Abstract: After centuries of scientific study, nonlinear dynamical systems still pose seemingly insurmountable challenges to analysts today. Chaotic and complex phenomena, even in the simplest nonlinear systems, bring the reductionist approach into question. Surprisingly, however, microscopic disorder often underlies collective macroscopic order, and system symmetry dictates universal routes to pattern formation. While complexity is currently intractable in general, some understanding can be gleaned from a few, often idealized, paradigmatic models. Yet, the impact of asymmetries and heterogeneities in these models is still largely unknown.
In this seminar, I will discuss our recent studies on symmetry breaking and asymmetry-induced symmetry in several classical oscillatory systems. First, I will detail the emergence massively multistable chimera states in networks of Janus oscillators with antiferromagnetic network structure as well as the enhanced synchronization observed when this structure is perturbed. Next, I will describe the appearance of localized analogs of chimera states in oscillating chemical reactions and emphasize open questions about the transport of related topological defects. Finally, I will note surprising new features of instabilities in media driven by vibrations in heterogeneous geometries, including heterogeneity-stabilized homogeneous states in fluid surfaces and anharmonic instabilities in coupled pendulum arrays.