We study bifurcations leading to the appearance of elliptic orbits in the case of four-dimensional symplectic diffeomorphisms (and Hamiltonian flows with three degrees of freedom) with a homoclinic tangency to a saddle-focus periodic orbit.

We summarize recent results on the construction of Lax pairs with spectral parameter for the twisted and untwisted elliptic Calogero–Moser systems associated with arbitrary simple Lie algebras, their scaling limits to Toda systems, and their role in Seiberg–Witten theory. We extend part of this work by presenting a new parametrization for the spectral curves for elliptic spin Calogero–Moser systems associated with $SL(N)$.

The paper is devoted to the problem of analytical classification of conformal maps of the form $f : z \mapsto z + z^2 +\ldots$ in a neighborhood of the degenerate fixed point $z=0$. It is shown that the analytical invariants, constructed in the works of Voronin and Ecalle, may be considered as a measure of splitting for stable and unstable (semi-) invariant foliations associated with the fixed point. This splitting is exponentially small with respect to the distance to the fixed point.

The results of Morse and Hedlund about minimal heteroclinic geodesics on surfaces are generalized to a class of Finsler manifolds possessing a symmetry. The existence of minimal heteroclinic geodesics is established. Under an assumption that the set of such geodesics has certain compactness properties, multibump chaotic geodesics are constructed.

In this paper we examine an integrable and a non-integrable class of the first order nonlinear ordinary differential equations of the type $\dot{x}=x - x^n + \varepsilon g(t)$, $x \in \mathbb{C}$, $n \in \mathbb{N}$. We exploit, using the analysis proposed in [1], the asymptotic formulas which give the location of the singularities in the complex plane and show that there is an essential difference regarding the formation and the density of the singularities between the cases $g(t)=1$ and $g(t)=t$. Our analytical results are combined with a numerical study of the solutions in the complex time plane.

In this article we considered the integrable problems of three vortices on a plane and sphere for noncompact case. We investigated explicitly the problems of a collapse and scattering of vortices and obtained the conditions of realization. We completed the bifurcation analysis and investigated the dependence of stability in linear approximation and frequency of rotation in relative coordinates for collinear and Thomson's configurations from value of a full moment and indicated the geometric interpretation for characteristic situations. We constructed a phase portrait and geometric projection for an integrable configuration of four vortices on a plane.

The classification of motion of two point vortices in a cylindrical pipe is resulted. The research is based on construction of the bifurcation diagram of the problem. A possibility of dynamic collapse of vortices is discussed.