Related papers: Planar Kolmogorov systems with infinitely many sin…

For a continuously differentiable Kolmogorov map defined from the nonnegative orthant to itself, a type-K competitive system is defined. Under the assumptions that the system is dissipative and the origin is a repeller, the global dynamics…

For the Schlesinger-type equation related to the fifth Painlev\'e equation (V) via isomonodromy deformation, we present a three-parameter family of matrix solutions along the imaginary axis near the point at infinity, and also the…

For the fifth Painlev\'e equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse…

A Fuchsian system of rank 8 in 3 variables with 4 parameters is presented. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential…

We study a particular return map for a class of low dimensional chaotic models called Kolmogorov Lorenz systems, which received an elegant general Hamiltonian description and includes also the famous Lorenz63 case, from the viewpoint of…

We consider the planar family of rigid systems of the form $x'=-y+xP(x,y), y'=x+yP(x,y)$, where $P$ is any polynomial with monomials of degree one and three. This is the simplest non-trivial family of rigid systems with no rotatory…

Given a planar polynomial vector field $X$ with a fixed Newton polytope $\mathcal{P}$, we prove (under some non degeneracy conditions) that the monomials associated to the upper boundary of $\mathcal{P}$ determine (under topological…

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly…

A generalized two-dimensional cubic Lotka-Volterra model with infinitesimal parameters is studied. Three different cases have been considered, one non-degenerate and two degenerate. The local behavior of the model has been studied in the…

Simple finite dimensional Kantor triple systems over the complex numbers are classified in terms of Satake diagrams. We prove that every simple and linearly compact Kantor triple system has finite dimension and give an explicit presentation…

Dynamical system techniques are extremely useful to study cosmology. It turns out that in most of the cases, we deal with finite isolated fixed points corresponding to a given cosmological epoch. However, it is equally important to analyse…

We study the class of continuous polynomial Volterra processes, which we define as solutions to stochastic Volterra equations driven by a continuous semimartingale with affine drift and quadratic diffusion matrix in the state of the…

In this paper, a class of Kolmogorov systems with delays are studied. Sufficient conditions are provided for a system to have a compact uniform attractor. Then Jansen's result (J. Math. Biol. Vol. 25 (1987) 411-422) for autonomous…

To any tree on $n$ vertices we associate an $n$-dimensional Lotka-Volterra system with $3n-2$ parameters and, for generic values of the parameters, prove it is superintegrable, i.e. it admits $n-1$ functionally independent integrals. We…

This study analyzes the Collatz map through nonlinear dynamics. By embedding integers in Sharkovsky's ordering, we show that odd initial values suffice for full dynamical characterization. We introduce ``direction phases'' to partition…

In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model…

The maximally chaotic K-systems are dynamical systems which have nonzero Kolmogorov entropy. On the other hand, the hyperbolic dynamical systems that fulfil the Anosov C-condition have exponential instability of phase trajectories, mixing…

We associate parametric classes of $n$-component Lotka-Volterra systems which admit $k$ additional linear Darboux polynomials, with admissible loopless hypergraphs of order $n$ and size $k$. We study the equivalence relation on admissible…

We consider the six-dimensional dynamical system in three components introduced by Ryan to describe the scenario of Belinskii, Khalatnikov and Lifshitz to the cosmological singularity when the spatial metric tensor is not diagonal. Despite…

We study bifurcations in finite-parameter families of vector fields on $S^2$. Recent papers by Yu. Ilyashenko, N. Goncharuk, Yu. Kudryashov, I. Schurov, and N. Solodovnikov provide examples of (locally generic) structurally unstable…