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Mechanisms for extreme events in position-dependent mass systems

M. Wasif Ahamed1,2,*, V. Chithiika Ruby3,4,†, M. Sathish Aravindh2,‡, A. Venkatesan1,§, and M. Lakshmanan2,∥

  • 1PG & Research Department of Physics, Nehru Memorial College (Autonomous), Affiliated to Bharathidasan University, Puthanampatti, Tiruchirappalli 621 007, Tamil Nadu, India
  • 2Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
  • 3Center for Nonlinear and Complex Networks, SRM TRP Engineering College, Tiruchirappalli 621 105, Tamil Nadu, India
  • 4Center for Research, Easwari Engineering College, Chennai 600 089, Tamil Nadu, India

  • *Contact author: ahamedw019@gmail.com
  • †Contact author: chithiikaruby.v@trp.srmtrichy.edu.in
  • ‡Contact author: sathisharavindhm@gmail.com
  • §Contact author: av.phys@gmail.com
  • ∥Contact author: lakshman.cnld@gmail.com
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Phys. Rev. E 114, 044204 – Published 8 October, 2026

DOI: https://doi.org/10.1103/19cl-x33l

Abstract

Oscillators defined on curved spaces provide a natural framework for exploring geometry-induced nonlinear dynamics. The two-dimensional Mathews-Lakshmanan oscillator and Higgs oscillator are prominent examples of geometrically induced nonlinear oscillators. The two-dimensional Higgs oscillator in flat space arises from a conformal (stereographic) projection of the harmonic oscillator on the sphere S2 onto the two-dimensional plane. Its one-dimensional counterpart, defined by a non-Euclidean geometry with curvature parameter, exhibits chaotic dynamics and extreme events (EEs) when subjected to damping and external driving forces [Ahamed et al., Chaos 35, 023119 (2025)]. In this work, we introduce coupling through mass interaction among multiple one-dimensional damped and forced Higgs oscillators, and we investigate the resulting collective nonlinear dynamics within a many-particle framework. More importantly, we establish the connection between the conformal mass m(x) and the extrema of velocity through a power-law relation, demonstrating how the conformal mass regulates the turning points leading to the emergence of extreme events EEs. Furthermore, to provide additional insight into the underlying mechanism, we quantify the energy synchronization error among the oscillators, thereby revealing how energy synchronization evolves during the onset of EEs.

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References (52)

  1. V. Lucarini, D. Faranda, J. M. M. de Freitas, M. Holland, T. Kuna, M. Nicol, M. Todd, S. Vaienti, et al., Extremes and Recurrence in Dynamical Systems (Wiley Inc., Hoboken, New Jersey, 2016).
  2. S. Albeverio, V. Jentsch, and H. Kantz, Extreme Events in Nature and Society (Springer Science & Business Media, Berlin Heidelberg, 2006).
  3. S. N. Chowdhury, A. Ray, S. K. Dana, and D. Ghosh, Extreme events in dynamical systems and random walkers: A review, Phys. Rep. 966, 1 (2022).
  4. C. Grebogi, E. Ott, and J. A. Yorke, Crises, sudden changes in chaotic attractors, and transient chaos, Physica D 7, 181 (1983).
  5. C. Grebogi, E. Ott, F. Romeiras, and J. A. Yorke, Critical exponents for crisis-induced intermittency, Phys. Rev. A 36, 5365 (1987).
  6. Y. Pomeau and P. Manneville, Universality in Chaos (Routledge, New York, 2017), pp. 327–335.
  7. S. L. Kingston, K. Thamilmaran, P. Pal, U. Feudel, and S. K. Dana, Extreme events in the forced Liénard system, Phys. Rev. E 96, 052204 (2017).
  8. A. Saha and U. Feudel, Extreme events in FitzHugh-Nagumo oscillators coupled with two time delays, Phys. Rev. E 95, 062219 (2017).
  9. A. Saha and U. Feudel, Riddled basins of attraction in systems exhibiting extreme events, Chaos 28, 033610 (2018).
  10. S. D. Vijay, K. Thamilmaran, and A. I. Ahamed, Superextreme spiking oscillations and multistability in a memristor-based Hindmarsh–Rose neuron model, Nonlin. Dyn. 111, 789 (2023).
  11. B. Thangavel, S. Srinivasan, and T. Kathamuthu, Extreme events in a forced BVP oscillator: Experimental and numerical studies, Chaos, Solitons Fractals 153, 111569 (2021).
  12. A. Ray, A. Mishra, D. Ghosh, T. Kapitaniak, S. K. Dana, and C. Hens, Extreme events in a network of heterogeneous Josephson junctions, Phys. Rev. E 101, 032209 (2020).
  13. G. Fotopoulos, D. J. Frantzeskakis, N. I. Karachalios, P. G. Kevrekidis, V. Koukouloyannis, and K. Vetas, Extreme wave events for a nonlinear Schrödinger equation with linear damping and Gaussian driving, Commun. Nonlin. Sci. Numer. Simul. 82, 105058 (2020).
  14. S. Kumarasamy and A. N. Pisarchik, Extreme events in systems with discontinuous boundaries, Phys. Rev. E 98, 032203 (2018).
  15. B. Kaviya, R. Suresh, and V. Chandrasekar, Extreme bursting events via pulse-shaped explosion in mixed Rayleigh-Liénard nonlinear oscillator, Eur. Phys. J. Plus 137, 844 (2022).
  16. S. Sudharsan, T. K. Pal, D. Ghosh, and J. Kurths, Extreme events in two coupled chaotic oscillators, Phys. Rev. E 111, 034214 (2025).
  17. D. Zhao, Y. Li, Y. Xu, Q. Liu, and J. Kurths, Extreme events in a class of nonlinear Duffing-type oscillators with a parametric periodic force, Eur. Phys. J. Plus 137, 314 (2022).
  18. S. Sudharsan, A. Venkatesan, P. Muruganandam, and M. Senthilvelan, Emergence and mitigation of extreme events in a parametrically driven system with velocity-dependent potential, Eur. Phys. J. Plus 136, 129 (2021).
  19. P. Durairaj, S. Kanagaraj, S. Kumarasamy, and K. Rajagopal, Emergence of extreme events in a quasiperiodic oscillator, Phys. Rev. E 107, L022201 (2023).
  20. R. Venkatesh, S. Dinesh Vijay, M. Sathish Aravindh, A. Venkatesan, and M. Lakshmanan, Superextreme and transient dynamics in forced doubly clamped silicon flexural resonator, Chaos 36, 033148 (2026).
  21. P. W. Higgs, Dynamical symmetries in a spherical geometry. I, J. Phys. A 12, 309 (1979).
  22. R. Delbourgo, A. Salam, and J. Strathdee, Infinities of nonlinear and Lagrangian theories, Phys. Rev. 187, 1999 (1969).
  23. M. Daniel, M. Kruskal, M. Lakshmanan, and K. Nakamura, Singularity structure analysis of the continuum Heisenberg spin chain with anisotropy and transverse field: Nonintegrability and chaos, J. Math. Phys. 33, 771 (1992).
  24. J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. 158, 142 (1984).
  25. V. Chithiika Ruby and M. Lakshmanan, Liénard type nonlinear oscillators and quantum solvability, Phys. Scr. 99, 062004 (2024).
  26. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Wiley, New York, 1991).
  27. T. Gora and F. Williams, Theory of electronic states and transport in graded mixed semiconductors, Phys. Rev. 177, 1179 (1969).
  28. R. Khordad, Effect of position-dependent effective mass on linear and nonlinear optical properties of a cubic quantum dot, Physica B 406, 3911 (2011).
  29. A. Alaakol, B. H. Azeez, and N. M. Hameed, Physical tuning of nonlinear optical responses in position-dependent mass quantum dots via ordering ambiguity, Physica E 183, 116600 (2026).
  30. G. Ansmann, R. Karnatak, K. Lehnertz, and U. Feudel, Extreme events in excitable systems and mechanisms of their generation, Phys. Rev. E 88, 052911 (2013).
  31. A. Roy and S. Sinha, Impact of random links on neuronal extreme events, Chaos, Solitons Fractals 180, 114568 (2024).
  32. R. S. Ardhanareeswaran, S. Sudharsan, M. Senthilvelan, and D. Ghosh, Intermittent cluster synchronization in a unidirectional ring of bursting neurons, Phys. Rev. E 111, 014215 (2025).
  33. R. Shashangan, S. Sudharsan, D. Ghosh, and M. Senthilvelan, Propagation of extreme events in multiplex neuronal networks, Phys. Rev. E 112, 014310 (2025).
  34. V. Varshney, S. Kumarasamy, A. Mishra, B. Biswal, and A. Prasad, Traveling of extreme events in network of counter-rotating nonlinear oscillators, Chaos 31, 093136 (2021).
  35. R. Suresh and V. K. Chandrasekar, Influence of time-delay feedback on extreme events in a forced Liénard system, Phys. Rev. E 98, 052211 (2018).
  36. M. Wasif Ahamed, R. Kavitha, V. Chithiika Ruby, M. Sathish Aravindh, A. Venkatesan, and M. Lakshmanan, Extreme events in the Higgs oscillator: A dynamical study and forecasting approach, Chaos 35, 023119 (2025).
  37. J. Mycielski, G. Bastard, and C. Rigaux, Intraband magnetoabsorption in mixed semiconductors with composition fluctuations and in imperfect semiconductors, Phys. Rev. B 16, 1675 (1977).
  38. P. M. Mathews and M. Lakshmanan, On a unique nonlinear oscillator, Q. Appl. Math. 32, 215 (1974).
  39. P. M. Mathews and M. Lakshmanan, A quantum-mechanically solvable nonpolynomial Lagrangian with velocity-dependent interaction, Nuov. Cim. A 26, 299 (1975).
  40. J. F. Carinena, M. F. Ranada, M. Santander, and M. Senthilvelan, A non-linear oscillator with quasi-harmonic behaviour: Two- and n -dimensional oscillators, Nonlinearity 17, 1941 (2004).
  41. V. Chithiika Ruby and M. Lakshmanan, On the classical and quantum dynamics of a class of nonpolynomial oscillators, J. Phys. A 54, 385301 (2021).
  42. M. Lakshmanan and K. Eswaran, Quantum dynamics of a solvable nonlinear chiral model, J. Phys. A 8, 1658 (1975).
  43. M. F. Rañada and M. Santander, On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2, J. Math. Phys. 43, 431 (2002).
  44. M. Lakshmanan and V. K. Chandrasekar, Generating finite dimensional integrable nonlinear dynamical systems, Eur. Phys. J.: Spec. Top. 222, 665 (2013).
  45. C. Quesne, Deformed shape invariance symmetry and potentials in curved space with two known eigenstates, J. Math. Phys. 59, 042104 (2018).
  46. A. Ballesteros, A. Enciso, F. J. Herranz, and O. Ragnisco, Superintegrable anharmonic oscillators on N-dimensional curved spaces, J. Nonlin. Math. Phys. 15, 43 (2008).
  47. V. Mohammadi, S. Aghaei, and A. Chenaghlou, Dirac equation in presence of the Hartmann and Higgs oscillator superintegrable potentials with the spin and pseudospin symmetries, Int. J. Mod. Phys. A 31, 1650190 (2016).
  48. O. Evnin and R. Nivesvivat, Hidden symmetries of the Higgs oscillator and the conformal algebra, J. Phys. A: Math. Theor. 50, 015202 (2017).
  49. A. Mahdifar, R. Roknizadeh, and M. Naderi, Geometric approach to nonlinear coherent states using the Higgs model for harmonic oscillator, J. Phys. A 39, 7003 (2006).
  50. V. Chithiika Ruby and M. Lakshmanan, Algebraic analysis of the two-dimensional Higgs oscillator and nonlinear coherent states, Phys. Scr. 100, 095239 (2025).
  51. A. Venkatesan and M. Lakshmanan, Nonlinear dynamics of damped and driven velocity-dependent systems, Phys. Rev. E 55, 5134 (1997).
  52. C. Bonatto and A. Endler, Extreme and superextreme events in a loss-modulated CO2 laser: Nonlinear resonance route and precursors, Phys. Rev. E 96, 012216 (2017).

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Phys. Rev. E 114, 044204– Published 8 October, 2026
  • Received 9 October 2025
  • Revised 11 August 2026
  • Accepted 8 September 2026
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