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A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will be to analyze the…
The infinite-dimensional mechanics of fluids and plasmas can be formulated as "noncanonical" Hamiltonian systems on a phase space of Eulerian variables. Singularities of the Poisson bracket operator produce singular Casimir elements that…
We define a Gaussian invariant measure for the two-dimensional averaged-Euler equation and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the…
A perturbative way to investigate superfluid properties of various systems under nonuniform potential is presented. We derive the perturbation expansion of the superfluid fraction, which indicates how liquid exhibits nonclassical rotational…
Analytical perturbations of a family of finite-dimensional Poisson systems are considered. It is shown that the family is analytically orbitally conjugate in $U \subset \mathbb{R}^n$ to a planar harmonic oscillator defined on the symplectic…
We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the…
This article discusses the search procedure for the Poincar\'e recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system using a previously developed high-precision numerical method. For the resulting…
It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold.…
Topological constraints play a key role in the self-organizing processes that create structures in macro systems. In fact, if all possible degrees of freedom are actualized on equal footing without constraint, the state of "equipartition"…
This paper concerns supersonic flows with nonzero vorticity governed by the steady Euler-Poisson system, under the coupled effects of the electric potential and the geometry of a convergent nozzle. By the coordinate rotation, the existence…
The results of this study are continuation of the research of Poincar\'e chaos initiated in papers (Akhmet M, Fen MO. Commun Nonlinear Sci Numer Simulat 2016;40:1-5; Akhmet M, Fen MO. Turk J Math, doi:10.3906/mat-1603-51, accepted). We…
We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a…
A discontinuous area-preserving mapping derived from a sinusoidally-forced impacting system is studied. This system, the elastic impact oscillator, is very closely related to the accelerator models of particle physics such as the Fermi map.…
Our first purpose is to study the stability of linear flows on real, connected, compact, semisimple Lie groups. After, we study and classify periodic orbits of linear and invariant flows. In particular, we obtain a version of…
We study an ensemble of random matrices (the Rosenzweig-Porter model) which, in contrast to the standard Gaussian ensemble, is not invariant under changes of basis. We show that a rather complete understanding of its level correlations can…
In an overall framework of quantum mechanics of unitary systems a rather sophisticated new version of perturbation theory is developed. What is assumed is, firstly, that the perturbed Hamiltonians $H=H_0+\lambda V$ are non-Hermitian and lie…
Application of the intersection theory to construction of n-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few new discretizations of motion of the Euler top sharing…
This paper concerns the well-posedness of subsonic Euler-Poisson flows in a convergent nozzle. Due to the geometry of the nozzle, we first introduce a coordinate transformation to prove the existence of radially symmetric subsonic solutions…
The Euler-Poincar\'e approach to complex fluids is used to derive multiscale equations for computationally modelling Euler flows as a basis for modelling turbulence. The model is based on a \emph{kinematic sweeping ansatz} (KSA) which…
We investigate the singularly perturbed monotone systems with respect to cones of rank $2$ and obtain the so called Generic Poincar\'{e}-Bendixson theorem for such perturbed systems, that is, for a bounded positively invariant set, there…