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Complex localization mechanisms in networks of coupled oscillators: two case studies (2309.08547v3)

Published 15 Sep 2023 in nlin.PS

Abstract: Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit a coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the band gap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.

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References (58)
  1. J. J. Bramburger and B. Sandstede, “Spatially localized structures in lattice dynamical systems,” Journal of Nonlinear Science 30, 603–644 (2020a).
  2. J. J. Bramburger, “Isolas of multi-pulse solutions to lattice dynamical systems,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics 151, 916–952 (2021).
  3. M. Beck, J. Knobloch, D. J. Lloyd, B. Sandstede,  and T. Wagenknecht, “Snakes, ladders, and isolas of localized patterns,” SIAM Journal on Mathematical Analysis 41, 936–972 (2009).
  4. M. Tian, J. J. Bramburger,  and B. Sandstede, “Snaking bifurcations of localized patterns on ring lattices,” IMA Journal of Applied Mathematics 86, 1112–1140 (2021).
  5. J. J. Bramburger and B. Sandstede, “Localized patterns in planar bistable weakly coupled lattice systems,” Nonlinearity 33, 3500 (2020b).
  6. H. Shi and Y. Zhang, “Existence of breathers for discrete nonlinear Schrödinger equations,” Applied Mathematics Letters 50, 111–118 (2015).
  7. R. Parker, P. Kevrekidis,  and B. Sandstede, “Existence and spectral stability of multi-pulses in discrete hamiltonian lattice systems,” Physica D: Nonlinear Phenomena 408, 132414 (2020).
  8. S. Flach and C. R. Willis, “Discrete breathers,” Physics Reports 295, 181–264 (1998).
  9. J. Cuevas-Maraver and P. G. Kevrekidis, “Discrete breathers in ϕ4superscriptitalic-ϕ4\phi^{4}italic_ϕ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT and related models,” in A Dynamical Perspective on the ϕ4superscriptitalic-ϕ4\phi^{4}italic_ϕ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT Model (Springer, 2019) pp. 137–162.
  10. R. Parker and A. Aceves, “Standing-wave solutions in twisted multicore fibers,” Physical Review A 103, 053505 (2021).
  11. M. Clerc, S. Coulibaly, M. Ferré,  and R. Rojas, “Chimera states in a duffing oscillators chain coupled to nearest neighbors,” Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 083126 (2018).
  12. F. Fontanela, A. Grolet, L. Salles, A. Chabchoub,  and N. Hoffmann, “Dark solitons, modulation instability and breathers in a chain of weakly nonlinear oscillators with cyclic symmetry,” Journal of Sound and Vibration 413, 467–481 (2018).
  13. B. Niedergesäß, A. Papangelo, A. Grolet, A. Vizzaccaro, F. Fontanela, L. Salles, A. Sievers,  and N. Hoffmann, “Experimental observations of nonlinear vibration localization in a cyclic chain of weakly coupled nonlinear oscillators,” Journal of Sound and Vibration 497, 115952 (2021).
  14. A. Papangelo, N. Hoffmann, A. Grolet, M. Stender,  and M. Ciavarella, “Multiple spatially localized dynamical states in friction-excited oscillator chains,” Journal of Sound and Vibration 417, 56–64 (2018).
  15. I. Shiroky, A. Papangelo, N. Hoffmann,  and O. Gendelman, “Nucleation and propagation of excitation fronts in self-excited systems,” Physica D: Nonlinear Phenomena 401, 132176 (2020).
  16. C.-H. Chiu, W.-W. Lin,  and C.-S. Wang, “Synchronization in lattices of coupled oscillators with various boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications 46, 213–229 (2001).
  17. A. Papangelo, A. Grolet, L. Salles, N. Hoffmann,  and M. Ciavarella, “Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry,” Communications in Nonlinear Science and Numerical Simulation 44, 108–119 (2017).
  18. M. A. Schwemmer and T. J. Lewis, “The theory of weakly coupled oscillators,” Phase response curves in neuroscience: theory, experiment, and analysis , 3–31 (2012).
  19. E. M. Izhikevich, Y. Kuramoto, et al., “Weakly coupled oscillators,” Encyclopedia of mathematical physics 5, 448 (2006).
  20. F. C. Hoppensteadt and E. M. Izhikevich, Weakly connected neural networks, Vol. 126 (Springer Science & Business Media, 1997).
  21. J. Hale, Ordinary Differential Equations, Dover Books on Mathematics Series (Dover Publications, 2009).
  22. G. B. Ermentrout and N. Kopell, “Multiple pulse interactions and averaging in systems of coupled neural oscillators,” Journal of Mathematical Biology 29, 195–217 (1991).
  23. K. C. Wedgwood, K. K. Lin, R. Thul,  and S. Coombes, “Phase-amplitude descriptions of neural oscillator models,” The Journal of Mathematical Neuroscience 3, 1–22 (2013).
  24. Y. Kuramoto and Y. Kuramoto, Chemical turbulence (Springer, 1984).
  25. P. Ashwin, S. Coombes,  and R. Nicks, “Mathematical frameworks for oscillatory network dynamics in neuroscience,” The Journal of Mathematical Neuroscience 6, 1–92 (2016).
  26. C. Bick, M. Goodfellow, C. R. Laing,  and E. A. Martens, “Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review,” The Journal of Mathematical Neuroscience 10, 9 (2020).
  27. Y. Kuramoto and D. Battogtokh, “Coexistence of coherence and incoherence in nonlocally coupled phase oscillators,” arXiv preprint cond-mat/0210694  (2002).
  28. D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Physical review letters 93, 174102 (2004).
  29. O. E. Omel’chenko, “The mathematics behind chimera states,” Nonlinearity 31, R121 (2018).
  30. C. R. Laing, “Chimeras in networks with purely local coupling,” Physical Review E 92, 050904 (2015).
  31. Z. G. Nicolaou, D. Eroglu,  and A. E. Motter, “Multifaceted dynamics of janus oscillator networks,” Physical Review X 9, 011017 (2019).
  32. J. Xie, E. Knobloch,  and H.-C. Kao, “Multicluster and traveling chimera states in nonlocal phase-coupled oscillators,” Physical Review E 90, 022919 (2014).
  33. M. Bataille-Gonzalez, M. Clerc, E. Knobloch,  and O. Omel’chenko, “Traveling spiral wave chimeras in coupled oscillator systems: emergence, dynamics, and transitions,” New Journal of Physics 25, 103023 (2023).
  34. J. Yang, B. A. Malomed, D. J. Kaup,  and A. Champneys, “Embedded solitons: a new type of solitary wave,” Mathematics and computers in Simulation 56, 585–600 (2001).
  35. T. Wagenknecht and A. Champneys, “When gap solitons become embedded solitons: a generic unfolding,” Physica D: Nonlinear Phenomena 177, 50–70 (2003).
  36. B. C. Ponedel and E. Knobloch, “Gap solitons and forced snaking,” Physical Review E 98, 062215 (2018).
  37. N. Pernet, P. St-Jean, D. D. Solnyshkov, G. Malpuech, N. Carlon Zambon, Q. Fontaine, B. Real, O. Jamadi, A. Lemaitre, M. Morassi, et al., “Gap solitons in a one-dimensional driven-dissipative topological lattice,” Nature Physics 18, 678–684 (2022).
  38. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter,  and D. N. Christodoulides, “Non-hermitian physics and pt symmetry,” Nature Physics 14, 11–19 (2018).
  39. Z. G. Nicolaou, D. J. Case, E. B. v. d. Wee, M. M. Driscoll,  and A. E. Motter, “Heterogeneity-stabilized homogeneous states in driven media,” Nature communications 12, 4486 (2021).
  40. Z. G. Nicolaou and A. E. Motter, “Anharmonic classical time crystals: A coresonance pattern formation mechanism,” Physical Review Research 3, 023106 (2021).
  41. E. Doedel, H. B. Keller,  and J. P. Kernevez, “Numerical analysis and control of bifurcation problems (i): bifurcation in finite dimensions,” International journal of bifurcation and chaos 1, 493–520 (1991a).
  42. See our Github repository at https://github.com/znicolaou/snakingoscillators for source code reproducing the results in this paper.
  43. J. Burke and E. Knobloch, “Snakes and ladders: localized states in the swift–hohenberg equation,” Physics Letters A 360, 681–688 (2007).
  44. J. P. Gaivão and V. Gelfreich, “Splitting of separatrices for the hamiltonian-hopf bifurcation with the swift–hohenberg equation as an example,” Nonlinearity 24, 677 (2011).
  45. E. J. Doedel, “Auto: A program for the automatic bifurcation analysis of autonomous systems,” Congr. Numer 30, 25–93 (1981).
  46. E. Doedel, H. B. Keller,  and J. P. Kernevez, “Numerical analysis and control of bifurcation problems (ii): Bifurcation in infinite dimensions,” International Journal of Bifurcation and Chaos 1, 745–772 (1991b).
  47. P. J. Olver, Applications of Lie groups to differential equations, Vol. 107 (Springer Science & Business Media, 1993).
  48. T. F. Fairgrieve and A. D. Jepson, “Ok floquet multipliers,” SIAM journal on numerical analysis 28, 1446–1462 (1991).
  49. A. Bergner, M. Frasca, G. Sciuto, A. Buscarino, E. J. Ngamga, L. Fortuna,  and J. Kurths, “Remote synchronization in star networks,” Physical Review E 85, 026208 (2012).
  50. H. Chaté, “Spatiotemporal intermittency regimes of the one-dimensional complex ginzburg-landau equation,” Nonlinearity 7, 185 (1994).
  51. M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Reviews of modern physics 65, 851 (1993).
  52. O. Batiste, E. Knobloch, I. Mercader,  and M. Net, “Simulations of oscillatory binary fluid convection in large aspect ratio containers,” Physical Review E 65, 016303 (2001).
  53. E. Ott, Chaos in dynamical systems (Cambridge university press, 2002).
  54. G. Ansmann, K. Lehnertz,  and U. Feudel, “Self-induced switchings between multiple space-time patterns on complex networks of excitable units,” Physical Review X 6, 011030 (2016).
  55. C. Bick, “Heteroclinic switching between chimeras,” Physical Review E 97, 050201 (2018).
  56. Y. Zhang, Z. G. Nicolaou, J. D. Hart, R. Roy,  and A. E. Motter, “Critical switching in globally attractive chimeras,” Physical Review X 10, 011044 (2020).
  57. A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman,  and J. D. Rademacher, “Unfolding a tangent equilibrium-to-periodic heteroclinic cycle,” SIAM Journal on Applied Dynamical Systems 8, 1261–1304 (2009).
  58. B. Sandstede and A. Scheel, “Defects in oscillatory media: toward a classification,” SIAM Journal on Applied Dynamical Systems 3, 1–68 (2004).
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