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Escape of a particle from two-dimensional potential well

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Abstract

The paper addresses the escape of a classical particle from a two-dimensional potential well. For the case of zero initial velocities, the escape basin on the configuration plane is explored. Numeric findings are analyzed versus analytic predictions. To this point, two complementary approaches are devised. The first one involves analysis of gradient system with potential relief of the considered well. The other analytic approach takes advantage of an appropriate modification of the isolated resonance approximation. The combination of two aforementioned analytic approaches provides an outer boundary for the stability basin. The mechanism of erosion of the predicted stability boundary via the secondary resonances is explored numerically.

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References

  1. Maki, A., Miino, Y., Umeda, N., Sakai, M., Ueta, T., Kawakami, H.: Nonlinear dynamics of ship capsizing at sea. Nonlinear Theory Appl. IEICE 13, 2–24 (2022). https://doi.org/10.1587/NOLTA.13.2

    Article  Google Scholar 

  2. Naik, S., Ross, S.D.: Geometry of escaping dynamics in nonlinear ship motion. Commun. Nonlinear Sci. Numer. Simul. 47, 48–70 (2017). https://doi.org/10.1016/J.CNSNS.2016.10.021

    Article  MathSciNet  Google Scholar 

  3. Arnold, L., Chueshov, I., Ochs, G.: Stability and capsizing of ships in random sea-a survey. Nonlinear Dyn. 36, 135–179 (2004)

    Article  MathSciNet  Google Scholar 

  4. Lin, H., Yim, S.C.S.: Chaotic roll motion and capsize of ships under periodic excitation with random noise. Appl. Ocean Res. 17, 185–204 (1995). https://doi.org/10.1016/0141-1187(95)00014-3

    Article  Google Scholar 

  5. Harne, R.L., Thota, M., Wang, K.W.: Bistable energy harvesting enhancement with an auxiliary linear oscillator. Smart Mater. Struct. 22, 125028 (2013). https://doi.org/10.1088/0964-1726/22/12/125028

    Article  Google Scholar 

  6. Harne, R.L., Wang, K.W.: A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater. Struct. 22(2), 023001 (2013)

    Article  Google Scholar 

  7. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, vol. 3 (2006). https://doi.org/10.1007/978-3-540-48926-9

  8. Zhang, W.M., Yan, H., Peng, Z.K., Meng, G.: Electrostatic pull-in instability in MEMS/NEMS: a review. Sens. Actuators A Phys. 214, 187–218 (2014). https://doi.org/10.1016/J.SNA.2014.04.025

    Article  Google Scholar 

  9. Barone, A., Paternò, G.: Physics and applications of the josephson effect. Phys. Appl. Josephson Effect. (1982). https://doi.org/10.1002/352760278X

    Article  Google Scholar 

  10. Farid, M., Gendelman, O.V.: Escape of a forced-damped particle from weakly nonlinear truncated potential well. Nonlinear Dyn. 103, 63–78 (2021). https://doi.org/10.1007/s11071-020-05987-8

    Article  Google Scholar 

  11. Gottwald, J.A., Virgin, L.N., Dowell, E.H.: Routes to escape from an energy well. J. Sound Vib. 187, 133–144 (1995). https://doi.org/10.1006/JSVI.1995.0506

    Article  MathSciNet  Google Scholar 

  12. Nieto, A.R., Seoane, J.M., Sanjuán, M.A.F.: Noise activates escapes in closed Hamiltonian systems. Commun. Nonlinear Sci. Numer. Simul. 105, 106074 (2022). https://doi.org/10.1016/J.CNSNS.2021.106074

    Article  Google Scholar 

  13. Alfaro, G., Capeáns, R., Sanjuán, M.A.F.: Forcing the escape: Partial control of escaping orbits from a transient chaotic region. Nonlinear Dyn. 104, 1603–1612 (2021). https://doi.org/10.1007/s11071-021-06331-4

    Article  Google Scholar 

  14. Gendelman, O.V., Karmi, G.: Basic mechanisms of escape of a harmonically forced classical particle from a potential well. Nonlinear Dyn. 98(4), 2775–2792 (2019). https://doi.org/10.1007/s11071-019-04985-9

    Article  Google Scholar 

  15. Alhussein, H., Daqaq, M.F.: Potential well escape via vortex-induced vibrations: a single-degree-of-freedom analysis. Physica D 426, 133001 (2021). https://doi.org/10.1016/J.PHYSD.2021.133001

    Article  MathSciNet  Google Scholar 

  16. Engel, A., Ezra, T., Gendelman, O.V., Fidlin, A.: Escape of two-DOF dynamical system from the potential well. Nonlinear Dyn. (2022). https://doi.org/10.1007/S11071-022-08000-6/FIGURES/14

    Article  Google Scholar 

  17. Genda, A., Fidlin, A., Gendelman, O.V.: On the escape of a resonantly excited couple of particles from a potential well. Nonlinear Dyn. 104, 91–102 (2021). https://doi.org/10.1007/s11071-021-06312-7

    Article  Google Scholar 

  18. Kandrup, H.E., Siopis, C., Contopoulos, G., Dvorak, R.: Diffusion and scaling in escapes from two-degrees-of-freedom Hamiltonian systems. Chaos Interdiscip. J. Nonlinear Sci. 9, 381 (1999). https://doi.org/10.1063/1.166415

    Article  MathSciNet  Google Scholar 

  19. Zotos, E.E.: Fractal basin boundaries and escape dynamics in a multiwell potential. Nonlinear Dyn. 85, 1613–1633 (2016). https://doi.org/10.1007/S11071-016-2782-5/FIGURES/16

    Article  MathSciNet  Google Scholar 

  20. Han, S., Lapointe, J., Lukens, J.E.: Effect of a two-dimensional potential on the rate of thermally induced escape over the potential barrier. Phys. Rev. B 46, 6338–6345 (1992)

    Article  Google Scholar 

  21. Zotos, E.E.: Elucidating the escape dynamics of the four hill potential. Nonlinear Dyn. 89, 135–151 (2017). https://doi.org/10.1007/S11071-017-3441-1/FIGURES/12

    Article  Google Scholar 

  22. Treschev, D., Zubelevich, O.: Introduction to the Perturbation Theory of Hamiltonian Systems. Springer, Berlin (2010). https://doi.org/10.1007/978-3-642-03028-4

    Book  Google Scholar 

  23. Zaslavsky, G.M.: The Physics of Chaos in Hamiltonian Systems. Imperial College Press, London (2007)

    Book  Google Scholar 

  24. Abad, A., Elipe, A., Vallejo, M.: Normalization of perturbed non-linear Hamiltonian oscillators. Int. J. Non-Linear Mech. 32, 443–453 (1997). https://doi.org/10.1016/S0020-7462(96)00075-3

    Article  MathSciNet  Google Scholar 

  25. Gendelman, O.V., Sapsis, T.P.: Energy exchange and localization in essentially nonlinear oscillatory systems: Canonical formalism. ASME J. Appl. Mech. 84, 011009 (2017)

    Article  Google Scholar 

  26. Broer, H.W.: KAM theory: the legacy of Kolmogorov’s 1954 paper. Bull. Am. Math. Soc. 41, 507–521 (2004). https://doi.org/10.1090/S0273-0979-04-01009-2

    Article  Google Scholar 

  27. Wayne, C.E.: An Introduction to KAM Theory. (2008) http://math.bu.edu/people/cew/preprints/introkam.pdf

  28. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes, part I: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23(1), 170–194 (2009)

    Article  Google Scholar 

  29. Pak, C.H., Rosenberg, R.M.: On the existence of normal mode vibrations in nonlinear systems. Q. Appl. Math. 26(3), 403–416 (1968)

    Article  MathSciNet  Google Scholar 

  30. Yabuno, H., & Nayfeh, A.H.: Nonlinear normal modes of a parametrically excited cantilever beam. In: Normal Modes and Localization in Nonlinear Systems, pp. 65–77 (2001)

  31. Farid, M.: Escape of a harmonically forced classical particle from asymmetric potential well. Commun. Nonlinear Sci. Numer. Simul. 84, 105182 (2020). https://doi.org/10.1016/J.CNSNS.2020.105182

    Article  MathSciNet  Google Scholar 

  32. Moskovitch, O., Gendelman, O.V.: Resonance and energy transfer in forced vibro-impact systems with linear compliance. Int. J. Non-Linear Mech. 145, 104104 (2022). https://doi.org/10.1016/J.IJNONLINMEC.2022.104104

    Article  Google Scholar 

  33. Karmi, G., Kravetc, P., Gendelman, O.V.: Analytic exploration of safe basins in a benchmark problem of forced escape. Nonlinear Dyn. 106, 1573–1589 (2021)

    Article  Google Scholar 

  34. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.,: Integrals and series [Vol 1 - Elem. Functions] (1986)

  35. Kiper, A.: Fourier series coefficients for powers of the jacobian elliptic functions. Math. Comput. 43, 247–259 (1984)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to Professor Yuli Starosvetsky for useful discussions and, in particular, for pointing on the integrability in the case \(C = 3/2\). The authors are also very grateful to anonymous Reviewers for outstanding help in refining the manuscript.

Funding

The authors are very grateful to the German Research Foundation (DFG) for the financial support within the Project Number 508244284.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, Technion—Israel Institute of Technology, Haifa, Israel

    A. Engel & O. V. Gendelman

  2. Karlsruhe Institute of Technology, Karlsruhe, Germany

    A. Fidlin

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  1. A. Engel
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  2. O. V. Gendelman
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  3. A. Fidlin
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Correspondence to O. V. Gendelman.

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Engel, A., Gendelman, O.V. & Fidlin, A. Escape of a particle from two-dimensional potential well. Nonlinear Dyn 112, 1601–1618 (2024). https://doi.org/10.1007/s11071-023-09154-7

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