We discuss the Arhangel’skĭı-Tall problem and related questions in models obtained by forcing with a coherent Souslin tree.

We consider planar polynomial vector fields. We aim to find the (asymptotic) upper and lower bounds for the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert’s 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. Both upper and lower bounds can be obtained for ...

Two-dimensional polynomial dynamical systems are mainly considered. We develop Erugin’s two-isocline method for the global analysis of such systems, construct canonical systems with field-rotation parameters and study various limit cycle bifurcations. In particular, we show how to carry out the classification of separatrix cycles and consider the most complicated limit cycle bifurcation: the bi...

In a wide variety of physical systems cooperative phenomena resulting from interactions at the atomic or molecular levels give rise to structures on mesoscopic to macroscopic length scales. The problem of calculating the properties of such systems from simulations based on mathematical models is computationally intense because of the range of length scales and length of time that must be includ...

In this paper, we consider the weakened Hilbert’s 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. In particular, a perturbed Hamiltonian polynomial vector field of degree 9 is studied, and an example of Z10-equivariant planar perturbed Hamiltonian systems is constructed. With maximal number of closed orbits, it gives rise to different configurations of limit cycles. ...

A part of the well-known Hilbert’s 16th problem is to consider the existence of maximal number of limit cycles for a general planar polynomial system. In general, this is a very difficult question and it has been studied by many mathematicians (see e.g. [Bautin, 1952; Zhang et al., 2004]). By [Ye, 1986] we know that there exists a quadratic system having four limit cycles. [Bautin, 1952] proved...