Recent zbMATH articles in MSC 37J25https://www.zbmath.org/atom/cc/37J252022-04-28T18:15:44.554837ZWerkzeugLyapunov functions and asymptotics at infinity of solutions of equations that are close to Hamiltonian equationshttps://www.zbmath.org/1482.370532022-04-28T18:15:44.554837Z"Sultanov, O. A."https://www.zbmath.org/authors/?q=ai:sultanov.oskar-aThe author considers perturbed Hamiltonian systems of second order: \[ \frac{dx}{dt} = \partial_y H(x,y,t)\ ,\ \frac{dy}{dt} = - \partial_x H(x,y,t) + F(x,y,t), \] displaying the asymptotic properties for \(t\rightarrow\infty\) \[ H(x,y) = H_0(x,y) + \sum_{k=1}^\infty t^{-k/q}H_k(x,y), \; F(x,y,t)=\sum_{k=1}^\infty t^{-k/q}F_k(x,y)\ ,\ q\in \mathbb{N}, \] and having \((0,0)\) as center-type fixed point.
The asymptotic properties are studied by considering a Lyapunov function of the form \[ V(x,y,t)=2H(x,y,t) + \sum_{k=1}^p t^{-k/q}v_k(x,y) ,\] where the \(v_k\)'s are subject to \[ (\partial_y H_0)\partial_x v_k - (\partial_x H_0)\partial_y v_k = G_k(x,y) .\] The right hand side of the above equation is defined in terms of the \(v_k\)'s as follows: \[ G_1(x,y) = - (2F_1)\partial_y H_0,\]
\[ G_2(x,y) = - (2F_2)\partial_y H_0 - \partial_y v_1 F_1 - [(\partial_y H_1)\partial_x v_1 - (\partial_x H_1)\partial_y v_1], \]
etc.
If for some \(p\in \mathbb{N}\) the system is solved with \(v_k= O(x^2+y^2)\) and \[ G_p(x,y)=-\gamma(x^2+y^2) + O((x^2+y^2)^{3/2}), \] the Lyapunov function is constructed. Several examples are considered, e.g., the 4-th Painlevé equation, auto-resonance systems, etc.
Reviewer: Vladimir Răsvan (Craiova)On the existence of focus singularities in one model of a Lagrange top with a vibrating suspension pointhttps://www.zbmath.org/1482.370542022-04-28T18:15:44.554837Z"Borisov, A. V."https://www.zbmath.org/authors/?q=ai:borisov.alexey-v"Ryabov, P. E."https://www.zbmath.org/authors/?q=ai:ryabov.p-e"Sokolov, S. V."https://www.zbmath.org/authors/?q=ai:sokolov.sergei-vSummary: We consider a completely integrable Hamiltonian system with two degrees of freedom that describes the dynamics of a Lagrange top with a vibrating suspension point. The results of a stability analysis of equilibrium positions are clearly presented. It turns out that, in the case of a vibrating suspension point, both equilibrium positions can be unstable, which corresponds to the existence of focus singularities in the considered model.Two-body problem on a sphere in the presence of a uniform magnetic fieldhttps://www.zbmath.org/1482.700072022-04-28T18:15:44.554837Z"Balabanova, Nataliya A."https://www.zbmath.org/authors/?q=ai:balabanova.nataliya-a"Montaldi, James A."https://www.zbmath.org/authors/?q=ai:montaldi.james-aSummary: We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles interacting via a cotangent potential, we show there are two families of relative equilibria, called Type I and Type II. The Type I relative equilibria exist for all strengths of the magnetic field, while those of Type II exist only if the field is sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We also determine the stability of the two families of relative equilibria.Fixed-time \(\mathcal {H}_{\infty }\) control for port-controlled Hamiltonian systemshttps://www.zbmath.org/1482.931882022-04-28T18:15:44.554837Z"Liu, Xinggui"https://www.zbmath.org/authors/?q=ai:liu.xinggui"Liao, Xiaofeng"https://www.zbmath.org/authors/?q=ai:liao.xiaofengEditorial remark: No review copy delivered.