相关论文: Bifurcations and Structural Stability of Generic P…

We classify global bifurcations in generic one-parameter local families of \vfs on $S^2$ with a parabolic cycle. The classification is quite different from the classical results presented in monographs on the bifurcation theory. As a by…

We study stability and bifurcations in holomorphic families of polynomial automorphisms of C^2. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines…

We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two…

For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this…

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…

This paper consists in discussing some issues on generic local classification of typical singularities of $2D$ piecewise smooth vector fields when the switching set is an algebraic variety. The main focus is to obtain classification results…

In this paper we give the bifurcation diagram of the family of cubic vector fields $\dot z=z^3+ \epsilon_1z+\epsilon_0$ for $z\in \mathbb{C}\mathbb{P}^1$, depending on the values of $\epsilon_1,\epsilon_0\in\mathbb{C}$. The bifurcation…

In this paper, we present a criterion for pitchfork bifurcation of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale's result \cite{RS2} significantly. Based on our criterion, we present a class of…

The Poincare-Hopf theorem tells us that given a smooth, structurally stable vector field on a surface of genus g, the number of saddles is 2-2g less than the number of sinks and sources. We generalize this result by introducing a more…

Denote by $H_{pqm}$ the space of all planar $(p,q)$-quasihomogeneous vector fields of degree $m$ endowed with the coefficient topology. In this paper we characterize the set $\Omega_{pqm}$ of the vector fields in $H_{pqm}$ that are…

A family of periodic perturbations of an attracting robust heteroclinic cycle defined on the two-sphere is studied by reducing the analysis to that of a one-parameter family of maps on a circle. The set of zeros of the family forms a…

This paper introduces a novel approach to multi-parameter persistence using 2-categorical structures. We develop a framework that captures hierarchical interactions between filter parameters, overcoming fundamental limitations of…

We point out that any stable generalized complex structure on a sphere bundle over a closed surface of genus at least two must be of constant type.

This paper presents results concerning bifurcations of 2D piecewise-smooth dynamical systems governed by vector fields. Generic three-parameter families of a class of Non-Smooth Vector Fields are studied and the bifurcation diagrams are…

We study global bifurcations in generic 3-parameter families of vector fields on $S^2$. In the recent article [arXiv:1506.06797], Ilyashenko, Kudryashov, and Schurov show that 3-parameter unfoldings of vector fields with the polycycle…

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least…

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which…

We prove a sharp representation stability result for graph complexes with a distinguished vertex, and prove that the chains realizing this sharp bound pass to non-trivial families of graph homology classes. This result may be interpreted as…

The paper studies the complex 1-dimensional polynomial vector fields with real coefficients under topological orbital equivalence preserving the separatrices of the pole at infinity. The number of generic strata is determined, and a…

Let $C$ be an algebraic smooth complex curve of genus $g>1$. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on $C$ and the comparison of different type of…