Phase reduction explains chimera shape: when multi-body interaction matters (2401.05366v4)
Abstract: We present an extension of the Kuramoto-Sakaguchi model for networks, deriving the second-order phase approximation for a paradigmatic model of oscillatory networks - an ensemble of non-identical Stuart-Landau oscillators coupled pairwisely via an arbitrary coupling matrix. We explicitly demonstrate how this matrix translates into the coupling structure in the phase equations. To illustrate the power of our approach and the crucial importance of high-order phase reduction, we tackle a trendy setup of non-locally coupled oscillators exhibiting a chimera state. We reveal that our second-order phase model reproduces the dependence of the chimera shape on the coupling strength that is not captured by the typically used first-order Kuramoto-like model. Our derivation contributes to a better understanding of complex networks' dynamics, establishing a relation between the coupling matrix and multi-body interaction terms in the high-order phase model.
- F. Battiston and G. Petri, eds., Higher-Order Systems, Understanding Complex Systems (Springer International Publishing, Cham, 2022).
- P. S. Skardal and A. Arenas, Communications Physics 3, 218 (2020).
- C. Bick, T. Böhle, and C. Kuehn, SIAM Journal on Applied Dynamical Systems 22, 1590 (2023a).
- Y. Kuramoto, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, edited by H. Araki (Springer, Berlin, Heidelberg, 1975) pp. 420–422.
- H. Sakaguchi and Y. Kuramoto, Progress of Theoretical Physics 76, 576 (1986).
- H. Nakao, Contemporary Physics 57, 188 (2016).
- Y. Kuramoto and H. Nakao, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, 20190041 (2019).
- B. Pietras and A. Daffertshofer, Physics Reports 819, 1 (2019).
- Y. Kuramoto and D. Battogtokh, Nonlinear Phenomena in Complex Systems 5, 380 (2002).
- D. M. Abrams and S. H. Strogatz, Physical Review Letters 93, 174102 (2004).
- M. J. Panaggio and D. M. Abrams, Nonlinearity 28, R67 (2015).
- E. Schöll, The European Physical Journal Special Topics 225, 891 (2016).
- O. E. Omel’chenko, Nonlinearity 31, R121 (2018).
- G. C. Sethia, A. Sen, and G. L. Johnston, Physical Review E 88, 042917 (2013).
- E. T. K. Mau, M. Rosenblum, and A. Pikovsky, Chaos: An Interdisciplinary Journal of Nonlinear Science 33, 101101 (2023).
- D. Wilson and B. Ermentrout, Journal of Mathematical Biology 76, 37 (2018).
- N. Fenichel, Indiana Univ. Math. J. 21, 193 (1971).
- A similar derivation for a slightly different system has been done in [37], where the authors used the terminology of “virtual pairwise connections” for the pairwise term in S~ijksubscript~𝑆𝑖𝑗𝑘\tilde{S}_{ijk}over~ start_ARG italic_S end_ARG start_POSTSUBSCRIPT italic_i italic_j italic_k end_POSTSUBSCRIPT because of the mediated coupling, that does not need a structural connection between oscillators i𝑖iitalic_i and k𝑘kitalic_k to facilitate a functional coupling.
- Using a different approach, León and Pazó derived second and third-order phase models for globally coupled identical SL oscillators [38].
- M. Kumar and M. Rosenblum, Physical Review E 104, 054202 (2021).
- M. Komarov and A. Pikovsky, Phys. Rev. E 84, 016210 (2011).
- M. Komarov and A. Pikovsky, Phys. Rev. E 92, 012906 (2015).
- P. Ashwin and A. Rodrigues, Physica D 325, 14 (2016).
- A. Pikovsky and M. Rosenblum, in Higher-Order Systems, edited by F. Battiston and G. Petri (Springer, 2022) pp. 181–195.
- C. Bick, T. Böhle, and C. Kuehn, Higher-Order Interactions in Phase Oscillator Networks through Phase Reductions of Oscillators with Phase Dependent Amplitude (2023b).
- I. León and D. Pazó, Physical Review E 100, 012211 (2019).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.