Recent advances in symmetric and network dynamics

We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts René Thom's general viewpoint, one version of which became known as “catastrophe theory.” We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette–Taylor flow, flames, the Belousov–Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network

The significance of leg mass in modeling quadrupedal running gaits

Abstract. In modeling running gaits of biological and robotic quadrupeds, leg mass is often neglected. Analysis of the system angular momentum in a biological model and a robot model indicates that including leg mass is significant in capturing the roll motion in trotting and pacing. Leg mass has a more significant effect on the pitch motion in bounding and is most critical in accurately capturing the dynamics of galloping.

Toward a more temporally explicit framework for community ecology

Although ecologists have long understood the fundamentally dynamic nature of communities, ecology has until recently seemed to emphasize other aspects of ecological complexity, such as diversity and spatial structure, ahead of temporal variation. Climate change has made studies into the temporal dimensions of community ecology more immediate and urgent, and has exposed the limits of our general understanding about how species interactions change over time. Here, I suggest four specific ways to continue building toward a more temporally explicit understanding of community ecology: (a) by increasing the representation of temporal change in interaction networks, (b) by developing both specific and general insights into event-driven dynamics, (c) by developing and testing sequential hypotheses to describe proposed explanations that unfold over time and (d) by characterizing seasonal windows of opportunity. A great deal about the temporal dynamics of communities remains uncertain, but temporally explicit studies have the potential to improve our fundamental understanding of how communities function