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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 topological generalisation of Lyapunovs result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the non-Hausdorff nature of the symplectic leaf space at that point. This criterion is applied to generalise the Energy-Momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a $G$-stable relative equilibrium satisfies the stronger condition of being $A$-stable, where $A$ is a specific group-theoretically defined subset of $G$ which contains the momentum isotropy subgroup of the relative equilibrium.
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
Dynamical system models of complex biochemical reaction networks are usually high-dimensional, nonlinear, and contain many unknown parameters. In some cases the reaction network structure dictates that positive equilibria must be unique for all value
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
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 sigmoida