Phase portrait of Hamiltonian systems with homogeneous nonlinearities
This nice paper is concerned with the Hamiltonian system of the form \[ \dot{x}=-H_y(x,y), \qquad \dot{y}=H_x(x,y), \tag{1} \] with \(H(x,y)=(x^2+y^2)/2+H_{n+1}(x,y),\) and \(H_{n+1}\) is a homogeneous polynomial of degree \(n+1.\) An algorithm which allows to depict the phase portrait of any system of type (1) in terms of the shape of \(g(\Theta)=H_{n+1}(\cos\Theta, \sin\Theta)\) is presented. This approach can be also used for obtaining the full bifurcation diagram of (1) for the case when some parameters \(\Lambda\) are involved in terms of the associated function \(g_\Lambda(\Theta).\) In order to get a complete description of the topological behaviour of the flow in the neighbourhood of a finite or infinite critical point, the authors use a variation of the Poincaré compactification and a corresponding classification of finite and infinite critical points in this compactification. Then the new algorithm is applied to obtain all phase portraits of (1) for \(n=2\) and \(n=3,\) as well as two bifurcation diagrams: the complete one for \(n=2\) and the one corresponding to reversible systems of type (1) for \(n=3.\) The results obtained are in complete correspondence with those known in the literature.
scientific article; zbMATH DE number 1513368