duffing Oscillator System with Monostable Potentials Driven by an Amplitude-Modulated Signal

K. Suddalai Kannan ; Sh. M. Abdul Kader ; V. Chinnathambi ; M. V. Sethu Meenakshi ; Sh. Rajasekar

Russian journal of nonlinear dynamics, Tome 19 (2023) no. 3, pp. 389-408.

Résumé

This study examines the phenomenon of vibrational resonance (VR) in a classical position-dependent mass (PDM) system characterized by three types of single-well potentials. These potentials are influenced by an amplitude-modulated (AM) signal with $\Omega\gg\omega$. Our analysis is limited to the following parametric choices: (i) $\omega_{0}^{2}, \,\beta,\,m_0^{},\, \lambda > 0$ (type-1 single-well), (ii) $\omega_{0}^{2}>0$, $\beta 0$, $2$, $1\lambda2$ (type-2 single-well), (iii) $\omega_{0}^{2}>0$, $\beta0$, $0$, $0\lambda1$ (type-3 single-well). The system presents an intriguing scenario in which the PDM function significantly contributes to the occurrence of VR. In addition to the analytical derivation of the equation for slow motions of the system based on the high-frequency signal's parameters using the method of direct separation of motion, numerical evidence is presented for VR and its basic dynamical behaviors are investigated. Based on the findings presented in this paper, the weak low-frequency signal within the single-well PDM system can be either attenuated or amplified by manipulating PDM parameters, such as mass amplitude ($m_0^{}$) and mass spatial nonlinearity $\lambda$. The outcomes of the analytical investigations are validated and further supported through numerical simulations.

Détail

Keywords: position-dependent mass system, amplitude-modulated signal, vibrational reso- nance, hysteresis
Mots-clés : chaos.

Bibliographie

[1] Rajasekar, S. and Sanjuán, M. A. F., Nonlinear Resonances, Springer, Cham, 2016, xvii+409 pp. | MR

[2] Xu, P., Jin, Y., and Xiao, S., “Stochastic Resonance in a Delayed Triple-Well Potential Driven by Correlated Noises”, Chaos, 27:11 (2017), 113109, 9 pp. | DOI | MR

[3] Arathi, S. and Rajasekar, S., “Impact of the Depth of the Wells and Multifractal Analysis on Stochastic Resonance in a Triple-Well System”, Phys. Scr., 84:6 (2011), 065011, 7 pp. | DOI | MR

[4] Jeyakumari, S., Chinnathambi, V., Rajasekar, S., and Sanjuán, M. A. F., “Single and Multiple Vibrational Resonance in a Quintic Oscillator with Monostable Potentials”, Phys. Rev. E, 80:4 (2009), 046608, 8 pp. | DOI

[5] Jeyakumari, S., Chinnathambi, V., Rajasekar, S., and Sanjuán, M. A. F., “Analysis of Vibrational Resonance in a Quintic Oscillator”, Chaos, 19:4 (2009), 043128, 8 pp. | DOI

[6] Rajamani, S., Rajasekar, S., and Sanjuán, M. A. F., Ghost-Vibrational Resonance, 2014, arXiv: 1404.5394 [nlin.CD] | MR | Zbl

[7] Van der Sande, G., Verschaffelt, G., Danckaert, J., and Mirasso, C. R., “Ghost Stochastic Resonance in Vertical-Cavity Surface-Emitting Lasers: Experiment and Theory”, Phys. Rev. E, 72:1 (2005), 016113, 9 pp. | DOI

[8] Kozma, R. and Freeman, W. J., “Chaotic Resonance — Methods and Applications for Robust Classification of Noisy and Variable Patterns”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11:6 (2001), 1607–1629 | DOI

[9] He, Y., Fu, Y., Qiao, Z., and Kang, Y., “Chaotic Resonance in a Fractional-Order Oscillator System with Application to Mechanical Fault Diagnosis”, Chaos Solitons Fractals, 142 (2021), 110536, 9 pp. | DOI | MR

[10] Bogatenko, T. R. and Semenov, V. V., “Coherence Resonance in an Excitable Potential Well”, Phys. Lett. A, 382:37 (2018), 2645–2649 | DOI | MR

[11] Yamakou, M. E. and Inack, E. M., Coherence Resonance and Stochastic Synchronization in a Small-World Neural Network: An Interplay in the Presence of Spike-Timing-Dependent Plasticity, 2022, arXiv: 2201.05436 [q-bio.NC] | MR

[12] Kurniawan, A., Toan Tran, T., Brown, S. A., Eskilsson, C., Orszaghova, J., and Greaves, D., “Numerical Simulation of Parametric Resonance in Point Absorbers Using a Simplified Model”, IET Renew. Power Gener., 15:14 (2021), 3186–3205 | DOI

[13] Li, Q., Liu, W., Zhang, Z., and Yue, Z., “Parametric Resonance of Pipes with Soft and Hard Segments Conveying Pulsating Fluids”, Int. J. Struct. Stab. Dyn., 18:10 (2018), 1850119, 28 pp. | DOI | MR

[14] Sultanov, O., Autoresonance in Oscillating Systems with Combined Excitation and Weak Dissipation, 2020, arXiv: 2009.08254 [math-ph] | MR

[15] Kovaleva, A., “Excitation and Control of Autoresonance in an Oscillator Chain”, IFAC-PapersOnLine, 48:11 (2015), 1037–1042 | DOI

[16] Landa, P. S. and McClintock, P. V. E., “Vibrational Resonance”, J. Phys. A, 33:45 (2000), L433–L438 | DOI | MR | Zbl

[17] Gitterman, M., “Bistable Oscillator Driven by Two Periodic Fields”, J. Phys. A, 34:24 (2001), L355–L357 | DOI | MR | Zbl

[18] Zaikin, A. A., López, L., Baltanás, J. P., Kurths, J., and Sanjuán, M. A. F., “Vibrational Resonance in a Noise-Induced Structure”, Phys. Rev. E, 66:1 (2002), 011106, 4 pp. | DOI

[19] Ullner, E., Zaikin, A., García-Ojalvo, J., Báscones, R., and Kurths, J., “Vibrational Resonance and Vibrational Propagation in Excitable Systems”, Phys. Lett. A, 312:5–6 (2003), 348–354 | DOI | MR

[20] Baltanás, J. P., López, L., Blechman, I. I., Landa, P. S., Zaikin, A., Kurths, J., and Sanjuán, M. A. F., “Experimental Evidence, Numerics, and Theory of Vibrational Resonance in Bistable Systems”, Phys. Rev. E, 67:6 (2003), 066119, 7 pp. | DOI | MR

[21] Chizhevsky, V. N. and Giacomelli, G., “Experimental and Theoretical Study of the Noise-Induced Gain Degradation in Vibrational Resonance”, Phys. Rev. E, 70:6 (2004), 062101, 4 pp. | DOI

[22] Chizhevsky, V. N., “Analytical Study of Vibrational Resonance in an Overdamped Bistable Oscillator”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18:6 (2008), 1767–1773 | DOI

[23] Gandhimathi, V. M., Rajasekar, S., and Kurths, J., “Vibrational and Stochastic Resonances in Two Coupled Overdamped Anharmonic Oscillators”, Phys. Lett. A, 360:2 (2006), 279–286 | DOI | Zbl

[24] Gandhimathi, V. M., Rajasekar, S., and Kurths, J., “Effects of the Shape of Periodic Forces on Stochastic Resonance”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18:7 (2008), 2073–2088 | DOI

[25] Casado-Pascual, J. and Baltanás, J. P., “Effects of Additive Noise on Vibrational Resonance in a Bistable System”, Phys. Rev. E, 69:4 (2004), 046108, 7 pp. | DOI

[26] Chizhevsky, V. N. and Giacomelli, G., “Improvement of Signal-to-Noise Ratio in a Bistable Optical System: Comparison between Vibrational and Stochastic Resonance”, Phys. Rev. A, 71:1 (2005), 011801(R), 4 pp. | DOI

[27] Chizhevsky, V. N., Smeu, E., and Giacomelli, G., “Experimental Evidence of “Vibrational Resonance” in an Optical System”, Phys. Rev. Lett., 91:22 (2003), 220602, 4 pp. | DOI

[28] Jüngling, Th., Benner, H., Stemler, Th., and Just, W., “Noise-Free Stochastic Resonance at an Interior Crisis”, Phys. Rev. E, 77:3 (2008), 036216, 6 pp. | DOI

[29] Yao, C. and Zhan, M., “Signal Transmission by Vibrational Resonance in One-Way Coupled Bistable Systems”, Phys. Rev. E, 81:6 (2010), 061129, 8 pp. | DOI

[30] Yang, J. H. and Liu, X. B., “Delay-Improved Signal Propagation in Globally Coupled Bistable Systems”, Phys. Scr., 83:6 (2011), 065008, 7 pp. | DOI | MR | Zbl

[31] Borromeo, M. and Marchesoni, F., “Vibrational Ratchets”, Phys. Rev. E, 73:1 (2006), 016142, 8 pp. | DOI

[32] Borromeo, M. and Marchesogu, F., “Artificial Sieves for Quasimassless Particles”, Phys. Rev. Lett., 99:15 (2007), 150605, 4 pp. | DOI

[33] Yang, J. H. and Liu, X. B., “Delay Induces Quasi-Periodic Vibrational Resonance”, J. Phys. A, 43:12 (2010), 122001 | DOI

[34] Jeevarathinam, C., Rajasekar, S., and Sanjuán, M. A. F., “Theory and Numerics of Vibrational Resonance in Duffing Oscillator with Time-Delayed Feedback”, Phys. Rev. E, 83:6 (2011), 066205, 12 pp. | DOI

[35] Yang, J. H. and Liu, X. B., “Controlling Vibrational Resonance in a Multistable System by Time Delay”, Chaos, 20:3 (2010), 033124 | DOI

[36] Yang, J. H. and Liu, X. B., “Controlling Vibrational Resonance in a Delayed Multistable System Driven by an Amplitude-Modulated Signal”, Phys. Scr., 82:2 (2010), 025006 | DOI | Zbl

[37] Shi, J., Huang, C., Dong, T., and Zhang, X., “High-Frequency and Low-Frequency Effects on Vibrational Resonance in a Synthetic Gene Network”, Phys. Biol., 7:3 (2010), 036006 | DOI

[38] Deng, B., Wang, J., Wei, X., Tsang, K. M., and Chan, W. L., “Vibrational Resonance in Neuron Populations”, Chaos, 20:1 (2010), 013113 | DOI

[39] Yao, C., Liu, Y., and Zhau, M., “Frequency-Resonance-Enhanced Vibrational Resonance in Bistable Systems”, Phys. Rev. E, 83:6 (2011), 061122, 6 pp. | DOI | MR

[40] Shi, J.-Ch. and Dong, T., “Control of Vibrational Biresonance in a Calcium Ion System”, Acta Phys. Chim. Sin., 26:2 (2010), 487–494 | DOI

[41] Rajasekar, S., Abirami, K., and Sanjuán, M. A. F., “Novel Vibrational Resonance in Multistable Systems”, Chaos, 21:3 (2011), 033106, 7 pp. | DOI | MR | Zbl

[42] Jin, Y., Ma, Zh., and Xiao, S., “Coherence and Stochastic Resonance in a Periodic Potential Driven by Multiplicative Dichotomous and Additive White Noise”, Chaos Solitons Fractals, 103 (2017), 470–475 | DOI | MR | Zbl

[43] Mathews, P. M. and Lakshmanan, M., “On a Unique Nonlinear Oscillator”, Quart. Appl. Math., 32 (1974/75), 215–218 | DOI | MR | Zbl

[44] Cariñena, J. F., Rañada, M. F., and Santander, M., “Curvature-Dependent Formalism, Schrödinger Equation and Energy Levels for the Harmonic Oscillator on Three-Dimensional Spherical and Hyperbolic Spaces”, J. Phys. A, 45:26 (2012), 265303, 14 pp. | DOI | MR | Zbl

[45] Kozlov, V. V. and Harin, A. O., “Kepler's Problem in Constant Curvature Spaces”, Celestial Mech. Dynam. Astronom., 54:4 (1992), 393–399 | DOI | MR | Zbl

[46] Sławianowski, J. J., “Bertrand Systems on Spaces of Constant Sectional Curvature. The Action-Angle Analysis”, Rep. Math. Phys., 46:3 (2000), 429–460 | DOI | MR

[47] Ballesteros, \smash[t]{Á.}, Enciso, A., Herranz, F. J., Ragnisco, O., and Riglioni, D., “Quantum Mechanics on Spaces of Nonconstant Curvature: The Oscillator Problem and Superintegrability”, Ann. Physics, 326:8 (2011), 2053–2073 | DOI | MR | Zbl

[48] Ragnisco, O. and Riglioni, D., “A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant Curvature with Accidental Degeneracy in the Spectrum”, SIGMA Symmetry Integrability Geom. Methods Appl., 6 (2010), 097, 10 pp. | MR | Zbl

[49] Wolf, K. B., Geometric Optics on Phase Space, Springer, Berlin, 2004, xvi+373 pp. | MR | Zbl

[50] von Roos, O., “Forces Acting on Free Carriers in Semiconductors of Inhomogeneous Composition: 1, 2”, Appl. Phys. Comm., 2:1–2 (1982), 57–87

[51] Sommerfeld, A., Lectures on Theoretical Physics, v. 1, Mechanics, Acad. Press, New York, 1964, 304 pp. | MR

[52] Krane, K., “The Falling Raindrop: Variations on a Theme of Newton”, Am. J. Phys., 49:2 (1981), 113–117 | DOI

[53] Flores, J., Solovey, G., and Gil, S., “Variable Mass Oscillator”, Am. J. Phys., 71:7 (2003), 721–725 | DOI

[54] Aquino, N., Campoy, G., and Yee-Madeira, H., “The Inversion Potential for NH$_3^{}$ Using a DFT Approach”, Chem. Phys. Lett., 296:1–2 (1998), 111–116 | DOI

[55] Hadjidemetriou, J., “Secular Variation of Mass and the Evolution of Binary Systems”, Adv. Astron. Astrophys., 5 (1967), 131–188 | DOI

[56] Richstone, D. O. and Potter, M. D., “Galactic Mass Loss: A Mild Evolutionary Correction to the Angular Size Test”, Astrophys. J., 254:2, part 1 (1982), 451–455 | DOI

[57] Bethe, H. A., “Possible Explanation of the Solar Neutrino Puzzle”, Phys. Rev. Lett., 56:12 (1986), 1305–1308 | DOI

[58] Pesce, C. P., “The Application of Lagrange Equations to Mechanical Systems with Mass Explicitly Dependent on Position”, J. Appl. Mech., 70:5 (2003), 751–756 | DOI | Zbl

[59] Davidson, M., “Variable Mass Theories in Relativistic Quantum Mechanics As an Explanation for Anomalous Low Energy Nuclear Phenomena”, J. Phys. Conf. Ser., 615 (2015), 012016 | DOI

[60] Pinto, M. B., “Introducing the Notion of Bare and Effective Mass via Newton's Second Law of Motion”, Eur. J. Phys., 28:2 (2007), 171, 10 pp. | DOI

[61] Walmsley, P., Putzke, C., Malone, L., Guillamón, I., Vignolles, D., Proust, C., Badoux, S., Coldea, A. I., Watson, M. D., Kasahara, S., Mizukami, Y., Shibauchi, T., Matsuda, Y., and Carrington, A., “Quasiparticle Mass Enhancement Close to the Quantum Critical Point in BaFe$_2^{}$(As$_{1-x}^{}$P$_x^{}$)$_2^{}$”, Phys. Rev. Lett., 110:25 (2013), 257002, 5 pp. | DOI

[62] Grinenko, V., Iida, K., Kurth, F., Efremov, D. V., Drechsler, S.-L., Cherniavskii, I., Morozov, I., Hänisch, J., Förster, T., Tarantini, C., Jaroszynski, J., Maiorov, B., Jaime, M., Yamamoto, A., Nakamura, I., Fujimoto, R., Hatano, T., Ikuta, H., and Hühne, R., “Selective Mass Enhancement Close to the Quantum Critical Point in BaFe$_2^{}$(As$_{1-x}^{}$P$_x^{}$)$_2^{}$”, Sci. Rep., 7 (2017), 4589 | DOI

[63] Wilkes, J. and Muljarov, E. A., “Exciton Effective Mass Enhancement in Coupled Quantum Wells in Electric and Magnetic Fields”, New J. Phys., 18:2 (2016), 023032, 13 pp. | DOI

[64] Pari, N. A.Á., García, D. J., and Cornaglia, P. S., “Quasiparticle Mass Enhancement as a Measure of Entanglement in the Kondo Problem”, Phys. Rev. Lett., 125:21 (2020), 217601, 5 pp. | DOI | MR

[65] El-Nabulsi, R. A., “Emergence of Quasiperiodic Quantum Wave Functions in Hausdorff Dimensional Crystals and Improved Intrinsic Carrier Concentrations”, J. Phys. Chem. Solids, 127 (2019), 224–230 | DOI

[66] El-Nabulsi, R. A., “Nonlocal Uncertainty and Its Implications in Quantum Mechanics at Ultramicroscopic Scales”, Quantum Stud.: Math. Found., 6 (2019), 123–133 | DOI | MR | Zbl

[67] El-Nabulsi, R. A., “Modeling of Electrical and Mesoscopic Circuits at Quantum Nanoscale from Heat Momentum Operator”, Phys. E, 98 (2018), 90–104 | DOI

[68] Duffing, G., Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung, Vieweg, Braunschweig, 1918, 148 pp.

[69] Roy-Layinde, T. O., Vincent, U. E., Abolade, S. A., Popoola, O. O., Laoye, J. A., and McClintock, P. V. E., “Vibrational Resonances in Driven Oscillators with Position-Dependent Mass”, Philos. Trans. Roy. Soc. A, 379:2192 (2021), 20200227, 23 pp. | DOI | MR