Swarms on the 3-sphere with adaptive synapses: Hebbian and anti-Hebbian learning rule

Counterexamples in synchronization: pathologies of consensus seeking gradient descent flows on surfaces

“…Other results on generalized Kuramoto models include a literature on quantum synchronization on the Bloch sphere and generalizations thereof to SO(n) and U(n), see e.g., [Lohe, 2010, DeVille, 2018. We mention just a few of the works on Kuramoto models over manifolds: the sphere [Crnkić and Jaćimović, 2018], the ellipsoid [Zhu, 2014], and the hyperboloid . Similar results also appear in applications involving e.g., opinion consensus [Aydogdu et al, 2017], bio-inspired models of source-seeking and learning [Al-Abri et al, 2018, Crnkić andJaćimović, 2018], and computer science applications like time-series clustering [Crnkić and Jaćimović, 2019] and shape matching of polytopes [Ha and Park, 2020].…”

“…We mention just a few of the works on Kuramoto models over manifolds: the sphere [Crnkić and Jaćimović, 2018], the ellipsoid [Zhu, 2014], and the hyperboloid . Similar results also appear in applications involving e.g., opinion consensus [Aydogdu et al, 2017], bio-inspired models of source-seeking and learning [Al-Abri et al, 2018, Crnkić andJaćimović, 2018], and computer science applications like time-series clustering [Crnkić and Jaćimović, 2019] and shape matching of polytopes [Ha and Park, 2020]. Some of these results on local convergence to the consensus manifold [Lohe, 2010, Aydogdu et al, 2017, DeVille, 2018, Markdahl et al, 2018 are summarized by our stability result for analytic manifolds.…”

Consensus and coordination on groups SO(3) and S3 over constant and state-dependent communication graphs

“…Inspired by Neuroscience, evolving weighted edges of a graph are called adaptive synapses. Coordination problems on S 3 with various learning rules are studied in [23,24].…”

Extension of a PID control theory to Lie groups applied to synchronising satellites and drones

“…The Kuramoto model for a system of N elementary nodes appears as a system of first‐order differential equations, namely for , where is the phase angle of the node, is a coupling constant, is the natural frequency of the node (distributed according to a given probability), and the coefficients are elements of a connectivity matrix, depending only on their indexes. An interesting research by Crnkić and Jaćimović [7] introduced and analysed several models of dynamics with adaptive (state‐dependent) interactions between nodes. The equations that describe the interaction dynamics are variations of the classical Hebbian/anti‐Hebbian paradigm from neuroscience.…”