Bifurcation sets in the Kovalevskaya-Yehia problem (Q2901892)

This paper describes an investigation of a heavy gyrostat where the distribution of masses satisfies Kovalevskaya conditions. So the energy tensor is defined by a diagonal matrix \(\mathbb{A}= \text{diag}(A,A,A/2)\) and the vector \(\underline a= (a_1,a_2,0)\) pointing from the fixed point to the center of mass of the body lies in the equatorial plane of the inertial ellipsoid. The constant gyrostatic moment \(\underline\lambda= (0,0,\lambda)\) points along the axis of dynamical symmetry. The system has Hamiltonian \(H= {1\over2}\langle\mathbb{A}\underline\omega, \underline\omega\rangle+ \underline\nu\times\underline a\), and first integrals \(G= \langle\mathbb{A}\underline\omega+ \underline\lambda, \underline\nu\rangle\) and \(\Gamma= \langle\underline\nu, \underline\nu\rangle\), where \(\underline\omega, \underline\nu\in \mathbb{R}^3\) and the configuration space is \(\mathbb{R}^6(\underline\omega, \underline\nu)\). Yehia identified another integral \(K\) of degree 4 for this system.NEWLINENEWLINE A change of variables leads to a four-dimensional common level surface \(M^4_{g,\mu}\) for the first integrals that has two continuous parameters -- an area constant \(g\) and a gyrostatic moment \(\mu\). In the new variables the authors consider the moment map \(\widetilde K\times\widetilde H: M^4_{g,\mu}\to \mathbb{R}^2(k,h)\), where \(g,\mu> 0\). The set of critical values of this map, \(\Sigma_{g,\mu}\), defines the bifurcation diagrams. The authors develop a method for calculating the bifurcation set in the parameter space that corresponds to bifurcation diagrams in \(\Sigma_{g,\mu}\). An extensive set of illustrations show different types of bifurcation diagrams and their interrelationships.