On a deformed version of the two-disk dynamo system. (English) Zbl 07361059

Summary: 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 \(\mathrm{Im}(\mathcal{EC})=\mathbb{R}^2\). 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.


70H05 Hamilton’s equations
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics
70H14 Stability problems for problems in Hamiltonian and Lagrangian mechanics
70K20 Stability for nonlinear problems in mechanics
70K44 Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics
Full Text: DOI


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