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Stability of a Duffing oscillator with a position-dependent mass

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Abstract

This paper reports high-resolution stability diagrams classifying the different solutions of a driven Duffing oscillator with a position-dependent mass. The Duffing oscillator is a prototypical model to produce reference charts for experimentalists and to study stability phases normally present in nonlinear systems. The diagrams obtained reveal the size and organization of the oscillation phases present in the control plane defined by a mass index and the amplitude of the external drive. The range of values of the mass index and the force amplitude which were investigated display a variety of dynamical behaviors, as sequences of periodic orbits with number of spikes increasing by spike-adding and by spike-doubling routes, and spike-doubling routes ending in regions of chaotic dynamics. Chaotic situations reported in the literature are seen as particular cases of a complex scenario, which includes the occurrence of quint points, where five different stability phases meet. The phase organization is also investigated as a function of the angular frequency of the external force. The results show that the system is free of chaos for sufficiently small frequency of the driving force, and that chaotic regions increase in size and occur for higher values of the force amplitude, with the increase of the driving frequency.

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All needed data are contained in the paper.

References

  1. P.M. Mathews, M. Lakshmanan, Quart. Appl. Math. 32, 215 (1974). https://doi.org/10.1090/qam/430422

    Article  MathSciNet  Google Scholar 

  2. P.W. Higgs, J. Phys. A Math. Theor. 12, 309 (1979). https://doi.org/10.1088/0305-4470/12/3/006

    Article  ADS  Google Scholar 

  3. A.R. Plastino, A. Rigo, M. Casas, F. Garcias, A. Plastino, Phys. Rev. A 60, 4318 (1999). https://doi.org/10.1103/PhysRevA.60.4318

    Article  ADS  Google Scholar 

  4. J. Yu, S.H. Dong, G.H. Sun, Phys. Lett. A 322, 290 (2004). https://doi.org/10.1016/j.physleta.2004.01.039

    Article  ADS  MathSciNet  Google Scholar 

  5. S. Cruz y Cruz, J. Negro, L.M. Nieto, Phys. Lett. A 369, 400 (2007). https://doi.org/10.1016/j.physleta.2007.05.040

    Article  ADS  MathSciNet  Google Scholar 

  6. S. Cruz y Cruz, J. Negro, L.M. Nieto, J. Phys. Conf. Ser. 128, 012053 (2008). https://doi.org/10.1088/1742-6596/128/1/012053

    Article  Google Scholar 

  7. S. Cruz y Cruz, O. Rosas-Ortiz, J. Phys. A Math. Theor. (2009). https://doi.org/10.1088/1751-8113/42/18/185205

    Article  Google Scholar 

  8. S. Cruz y Cruz, O. Rosas-Ortiz, Int. J. Theor. Phys. 50, 2201 (2011). https://doi.org/10.1007/s10773-011-0728-8

    Article  Google Scholar 

  9. B. Bagchi, S. Das, S. Ghosh, S. Poria, J. Phys. A Math. Theor. (2013). https://doi.org/10.1088/1751-8113/46/3/032001

    Article  Google Scholar 

  10. B. Bagchi, S. Das, S. Ghosh, S. Poria, J. Phys. A Math. Theor. (2013). https://doi.org/10.1088/1751-8113/46/36/368002

    Article  Google Scholar 

  11. O. Mustafa, J. Phys. A Math. Theor. (2013). https://doi.org/10.1088/1751-8113/46/36/368001

    Article  Google Scholar 

  12. D. Ghosh, B. Roy, Ann. Phys. 353, 222 (2015). https://doi.org/10.1016/j.aop.2014.11.009

    Article  ADS  Google Scholar 

  13. O. Mustafa, J. Phys. A Math. Theor. (2015). https://doi.org/10.1088/1751-8113/48/22/225206

    Article  Google Scholar 

  14. B. Bagchi, S. Ghosh, B. Pal, S. Poria, J. Math. Phys. (2016). https://doi.org/10.1063/1.4939486

    Article  Google Scholar 

  15. R. Bravo, M.S. Plyushchay, Phys. Rev. D (2016). https://doi.org/10.1103/PhysRevD.93.105023

    Article  Google Scholar 

  16. O. Mustafa, Phys. Scr. 95, 065214 (2020). https://doi.org/10.1088/1402-4896/ab825b

    Article  ADS  Google Scholar 

  17. O. Mustafa, Phys. Scr. 96, 065205 (2021). https://doi.org/10.1088/1402-4896/abf06a

    Article  ADS  Google Scholar 

  18. O. Mustafa, Eur. Phys. J. Plus 136, 249 (2021). https://doi.org/10.1140/epjp/s13360-021-01250-0

    Article  Google Scholar 

  19. L.A. Hinvi, A.A. Koukpemedji, V.A. Monwanou, C.H. Miwadinou, V. Kamdoum Tamba, J.B. Chabi Orou, J. Korean Phys. Soc. 79, 755 (2021). https://doi.org/10.1007/s40042-021-00276-y

    Article  ADS  Google Scholar 

  20. T.O. Roy-Layinde, U.E. Vincent, S.A. Abolade, O.O. Popoola, J.A. Laoye, P.V.E. McClintock, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. (2021). https://doi.org/10.1098/rsta.2020.0227

    Article  Google Scholar 

  21. C. Ruby, M. Lakshmanan, J. Phys. A. Math. Theor. (2021). https://doi.org/10.1088/1751-8121/ac1b77

    Article  Google Scholar 

  22. S. Bouledjedj, A. Khodja, F. Benamira, L. Guechi, Can. J. Phys. (2022). https://doi.org/10.1139/cjp-2022-0096

    Article  Google Scholar 

  23. E.I. Jafarov, J. Van der Jeugt, Pramana-J. Phys. (2022). https://doi.org/10.1007/s12043-021-02279-7

    Article  Google Scholar 

  24. O. Mustafa, Ann. Phys. (2022). https://doi.org/10.1016/j.aop.2022.169124

    Article  Google Scholar 

  25. A.G. Nikitin, Ukr. Math. J. 74, 405 (2022). https://doi.org/10.1007/s11253-022-02072-8

    Article  Google Scholar 

  26. B. Bagchi, R. Ghosh, Dirac equation with Morse potential under the influence of position-dependent mass and local Fermi velocity, in International Conference on Quantum Phenomena, Quantum Control and Quantum Optics, Quantum Fest 2021, edited by S. Cruz y Cruz and M. Enriquez, volume 2448 of Journal of Physics Conference Series, Ctr Res & Adv Studies; Inst Politecnico Nacl, Unidad Profes Interdisciplinaria Ingn & Tecnologias Avanzadas; Tecnologico Monterrey; Fondo Apoyo Publicaciones Tecnologico Monterrey, 2023, Biennial International Conference on Quantum Phenomena, Quantum Control and Quantum Optics (Quantum Fest), Monterrey, MEXICO, Oct 25–29, (2021)

  27. B.G.G. da Costa, I.S.S. Gomez, B. Rath, J. Math. Phys. (2023). https://doi.org/10.1063/5.0094564

    Article  Google Scholar 

  28. F.M. Fernandez, Quantum Stud. Math. Found. (2023). https://doi.org/10.1007/s40509-023-00305-4

    Article  Google Scholar 

  29. Y. Gao, J. Mayfield, S. Luo, Numer. Methods Partial Differ. Equ. (2023). https://doi.org/10.1002/num.23006

    Article  Google Scholar 

  30. R.M. Lima, H.R. Christiansen, Phys. E-Low-dimens. Syst. Nanostruct. (2023). https://doi.org/10.1016/j.physe.2023.115688

    Article  Google Scholar 

  31. O. Mustafa, Z. Algadhi, Quantum Stud. Math. Found. 10, 263 (2023). https://doi.org/10.1007/s40509-023-00293-5

    Article  MathSciNet  Google Scholar 

  32. B. Rath, P. Mallick, J. Asad, R. Wannan, R. Jarrar, H. Shanak, Axioms (2023). https://doi.org/10.3390/axioms12040318

    Article  Google Scholar 

  33. A. Zeni, J. Gallas, Phys. D-Nonlinear Phenom. 89, 71 (1995). https://doi.org/10.1016/0167-2789(95)00215-4

    Article  ADS  Google Scholar 

  34. C. Quesne, J. Math. Phys. (2015). https://doi.org/10.1063/1.4906113

    Article  Google Scholar 

  35. Lyapunov, A. M., The General Problem of the Stability of Motion, Taylor and Francis, London, 1992, Edited by A.T. Fuller (English translation of a French translation of the 1892 Russian original)

  36. A. Pikovsky, A. Politi, Lyapunov Exponents, A Tool to Explore Complex Dynamics (Cambridge University Press, Cambridge, 2016)

    MATH  Google Scholar 

  37. J.C. Vallejo, M.A.F. Sanjuan, Predictability of Chaotic Dynamics, 2nd edn. (Springer Verlag, Berlin, 2011)

    MATH  Google Scholar 

  38. J.G. Freire, J.A.C. Gallas, Phys. Lett. A 375, 1097 (2011). https://doi.org/10.1016/j.physleta.2011.01.017

    Article  ADS  Google Scholar 

  39. J.G. Freire, J.A.C. Gallas, Phys. Chem. Chem. Phys. 13, 12191 (2011). https://doi.org/10.1039/c0cp02776f

    Article  Google Scholar 

  40. X.-B. Rao, Y.-D. Chu, Y.-X. Chang, J.-G. Zhang, Commun. Nonlinear Sci. Numer. Simul. 50, 330 (2017). https://doi.org/10.1016/j.cnsns.2017.03.016

    Article  ADS  MathSciNet  Google Scholar 

  41. L. Xu, Y.-D. Chu, Q. Yang, Chaos Solitons Fract. (2020). https://doi.org/10.1016/j.chaos.2020.109998

    Article  Google Scholar 

  42. C.S. Rodrigues, C.G.P. dos Santos, R.C.C. de Miranda, E. Parma, H. Varela, R. Nagao, Phys. Chem. Chem. Phys. 22, 21823 (2020). https://doi.org/10.1039/d0cp04238b

    Article  Google Scholar 

  43. J.A.C. Gallas, Braz. J. Phys. 51, 919 (2021). https://doi.org/10.1007/s13538-021-00865-z

    Article  ADS  Google Scholar 

  44. J.A.C. Gallas, J. Phys. Condens. Matter (2022). https://doi.org/10.1088/1361-648X/ac4b2b

    Article  Google Scholar 

  45. C. Bonatto, J.A.C. Gallas, Y. Ueda, Phys. Rev. E (2008). https://doi.org/10.1103/PhysRevE.77.026217

    Article  Google Scholar 

  46. J.G. Freire, M.R. Gallas, J.A.C. Gallas, EPL (2017). https://doi.org/10.1209/0295-5075/118/38003

    Article  Google Scholar 

  47. C.K. Volos, J.A.C. Gallas, Eur. Phys. J. Plus (2022). https://doi.org/10.1140/epjp/s13360-021-02318-7

    Article  Google Scholar 

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Acknowledgements

JACG was partially supported by the Max-Planck-Institut für Physik Komplexer Systeme, Dresden, Germany, and by CNPq, Brazil, Grant PQ-305305/2020-4. LFZ acknowledges support from CNPq, Brazil, Grant No. 303189/2022-3, and partial support by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. LFZ gratefully dedicates this paper to Dr. Jason A. C. Gallas, whose optimistic aptitude and enthusiasm, and dedication to scientific research, was a source of inspiration for friends and colleagues around the world. Jason Gallas started the present collaboration with LFZ, and actively participated in the preparation of the paper almost up to his untimely demise, at May 1st, 2023.

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Authors and Affiliations

  1. Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, Porto Alegre, RS, 91501-970, Brazil

    Luiz F. Ziebell

  2. Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187, Dresden, Germany

    Jason A. C. Gallas

  3. Instituto de Altos Estudos da Paraíba, Rua Silvino Lopes 419-2502, João Pessoa, 58039-190, Brazil

    Jason A. C. Gallas

  4. Complexity Sciences Center, 9225 Collins Avenue Suite 1208, Surfside, FL, 33154, USA

    Jason A. C. Gallas

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  1. Luiz F. Ziebell
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Contributions

JACG started the discussion about the subject, devised the flow, computed all bitmaps, and started to write the paper. LFZ participated in the discussions about the subject, contributed to the bibliographical research and to the verification of the mathematical structure involved in the model, and finished up the writing of the paper.

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Correspondence to Luiz F. Ziebell.

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Ziebell, L.F., Gallas, J.A.C. Stability of a Duffing oscillator with a position-dependent mass. Eur. Phys. J. Plus 138, 930 (2023). https://doi.org/10.1140/epjp/s13360-023-04569-y

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