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In this paper we investigate equilibria of continuous differential equation models of network dynamics. The motivation comes from gene regulatory networks where each directed edge represents either down- or up-regulation, and is modeled by a sigmoidal nonlinear function. We show that the existence and stability of equilibria of a sigmoidal system is determined by a combinatorial analysis of the limiting switching system with piece-wise constant non-linearities. In addition, we describe a local decomposition of a switching system into a product of simpler cyclic feedback systems, where the cycles in each decomposition correspond to a particular subset of network loops.
We introduce a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. Motivated by models of regulatory networks, we construct a state transition graph from a piecewise affine ordinary d
We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum methods. Using a top
The leading-order approximation to a Filippov system $f$ about a generic boundary equilibrium $x^*$ is a system $F$ that is affine one side of the boundary and constant on the other side. We prove $x^*$ is exponentially stable for $f$ if and only if
We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortice
For the Newtonian (gravitational) $n$-body problem in the Euclidean $d$-dimensional space, $dge 2$, the simplest possible periodic solutions are provided by circular relative equilibria, (RE) for short, namely solutions in which each body rigidly rot