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  • Open Access

Square matrix-based six-dimensional convergence map for nonlinear beam dynamics analysis

Jinyu Wan*

Yue Hao†

  • *Contact author: wanjinyu@ihep.ac.cn
  • †Contact author: haoy@frib.msu.edu
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Phys. Rev. Accel. Beams 29, 104701 – Published 8 October, 2026

DOI: https://doi.org/10.1103/3x6c-zqpt

Abstract

The square matrix-based convergence map (CM) method has proven effective in characterizing nonlinear dynamics in several 4D dynamical systems. However, when time-dependent perturbations, such as crabbing kicks in colliders, are present, a comprehensive 6D analysis becomes essential to accurately capture the coupling between transverse and longitudinal motions. In this work, we extend the CM method to the full 6D phase space by employing an eigen-decomposition-based formulation of the square matrix combined with iterative procedures. The proposed 6D CM approach is first validated using a simplified crabbing map and subsequently applied to the dynamic aperture study of the future Electron-Ion Collider. The CM results show qualitative consistency with frequency analysis in identifying major resonance structures and stability boundaries by using only one-turn map, suggesting that the 6D CM can serve as a fast indicator for nonlinear beam dynamics analysis.

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References (40)

  1. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd ed. (CRC Press, Boca Raton, FL, 1998).
  2. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, 2nd ed. Applied Mathematical Sciences, Vol. 38 (Springer, New York, NY, 2013).
  3. E. Todesco and M. Giovannozzi, Dynamic aperture estimates and phase-space distortions in nonlinear betatron motion, Phys. Rev. E 53, 4067 (1996).
  4. W. Chou and D. Ritson, Dynamic aperture studies during collisions in the LHC, in Proceedings of the 1997 Particle Accelerator Conference (Cat. No. 97CH36167) (IEEE, New York, 1997), pp. 113–115,10.1109/PAC.1997.749561
  5. J. Barranco García, R. De Maria, A. Grudiev, R. Tomás García, R. Appleby, and D. Brett, Long term dynamics of the high luminosity Large Hadron Collider with crab cavities, Phys. Rev. Accel. Beams 19, 101003 (2016).
  6. A. Alekou, R. B. Appleby, E. Cruz-Alaniz, H. Bartosik, M. Carlà, S. Kostoglou, and Y. Papaphilippou, Long term stability studies in the presence of crab cavities and high order multipoles in the CERN super proton synchrotron and high luminosity large hadron collider, Nucl. Instrum. Methods Phys. Res. A 1050, 168178 (2023).
  7. R. D. Ruth, Single particle dynamics in circular accelerators, Tech. Rep., Stanford Linear Accelerator Center, Menlo Park, CA, USA, 1986, osti.gov/biblio/6421344, https://www.osti.gov/biblio/6421344.
  8. G. Guignard, A general treatment of resonances in accelerators, Tech. Rep, European Organization for Nuclear Research, 1978, CERN-78-11,https://cds.cern.ch/record/132991/files/CERN-78-11.pdf.
  9. A. Schoch, Theory of linear and Non-linear perturbations of betatron oscillations in alternating-gradient synchrotrons, Tech. Rep., CERN, 1958, full text, https://cds.cern.ch/record/213137/files/full_document.pdf.
  10. A. J. Dragt, Lie algebraic methods for charged particle optics, AIP Conf. Proc. 177, 261 (1988).
  11. M. Berz, Differential algebra: A new tool, in Proceedings of the 1989 IEEE Particle Accelerator Conference: Accelerator Science and Technology (IEEE, New York, 1989), pp.–1419–1423,10.1109/PAC.1989.73468.
  12. A. W. Chao, Lecture notes on topics in accelerator physics, Tech. Rep., SLAC National Accelerator Laboratory, Menlo Park, CA, USA, 2002, PDF, https://wiki.jlab.org/ciswiki/images/f/f2/Alex_Chao_Lecture_Notes.pdf.
  13. A. Bazzani, G. Servizi, G. Turchetti, and E. Todesco, A normal form approach to the theory of nonlinear betatronic motion, CERN-94-02, CERN, 1994, https://cds.cern.ch/record/262179/files/CERN-94-02.pdf.
  14. E. Forest, J. Irwin, and M. Berz, Normal form methods for complicated periodic systems: A complete solution using differential algebra and Lie operators, Part. Accel. 24, 91 (1989). https://cds.cern.ch/record/1053511/files/p91.pdf
  15. M. Giovannozzi, Proposed scaling law for intensity evolution in hadron storage rings based on dynamic aperture variation with time, Phys. Rev. ST Accel. Beams 15, 024001 (2012).
  16. A. Bazzani, M. Giovannozzi, E. H. Maclean, C. E. Montanari, F. F. Van der Veken, and W. Van Goethem, Advances on the modeling of the time evolution of dynamic aperture of hadron circular accelerators, Phys. Rev. Accel. Beams 22, 104003 (2019).
  17. D. Robin, C. Steier, J. Laskar, and L. Nadolski, Global dynamics of the advanced light source revealed through experimental frequency map analysis, Phys. Rev. Lett. 85, 558 (2000).
  18. J. Laskar, Frequency map analysis and particle accelerators, in Proceedings of the 2003 Particle Accelerator Conference (IEEE, New York, 2003), Vol. 1, pp. 378–382, 10.1109/PAC.2003.1288929
  19. J. Laskar, C. Froeschlé, and A. Celletti, The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping, Phys. D 56, 253 (1992).
  20. M. Comunian, A. Pisent, A. Bazzani, G. Turchetti, and S. Rambaldi, Frequency map analysis of a three-dimensional particle in the core model of a high intensity linac, Phys. Rev. ST Accel. Beams 4, 124201 (2001).
  21. P. Brunelle, A. Loulergue, A. Nadji, L. S. Nadolski, and M. A. Tordeux, Non linear beam dynamics studies at SOLEIL using experimental frequency map analysis, in Proceedings of the 1st International Particle Accelerator Conference (IPAC’10), Kyoto, Japan (JACoW, Geneva, Switzerland, 2010), pp. 4653–4655, https://proceedings.jacow.org/IPAC10/papers/thpe061.pdf.
  22. D. Shatilov, E. Levichev, E. Simonov, and M. Zobov, Application of frequency map analysis to beam-beam effects study in crab waist collision scheme, Phys. Rev. ST Accel. Beams 14, 014001 (2011).
  23. Y. Li, J. Wan, A. Liu, Y. Jiao, and R. Rainer, Data-driven chaos indicator for nonlinear dynamics and applications on storage ring lattice design, Nucl. Instrum. Methods Phys. Res. A 1024, 166060 (2022).
  24. K. Hwang, C. Mitchell, and R. Ryne, Rapidly converging chaos indicator for studying dynamic aperture in a storage ring with space charge, Phys. Rev. Accel. Beams 23, 084601 (2020).
  25. C. E. Montanari, R. B. Appleby, A. Bazzani, A. Fornara, M. Giovannozzi, S. Redaelli, G. Sterbini, and G. Turchetti, Chaos indicators for nonlinear dynamics in circular particle accelerators, Eur. Phys. J. Plus 140, 603 (2025).
  26. J. Wan and Y. Jiao, Machine learning enabled fast evaluation of dynamic aperture for storage ring accelerators, New J. Phys. 24, 063030 (2022).
  27. M. Casanova, B. Dalena, L. Bonaventura, and M. Giovannozzi, Ensemble reservoir computing for dynamical systems: prediction of phase-space stable region for hadron storage rings, Eur. Phys. J. Plus 138, 559 (2023).
  28. L. H. Yu, Analysis of nonlinear dynamics by square matrix method, Phys. Rev. Accel. Beams 20, 034001 (2017).
  29. L. H. Yu, Y. Hidaka, V. Smaluk, K. Anderson, and Y. Hao, Convergence map with action-angle variables based on square matrix for nonlinear lattice optimization, Phys. Rev. Accel. Beams 26, 054002 (2023).
  30. K. J. Anderson, Y. Hao, and L. H. Yu, Study of nonlinear dynamics in the 4-D Hénon map using the square matrix method and iterative methods, in Proc. 5th North American Particle Accelerator Conf. (NAPAC’22), Albuquerque, NM, USA (JACoW Publishing, Geneva, Switzerland, 2022), pp. 232–235,10.18429/JACoW-NAPAC2022-MOPA81
  31. L. H. Yu, Y. Li, and L. Yang, Theory of square matrix for nonlinear optimization, and application to NSLS-II, in Nonlinear Dynamics and Collective Effects in Particle Beam Physics: Proceedings of the International Committee on Future Accelerators, Arcidosso, Italy, 2017 (World Scientific, Singapore, 2019), pp. 12–22,10.1142/9789813279612_0002
  32. L. H. Yu, Y. Hidaka, F. Plassard, V. Smaluk, and Y. Hao, Progress on convergence map based on square matrix for nonlinear lattice optimization, in Proc. 5th North American Particle Accelerator Conf. (NAPAC’22), Albuquerque, NM, USA (JACoW Publishing, Geneva, Switzerland, 2022), pp. 823–825,10.18429/JACoW-NAPAC2022-WEPA80
  33. J. Berg, C. Liu, H. Lovelace III, S. Peggs, G. Robert-Demolaize, N. Tsoupas, and D. Xu, Lattice design for the hadron storage ring of the electron-ion collider, Tech. Rep., Brookhaven National Laboratory, Upton, NY, USA, 2023, OSTI 1986052, https://www.osti.gov/servlets/purl/1986052.
  34. D. Xu, Y. Hao, Y. Luo, C. Montag, and J. Qiang, Study of harmonic crab cavity in EIC beam-beam simulations, in Proc. 12th International Particle Accelerator Conf. (IPAC’21), Campinas, SP, Brazil (JACoW Publishing, Geneva, Switzerland, 2021), pp. 2595–2597,10.18429/JACoW-IPAC2021-WEPAB009
  35. B. Kågström and A. Ruhe, An algorithm for numerical computation of the Jordan normal form of a complex matrix, ACM Trans. Math. Softw. 6, 398 (1980).
  36. H. W. Broer, KAM theory: The legacy of Kolmogorov’s 1954 paper, Bull. Amer. Math. Soc. 41, 507 (2004). https://www.ams.org/bull/2004-41-04/S0273-0979-04-01009-2/
  37. J. Wan, H. Alamprese, C. Ratcliff, J. Qiang, and Y. Hao, JuTrack: A Julia package for auto-differentiable accelerator modeling and particle tracking, Comput. Phys. Commun. 309, 109497 (2025).
  38. M. Harrison, S. Peggs, and T. Roser, The RHIC accelerator, Annu. Rev. Nucl. Part. Sci. 52, 425 (2002).
  39. Q. Wu, Y. Luo, B. P. Xiao, S. U. De Silva, and J. A. Mitchell, EIC crab cavity multipole analysis, in Proc. 12th International Particle Accelerator Conf. (IPAC’21), Campinas, SP, Brazil (JACoW Publishing, Geneva, Switzerland, 2021), pp. 2589–2591,10.18429/JACoW-IPAC2021-WEPAB006
  40. Q. Wu, B. P. Xiao, S. U. De Silva, Z. Li, and Y. Luo, EIC crab cavity multipole analysis and their effects on dynamic aperture, in Proc. 13th International Particle Accelerator Conf. (IPAC’22), Bangkok, Thailand (JACoW Publishing, Geneva, Switzerland, 2022), pp. 66–69,10.18429/JACoW-IPAC2022-MOPOST009

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Phys. Rev. Accel. Beams 29, 104701– Published 8 October, 2026
  • Received 4 February 2026
  • Accepted 30 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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