On a Hamiltonian Version of the Rikitake System

A stability criterion for non-degenerate equilibrium states of completely integrable systems

“…As the purpose of this work concerns the stability analysis of equilibrium states, let us recall from [14] that the equilibria of the vector field X β are the elements of the set E X β := {(M, 0, β) : M ∈ R} ∪ {(0, M, −β) : M ∈ R} ∪ {(0, 0, M) : M ∈ R}.…”

“…More precisely, the system we consider in the sequel, is the Hamiltonian version of the Rikitake two-disk dynamo system (see e.g. [10], [4]) analyzed in [14], and described by the equations:…”

On the symmetries of a Rikitake type system

“…Following [11], considering the Lie group O(Q) = {A ∈ GL(3, R)| A t QA = Q} generated by Q := diag(2, 1, 1) ∈ GL(3, R), the corresponding Lie algebra is o(Q) = {X ∈ gl(3, R)| X t Q + QX = O 3 }. As a real vector space o(Q) is generated by the base B o(Q) = {X 1 , X 2 , X 3 }, where…”

Hamilton-Poisson Realizations of the Integrable Deformations of the Rikitake System

“…Using the method given in [20], in this paper, we construct integrable deformations of the integrable version of the Rikitake system considered in [17]. We prove that these integrable deformations have Hamilton-Poisson realizations, which allows us to study them from some standard and nonstandard Poisson geometry points of view.…”

“…The study of an integrable version of the Rikitake system from some standard and nonstandard Poisson geometry points of view was given in [17]. Because this system is considered in our paper, we mention some of its dynamical properties: there are three families of equilibrium points and also periodic orbits around the stable equilibrium points and homoclinic and heteroclinic orbits.…”