On a deformed version of the two-disk dynamo system

Lăzureanu, Cristian ; Petrişor, Camelia ; Hedrea, Ciprian

Applications of Mathematics, Tome 66 (2021) no. 3, pp. 345-372.

Résumé

We give some deformations of the Rikitake two-disk dynamo system. Particularly, we consider an integrable deformation of an integrable version of the Rikitake system. The deformed system is a three-dimensional Hamilton-Poisson system. We present two Lie-Poisson structures and also symplectic realizations. Furthermore, we give a prequantization result of one of the Poisson manifold. We study the stability of the equilibrium states and we prove the existence of periodic orbits. We analyze some properties of the energy-Casimir mapping $\mathcal {EC}$ associated to our system. In many cases the dynamical behavior of such systems is related with some geometric properties of the image of the energy-Casimir mapping. These connections were observed in the cases when the image of $\mathcal {EC}$ is a convex proper subset of $\mathbb {R}^2$. In order to point out new connections, we choose deformation functions such that Im$(\mathcal {EC})=\mathbb {R}^2.$ Using the images of the equilibrium states through the energy-Casimir mapping we give parametric equations of some special orbits, namely heteroclinic orbits, split-heteroclinic orbits, and split-homoclinic orbits. Finally, we implement the mid-point rule to perform some numerical integrations of the considered system.

Détail

DOI : 10.21136/AM.2021.0303-19

Classification : 70H05, 70H06, 70H12, 70H14, 70K20, 70K44
Keywords: integrable deformation; Hamilton-Poisson system; stability; energy-Casimir mapping; periodic orbit; heteroclinic orbit; mid-point rule