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  • Letter
  • Open Access

Correspondence principle, dissipation, and Ginibre ensemble

David Villaseñor*, Hua Yan, Matic Orel, and Marko Robnik†

  • CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia, European Union

  • *Contact author: d.v.pcf.cu@gmail.com
  • †Contact author: marko.robnik@guest.um.si
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Phys. Rev. E 112, L042204 – Published 29 October, 2025

DOI: https://doi.org/10.1103/xhwm-jpdf

Abstract

The correspondence between quantum and classical behavior has been essential since the advent of quantum mechanics. This principle serves as a cornerstone for understanding quantum chaos, which has garnered increased attention due to its strong impact in various theoretical and experimental fields. When dissipation is considered, quantum chaos takes concepts from isolated quantum chaos to link classical chaotic motion with spectral correlations of Ginibre ensembles. This correspondence was first identified in periodically kicked systems with damping, but it has been shown to break down in dissipative atom-photon systems [Phys. Rev. Lett. 133, 240404 (2024)]. In this contribution, we revisit the original kicked model and perform a systematic exploration across a broad parameter space, reaching a genuine semiclassical limit. Our results demonstrate that the correspondence principle, as defined through this spectral connection, fails even in this prototypical system. These findings provide conclusive evidence that Ginibre spectral correlations are neither a robust nor a universal diagnostic of dissipative quantum chaos.

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References (145)

  1. N. Bohr, Über die Serienspektra der Elemente, Z. Phys. 2, 423 (1920).
  2. G. Q. Hassoun and D. H. Kobe, Synthesis of the Planck and Bohr formulations of the correspondence principle, Am. J. Phys. 57, 658 (1989).
  3. E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20, 130 (1963).
  4. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, UK, 2002).
  5. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations Of Vector fields (Springer, New York, 1983).
  6. V. S. Anishchenko, V. Astakhov, T. Vadivasova, A. Neiman, and L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems (Springer-Verlag Berlin, Heidelberg, 2007).
  7. B. K. Shivamoggi, Nonlinear Dynamics and Chaotic Phenomena: An Introduction (Springer, Dordrecht, 2014).
  8. F. Haake, Quantum Signatures of Chaos (Springer-Verlag, Berlin, 1991).
  9. G. Casati and B. Chirikov, Quantum Chaos: Between Order and Disorder (Cambridge University Press, New York, 1995).
  10. H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 2006).
  11. S. Wimberger, Nonlinear Dynamics and Quantum Chaos (Springer International Publishing, Switzerland, 2014).
  12. F. Borgonovi, F. M. Izrailev, L. F. Santos, and V. G. Zelevinsky, Quantum chaos and thermalization in isolated systems of interacting particles, Phys. Rep. 626, 1 (2016).
  13. L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  14. L. J. Fiderer and D. Braun, Quantum metrology with quantum-chaotic sensors, Nat. Commun. 9, 1351 (2018).
  15. B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94, 040302 (2016).
  16. K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Verified quantum information scrambling, Nature (London) 567, 61 (2019).
  17. É. Lantagne-Hurtubise, S. Plugge, O. Can, and M. Franz, Diagnosing quantum chaos in many-body systems using entanglement as a resource, Phys. Rev. Res. 2, 013254 (2020).
  18. N. Anand, G. Styliaris, M. Kumari, and P. Zanardi, Quantum coherence as a signature of chaos, Phys. Rev. Res. 3, 023214 (2021).
  19. K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117, 111601 (2016).
  20. J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106.
  21. J. M. Magán, Black holes, complexity and quantum chaos, J. High Energy Phys. 09 (2018) 043.
  22. V. Jahnke, Recent developments in the holographic description of quantum chaos, Adv. High Energy Phys. 2019, 9632708 (2019).
  23. J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Quantum chaos challenges many-body localization, Phys. Rev. E 102, 062144 (2020).
  24. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, NY, 2002).
  25. H. Carmichael, An Open Systems Approach to Quantum Optics. Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991 (Springer-Verlag, Berlin, 1993).
  26. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, Berlin, 2002).
  27. E. Ott, Strange attractors and chaotic motions of dynamical systems, Rev. Mod. Phys. 53, 655 (1981).
  28. J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617 (1985).
  29. Y. H. Ku and X. Sun, Chaos and limit cycle in Duffing's equation, J. Franklin Inst. 327, 165 (1990).
  30. Y. S. Smyrlis and D. T. Papageorgiou, Predicting chaos for infinite dimensional dynamical systems: The Kuramoto-Sivashinsky equation, a case study, Proc. Natl. Acad. Sci. USA 88, 11129 (1991).
  31. D. T. Papageorgiou and Y. S. Smyrlis, The route to chaos for the Kuramoto-Sivashinsky equation, Theoret. Comput. Fluid Dynamics 3, 15 (1991).
  32. G. Casati, F. Valz-Gris, and I. Guarnieri, On the connection between quantization of nonintegrable systems and statistical theory of spectra, Lett. Nuovo Cimento 28, 279 (1980).
  33. O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984).
  34. M. V. Berry, M. Tabor, and J. M. Ziman, Level clustering in the regular spectrum, Proc. Roy. Soc. London. A. Math. Phys. Sci. 356, 375 (1977).
  35. R. Grobe, F. Haake, and H.-J. Sommers, Quantum distinction of regular and chaotic dissipative motion, Phys. Rev. Lett. 61, 1899 (1988).
  36. J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys. 6, 440 (1965).
  37. R. Grobe and F. Haake, Universality of cubic-level repulsion for dissipative quantum chaos, Phys. Rev. Lett. 62, 2893 (1989).
  38. G. Akemann, M. Kieburg, A. Mielke, and T. Prosen, Universal signature from integrability to chaos in dissipative open quantum systems, Phys. Rev. Lett. 123, 254101 (2019).
  39. A. B. Jaiswal, A. Pandey, and R. Prakash, Universality classes of quantum chaotic dissipative systems, Europhys. Lett. 127, 30004 (2019).
  40. L. Sá, P. Ribeiro, and T. Prosen, Complex spacing ratios: A signature of dissipative quantum chaos, Phys. Rev. X 10, 021019 (2020).
  41. Á. Rubio-García, R. A. Molina, and J. Dukelsky, From integrability to chaos in quantum Liouvillians, SciPost Phys. Core 5, 026 (2022).
  42. A. M. García-García, L. Sá, and J. J. M. Verbaarschot, Universality and its limits in non-Hermitian many-body quantum chaos using the Sachdev-Ye-Kitaev model, Phys. Rev. D 107, 066007 (2023).
  43. G. Akemann, F. Balducci, A. Chenu, P. Päßler, F. Roccati, and R. Shir, Two transitions in complex eigenvalue statistics: Hermiticity and integrability breaking, Phys. Rev. Res. 7, 013098 (2025).
  44. R. Hamazaki, K. Kawabata, N. Kura, and M. Ueda, Universality classes of non-Hermitian random matrices, Phys. Rev. Res. 2, 023286 (2020).
  45. A. M. García-García, L. Sá, and J. J. M. Verbaarschot, Symmetry classification and universality in non-Hermitian many-body quantum chaos by the Sachdev-Ye-Kitaev model, Phys. Rev. X 12, 021040 (2022).
  46. K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, and S. Ryu, Symmetry of open quantum systems: Classification of dissipative quantum chaos, PRX Quantum 4, 030328 (2023).
  47. L. Sá, P. Ribeiro, and T. Prosen, Symmetry classification of many-body Lindbladians: Tenfold way and beyond, Phys. Rev. X 13, 031019 (2023).
  48. J. Li, T. Prosen, and A. Chan, Spectral statistics of non-Hermitian matrices and dissipative quantum chaos, Phys. Rev. Lett. 127, 170602 (2021).
  49. S. Shivam, A. De Luca, D. A. Huse, and A. Chan, Many-body quantum chaos and emergence of Ginibre ensemble, Phys. Rev. Lett. 130, 140403 (2023).
  50. J. Li, S. Yan, T. Prosen, and A. Chan, Spectral form factor in chaotic, localized, and integrable open quantum many-body systems, arXiv:2405.01641.
  51. K. Wold, Z. Zhu, F. Jin, X. Zhu, Z. Bao, J. Zhong, F. Shen, P. Zhang, H. Li, Z. Wang, C. Song, Q. Guo, S. Denisov, L. Sá, H. Wang, and P. Ribeiro, Experimental detection of dissipative quantum chaos, arXiv:2506.04325.
  52. D. Villaseñor, L. F. Santos, and P. Barberis-Blostein, Breakdown of the quantum distinction of regular and chaotic classical dynamics in dissipative systems, Phys. Rev. Lett. 133, 240404 (2024).
  53. L. Sá, P. Ribeiro, and T. Prosen, Spectral and steady-state properties of random Liouvillians, J. Phys. A: Math. Theor. 53, 305303 (2020).
  54. L. Sá, P. Ribeiro, T. Can, and T. Prosen, Spectral transitions and universal steady states in random Kraus maps and circuits, Phys. Rev. B 102, 134310 (2020).
  55. J. Costa, P. Ribeiro, A. D. Luca, T. Prosen, and L. Sá, Spectral and steady-state properties of fermionic random quadratic Liouvillians, SciPost Phys. 15, 145 (2023).
  56. J. Richter, L. Sá, and M. Haque, Integrability versus chaos in the steady state of many-body open quantum systems, Phys. Rev. E 111, 064103 (2025).
  57. F. Ferrari, L. Gravina, D. Eeltink, P. Scarlino, V. Savona, and F. Minganti, Dissipative quantum chaos unveiled by stochastic quantum trajectories, Phys. Rev. Res. 7, 013276 (2025).
  58. L. Peyruchat, F. Minganti, M. Scigliuzzo, F. Ferrari, V. Jouanny, F. Nori, V. Savona, and P. Scarlino, Landau–Zener without a qubit: Multiphoton sidebands interaction and signatures of dissipative quantum chaos, npj Quantum Inf. 11, 62 (2025).
  59. D. Mondal, L. F. Santos, and S. Sinha, Transient and steady-state chaos in dissipative quantum systems, arXiv:2506.05475.
  60. G. Rufo, S. Rufo, P. Ribeiro, and S. Chesi, Quantum and semi-classical signatures of dissipative chaos in the steady state, arXiv:2506.14961.
  61. Y. Li, Y. Wang, and Y.-C. Liu, Open-system analogy of Berry conjecture, arXiv:2509.14644.
  62. Y.-L. Zhang, Y. Huang, and X. Chen, Information scrambling in chaotic systems with dissipation, Phys. Rev. B 99, 014303 (2019).
  63. A. M. García-García, J. J. M. Verbaarschot, and J.-P. Zheng, Lyapunov exponent as a signature of dissipative many-body quantum chaos, Phys. Rev. D 110, 086010 (2024).
  64. R. Grobe and F. Haake, Dissipative death of quantum coherences in a spin system, Z. Phys. B 68, 503 (1987).
  65. P. Pepłowski and S. T. Dembiński, On comparison of classical and quantum kicked tops with dissipation, Z. Phys. B 83, 453 (1991).
  66. P. Pepłowski and K. Stefański, Kicked top with pumping and dissipation. Classical results, Phys. Lett. A 132, 408 (1988).
  67. P. A. Braun, D. Braun, F. Haake, and J. Weber, Semiclassical spin damping: Superradiance revisited, Eur. Phys. J. D 2, 165 (1998).
  68. P. A. Braun, D. Braun, and F. Haake, WKB for a damped spin, Eur. Phys. J. D 3, 1 (1998).
  69. D. Braun, Spectral properties of dissipative chaotic quantum maps, Chaos 9, 730 (1999).
  70. D. Braun, P. A. Braun, and F. Haake, Semiclassics for a dissipative quantum map, Physica D 131, 265 (1999).
  71. J. Iwaniszewski and P. Peplowski, Chaos and stochastic modeling of dissipation in a quantum kicked top, J. Phys. A: Math. Gen. 28, 2183 (1995).
  72. N. Gisin and I. C. Percival, The quantum-state diffusion model applied to open systems, J. Phys. A: Math. Gen. 25, 5677 (1992).
  73. T. P. Spiller and J. F. Ralph, The emergence of chaos in an open quantum system, Phys. Lett. A 194, 235 (1994).
  74. T. A. Brun, I. C. Percival, and R. Schack, Quantum chaos in open systems: A quantum state diffusion analysis, J. Phys. A: Math. Gen. 29, 2077 (1996).
  75. S. Mudute-Ndumbe and E.-M. Graefe, A non-Hermitian PT-symmetric kicked top, New J. Phys. 22, 103011 (2020).
  76. J. T. Chalker and B. Mehlig, Eigenvector statistics in non-Hermitian random matrix ensembles, Phys. Rev. Lett. 81, 3367 (1998).
  77. N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University Press, New York, 2011).
  78. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019).
  79. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  80. G. Akemann, A. Mielke, and P. Päßler, Spacing distribution in the two-dimensional Coulomb gas: Surmise and symmetry classes of non-Hermitian random matrices at noninteger β, Phys. Rev. E 106, 014146 (2022).
  81. G. Akemann, S.-S. Byun, and S. Oh, Spectral moments of complex and symplectic non-Hermitian random matrices, arXiv:2505.12271.
  82. E. M. Graefe, H. J. Korsch, and A. E. Niederle, Mean-Field dynamics of a non-Hermitian Bose-Hubbard dimer, Phys. Rev. Lett. 101, 150408 (2008).
  83. E.-M. Graefe, H. J. Korsch, and A. E. Niederle, Quantum-classical correspondence for a non-Hermitian Bose-Hubbard dimer, Phys. Rev. A 82, 013629 (2010).
  84. E.-M. Graefe, M. Höning, and H. J. Korsch, Classical limit of non-Hermitian quantum dynamics—a generalized canonical structure, J. Phys. A: Math. Theor. 43, 075306 (2010).
  85. K. Holmes, W. Rehman, S. Malzard, and E.-M. Graefe, Husimi dynamics generated by non-Hermitian Hamiltonians, Phys. Rev. Lett. 130, 157202 (2023).
  86. G. Passarelli, P. Lucignano, D. Rossini, and A. Russomanno, Chaos and magic in the dissipative quantum kicked top, Quantum 9, 1653 (2025).
  87. H. Frahm and H. J. Mikeska, On the dynamics of a quantum system which is classically chaotic, Z. Phys. B 60, 117 (1985).
  88. H. Frahm and H. J. Mikeska, Levelstatistics and stochasticity in a driven quantum system, Z. Phys. B 65, 249 (1986).
  89. F. Haake, M. Kuś, and R. Scharf, Classical and quantum chaos for a kicked top, Z. Phys. B 65, 381 (1987).
  90. M. Kuś, R. Scharf, and F. Haake, Symmetry versus degree of level repulsion for kicked quantum systems, Z. Phys. B 66, 129 (1987).
  91. M. Kus, J. Mostowski, and F. Haake, Universality of eigenvector statistics of kicked tops of different symmetries, J. Phys. A: Math. Gen. 21, L1073 (1988).
  92. R. Alicki, D. Makowiec, and W. Miklaszewski, Quantum chaos in terms of entropy for a periodically kicked top, Phys. Rev. Lett. 77, 838 (1996).
  93. V. Constantoudis and N. Theodorakopoulos, Lyapunov exponent, stretching numbers, and islands of stability of the kicked top, Phys. Rev. E 56, 5189 (1997).
  94. S. PG and M. S. Santhanam, Periodicity of dynamical signatures of chaos in quantum kicked top, Phys. Lett. A 550, 130596 (2025).
  95. Q. Wang and M. Robnik, Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space, Phys. Rev. E 108, 054217 (2023).
  96. H. Yan, Q. Wang, and M. Robnik, Further results on the power-law decay of the fraction of the mixed eigenstates in kicked-top model with mixed-type classical phase space, Phys. Rev. E 110, 064222 (2024).
  97. M. V. Berry and M. Robnik, Semiclassical level spacings when regular and chaotic orbits coexist, J. Phys. A: Math. Gen. 17, 2413 (1984).
  98. M. Robnik, Topics in quantum chaos of generic systems, Nonlinear Phenom. Complex Syst. 1, 1 (1998).
  99. M. Robnik, Recent developments in quantum chaos of mixed-type systems: A mini review, Nonlinear Phenom. Complex Syst. 26, 209 (2023).
  100. W.-M. Zhang, J.-M. Yuan, D. H. Feng, Q. Pan, and J. Tjon, Quantum fluctuations in classical chaos, Phys. Rev. A 42, 3646 (1990).
  101. M. Kuś, J. Zakrzewski, and K. Życzkowski, Quantum scars on a sphere, Phys. Rev. A 43, 4244 (1991).
  102. G. M. D'Ariano, L. R. Evangelista, and M. Saraceno, Classical and quantum structures in the kicked-top model, Phys. Rev. A 45, 3646 (1992).
  103. X. Wang, S. Ghose, B. C. Sanders, and B. Hu, Entanglement as a signature of quantum chaos, Phys. Rev. E 70, 016217 (2004).
  104. S. Ghose, R. Stock, P. Jessen, R. Lal, and A. Silberfarb, Chaos, entanglement, and decoherence in the quantum kicked top, Phys. Rev. A 78, 042318 (2008).
  105. S. Chaudhury, A. Smith, B. E. Anderson, S. Ghose, and P. S. Jessen, Quantum signatures of chaos in a kicked top, Nature (London) 461, 768 (2009).
  106. M. Lombardi and A. Matzkin, Entanglement and chaos in the kicked top, Phys. Rev. E 83, 016207 (2011).
  107. J. B. Ruebeck, J. Lin, and A. K. Pattanayak, Entanglement and its relationship to classical dynamics, Phys. Rev. E 95, 062222 (2017).
  108. A. Piga, M. Lewenstein, and J. Q. Quach, Quantum chaos and entanglement in ergodic and nonergodic systems, Phys. Rev. E 99, 032213 (2019).
  109. M. Kumari and S. Ghose, Untangling entanglement and chaos, Phys. Rev. A 99, 042311 (2019).
  110. A. Lerose and S. Pappalardi, Bridging entanglement dynamics and chaos in semiclassical systems, Phys. Rev. A 102, 032404 (2020).
  111. Z. Zou and J. Wang, Pseudoclassical dynamics of the kicked top, Entropy 24, 1092 (2022).
  112. B. Khalid and S. Kais, Classical analog of entanglement for a kicked top, Phys. Rev. A 112, 012201 (2025).
  113. M. C, U. Divakaran, and A. Lakshminarayan, Disordering a permutation symmetric system: Revivals, thermalization and chaos, arXiv:2505.24453.
  114. Q. Wang and M. Robnik, Statistics of phase space localization measures and quantum chaos in the kicked top model, Phys. Rev. E 107, 054213 (2023).
  115. K. Nakamura, Y. Okazaki, and A. R. Bishop, Periodically pulsed spin dynamics: Scaling behavior of semiclassical wave functions, Phys. Rev. Lett. 57, 5 (1986).
  116. Q. Wang and M. Robnik, Multifractality in quasienergy space of coherent states as a signature of quantum chaos, Entropy 23, 1347 (2021).
  117. M. Gonzalez, M. A. Bastarrachea-Magnani, and J. G. Hirsch, Phase space geometry of collective spin systems: Scaling and fractality, Phys. Rev. E 112, 024201 (2025).
  118. G. A. Smith, S. Chaudhury, A. Silberfarb, I. H. Deutsch, and P. S. Jessen, Continuous weak measurement and nonlinear dynamics in a cold spin ensemble, Phys. Rev. Lett. 93, 163602 (2004).
  119. G. A. Smith, A. Silberfarb, I. H. Deutsch, and P. S. Jessen, Efficient quantum-state estimation by continuous weak measurement and dynamical control, Phys. Rev. Lett. 97, 180403 (2006).
  120. S. Chaudhury, S. Merkel, T. Herr, A. Silberfarb, I. H. Deutsch, and P. S. Jessen, Quantum control of the hyperfine spin of a Cs atom ensemble, Phys. Rev. Lett. 99, 163002 (2007).
  121. Z. Zou, J. Gong, and W. Chen, Enhancing quantum metrology by quantum resonance dynamics, Phys. Rev. Lett. 134, 230802 (2025).
  122. R. Bonifacio, P. Schwendimann, and F. Haake, Quantum statistical theory of superradiance. I, Phys. Rev. A 4, 302 (1971).
  123. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
  124. S. Lieu, R. Belyansky, J. T. Young, R. Lundgren, V. V. Albert, and A. V. Gorshkov, Symmetry breaking and error correction in open quantum systems, Phys. Rev. Lett. 125, 240405 (2020).
  125. See Supplemental Material at http://link.aps.org/supplemental/10.1103/xhwm-jpdf for additional information about the kicked top with dissipation, which includes Refs. [141, 142, 143, 144, 145].
  126. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 1: Theory, Meccanica 15, 9 (1980).
  127. G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; A method for computing all of them. Part 2: Numerical application, Meccanica 15, 21 (1980).
  128. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D 16, 285 (1985).
  129. H. Markum, R. Pullirsch, and T. Wettig, Non-Hermitian random matrix theory and lattice QCD with chemical potential, Phys. Rev. Lett. 83, 484 (1999).
  130. M. F. Demers and L.-S. Young, Escape rates and conditionally invariant measures, Nonlinearity 19, 377 (2006).
  131. E. G. Altmann, J. S. E. Portela, and T. Tél, Leaking chaotic systems, Rev. Mod. Phys. 85, 869 (2013).
  132. E. G. Altmann, J. S. E. Portela, and T. Tél, Chaotic systems with absorption, Phys. Rev. Lett. 111, 144101 (2013).
  133. The maximal Lyapunov exponent with zero value identifies a limit cycle, while a negative value is associated to a point attractor.
  134. J. L. Kaplan and J. A. Yorke, Chaotic behavior of multidimensional difference equations, in Functional Differential Equations and Approximation of Fixed Points, edited by H.-O. Peitgen and H.-O. Walther (Springer Berlin Heidelberg, Berlin, Heidelberg, 1979), pp. 204–227.
  135. D. A. Russell, J. D. Hanson, and E. Ott, Dimension of strange attractors, Phys. Rev. Lett. 45, 1175 (1980).
  136. H. Mori, Fractal dimensions of chaotic flows of autonomous dissipative systems, Prog. Theor. Phys. 63, 1044 (1980).
  137. J. D. Farmer, E. Ott, and J. A. Yorke, The dimension of chaotic attractors, Physica D 7, 153 (1983).
  138. B. B. Mandelbrot, Fractals : Form, Chance, and Dimension (W. H. Freeman, San Francisco, 1977).
  139. S. Dutta, S. Zhang, and M. Haque, Quantum origin of limit cycles, fixed points, and critical slowing down, Phys. Rev. Lett. 134, 050407 (2025).
  140. Z.-T. Cai, H.-D. Li, and W. Chen, Quantum-classical correspondence of non-Hermitian symmetry breaking, Phys. Rev. Lett. 134, 240201 (2025).
  141. M. L. Mehta, Random Matrices (Academic Press, Boston, MA, 1991).
  142. T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Random matrix theories in quantum physics: Common concepts, Phys. Rep. 299, 189 (1998).
  143. V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  144. Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013).
  145. D. Viswanath, The fractal property of the Lorenz attractor, Physica D 190, 115 (2004).

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Phys. Rev. E 112, L042204– Published 29 October, 2025
  • Received 25 July 2025
  • Accepted 9 October 2025
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