A High-Order Melnikov Method for Heteroclinic Orbits in Planar Vector Fields and Heteroclinic Persisting Perturbations

This work extends the high-order Melnikov method established by FJ Chen and QD Wang to heteroclinic orbits, and it is used to prove, under a certain class of perturbations, the heteroclinic orbit in a planar vector field that remains unbroken. Perturbations which have this property together form the heteroclinic persisting space. The Van der Pol system is analysed as an application.

Planar vector fields with central symmetry: roughness and first degree of non-roughness

Рассматривается пространство гладких векторных полей, заданных в замкнутой области D на плоскости, инвариантных относительно центральной симметрии и трансверсальных границе D. Описано множество векторных полей, грубых относительно этого пространства; показано, что оно открыто и всюду плотно. Во множестве всех негрубых векторных полей выделено открытое всюду плотное подмножество, состоящее из векторных полей первой степени негрубости. We consider the space of smooth vector fields defined in a closed domain D on the plane, invariant under the central symmetry and transversal to the boundary D. The set of vector fields that are rough with respect to this space is described; it is shown that it is open and everywhere dense. In the set of all non-rough vector fields, an open everywhere dense subset consisting of vector fields of the first degree of non-roughness is distinguished.

Bifurcations of Heteroclinic Contours in Two-Parameter Planar Systems: Overview and Explicit Examples

Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of known results on bifurcations of these contours. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software matcont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.

Orbital Hypernormal Forms

In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can be carried out degree by degree using quasi-homogeneous expansions of the vector field of the system by means of reduced time-reparametrizations and near-identity transformations, achieving an important reduction in the computational effort. Moreover, although the orbital hypernormal form procedure is essentially nonlinear in nature, our results show that orbital hypernormal forms are characterized by means of linear operators. Some applications are considered: the case of planar vector fields, with emphasis on a case of the Takens–Bogdanov singularity.

A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

In this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.

On solutions of PDEs by using algebras

The components of complex differentiable functions define solutions for the Laplace’s equation. In this paper we generalize this result; for each PDE of the form $Au_{xx}+Bu_{xy}+Cu_{yy}=0$ we give an affine planar vector field $\varphi$ and an associative and commutative 2D algebra with unit $\mathbb A$, with respect to which the components of all functions of the form $\mathcal L\circ\varphi$ define solutions for this PDE, where $\mathcal L$ is differentiable in the sense of Lorch with respect to $\mathbb A$. In the same way, for each PDE of the form $Au_{xx}+Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu=0$, the components of the exponential function $e^{\varphi}$ defined with respect to $\mathbb A$, define solutions for this PDE. In the case of PDEs of the form $Au_{xx}+Bu_{xy}+Cu_{yy}+Fu=0$, sine, cosine, hyperbolic sine, and hyperbolic cosine functions can be used instead of the exponential function. Also, solutions for two dependent variables $3^{\text{th}}$ order PDEs and a $4^{\text{th}}$ order PDE are constructed.

On the structure of divergence-free measures on ℝ3

We consider the structure of divergence-free vector measures on the plane. We show that such measures can be decomposed into measures induced by closed simple curves. More generally, we show that if the divergence of a planar vector-valued measure is a signed measure, then the vector-valued measure can be decomposed into measures induced by simple curves (not necessarily closed). As an application we generalize certain rigidity properties of divergence-free vector fields to vector-valued measures. Namely, we show that if a locally finite vector-valued measure has zero divergence, vanishes in the lower half-space and the normal component of the unit tangent vector of the measure is bounded from below (in the upper half-plane), then the measure is identically zero.