The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space

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Nonlinear dynamics of networks: the groupoid formalism (2006, background): “The most immediate result is that both synchrony and phase-locking are natural consequences of network symmetry [20] [21] [35] [37] [41].”
Central pattern generators for locomotion control in animals and robots: A review (2008, background): “…and intrinsic frequencies (Cohen et al., 1982; Schoner et al., 1990; Kopell, 1995; Ermentrout & Kopell, 1991; Pikovsky, Rosenblum, & Kurths, 2001; Golubitsky & Stewart, 2002; Wang & Slotine, 2005; Buchli, Righetti, & Ijspeert, 2006; Pham & Slotine, 2007; Stan & Sepulchre, 2007; Seo & Slotine ,…”
Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks (2003, methodology): “The H/K theorem [4, 7] is an existence theorem; indeed, it states necessary and sufficient conditions for periodic solutions with a given spatio-temporal symmetry group to be possible.”
Geometric integrators for ODEs (2006, background): “We can distinguish linear/nonlinear, continuous/discrete, and ordinary/reversing symmetries [42, 68, 104].”