Affine transformations in Euclidean space generate a correspondence between integrable systems on cotangent bundles to a sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant transformations, we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.

The $2$-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the $2$-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the $2$-body problem on hyperbolic $3$-space and discuss their stability for a strictly attractive potential.

The space of $2$-jets of a real function of two real variables, denoted by $J^2(\mathbb{R}^2,\mathbb{R})$, admits the structure of a metabelian Carnot group, so $J^2(\mathbb{R}^2,\mathbb{R})$ has a normal abelian sub-group $\mathbb{A}$. As any sub-Riemannian manifold, $J^2(\mathbb{R}^2,\mathbb{R})$ has an associated Hamiltonian geodesic flow. The Hamiltonian action of $\mathbb{A}$ on $T^*J^2(\mathbb{R}^2,\mathbb{R})$ yields the reduced Hamiltonian $H_{\mu}$ on $T^*\mathcal{H} \simeq T^*(J^2(\mathbb{R}^2,\mathbb{R})/\mathbb{A})$, where $H_{\mu}$ is a two-dimensional Euclidean space. The paper is devoted to proving that the reduced Hamiltonian $H_{\mu}$ is non-integrable by meromorphic functions for some values of $\mu$. This result suggests the sub-Riemannian geodesic flow on $J^{2}(\mathbb{R}^2,\mathbb{R})$ is not meromorphically integrable.

We review a recent application of the ideas of normal form theory to systems (Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate variable. The main difference from the standard case consists in the non-uniqueness of the normal form and the total absence of the small divisors problem. The exposition is quite general, so as to allow extensions to the case of more non-periodic coordinates, and more functional settings. Here, for simplicity, we work in the real-analytic class.

In this paper, we obtain a classification of gradient-like flows on arbitrary surfaces by generalizing the circular Fleitas scheme. In 1975 he proved that such a scheme is a complete invariant of topological equivalence for polar flows on 2- and 3-manifolds. In this paper, we generalize the concept of a circular scheme to arbitrary gradient-like flows on surfaces.We prove that the isomorphism class of such schemes is a complete invariant of topological equivalence. We also solve exhaustively the realization problem by describing an abstract circular scheme and the process of realizing a gradient-like flow on the surface. In addition, we construct an efficient algorithm for distinguishing the isomorphism of circular schemes.

A general method is presented for constructing a nonlinear canonical transformation, which makes it possible to introduce local variables in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. This method can be used for investigating the behavior of the Hamiltonian system in the vicinity of its periodic trajectories. In particular, it can be applied to solve the problem of orbital stability of periodic motions.

This paper treats the problem of a spherical robot with an axisymmetric pendulum drive rolling without slipping on a vibrating plane. The main purpose of the paper is to investigate the stabilization of the upper vertical rotations of the pendulum using feedback (additional control action). For the chosen type of feedback, regions of asymptotic stability of the upper vertical rotations of the pendulum are constructed and possible bifurcations are analyzed. Special attention is also given to the question of the stability of periodic solutions arising as the vertical rotations lose stability.