Related papers: Formal Paths, Iterated Integrals and the Center Pr…
The classical Center-Focus Problem posed by H. Poincar\'e in 1880's is concerned on the characterization of planar polynomial vector fields $X=(-y+P(x,y))\dfrac{\partial}{\partial x}+(x+Q(x,y))\dfrac{\partial}{\partial y},$ with…
The Integration Theory of Linear Ordinary Differential Equation.
The Abel differential equation $y'=p(x)y^3 + q(x) y^2$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem…
The nonlinear differential system $ \dot{x}=\sum_{i=0}^{\ell}P_{m_i}(x,y),\ \dot{y}=\sum_{i=0}^{\ell}Q_{m_i}(x,y)$ is considered, where $P_{m_i}$ and $Q_{m_i}$ are homogeneous polynomials of degree $m_i\geq 1$ in $x$ and $y$, $m_0=1$. The…
We define a proper differential sequence of ordinary differential equations and introduce a method to derive an alternative sequence of integrals for such a sequence. We describe some general properties which are illustrated by several…
In this work we deal with analytic families of real planar vector fields $\mathcal{X}_\lambda$ having a monodromic singularity at the origin for any $\lambda \in \Lambda \subset \mathbb{R}^p$ and depending analytically on the parameters…
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…
Generalized diffusion type equations are considered and point symmetry analysis is applied to them. The equations with extremal order point symmetry algebras are described. Some old geometrical results are rederived in connection with…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to algebraic curves.
We study the moments finiteness problem for the class of Lipschitz maps $F: [a,b]\rightarrow\mathbb R^n$ with images in a compact Lipschitz triangulable curve $\Gamma$. We apply the obtained results to the center problem for ODEs describing…
This article examines a new approach to solving ordinary differential equations based on Fractional-Calculus theory. Poisson and Sturm-Liouville-type problems are studied, together with different boundary conditions. Each case is analyzed…
We shall study the equivalence problem for ordinary differential equations with respect to the affine transformations group.
When studying a general system of delay differential equation with a single constant delay, we encounter a certain lack of uniqueness in determining the coefficient of one of the third order terms of the series defining the center manifold.…
In the article it was shown the convergence of special integral of two dimensional Terry's problem. Main tools of the article are an investigation of real algebraic varieties and estimations of areas of algebraic surfaces.
We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering…
Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are…
We define a deterministic integral with respect to irregular paths as a limit of standard line integrals and completely describe a class of all paths for which this integral exists for functions with H\"older exponent in the range of (0,1].…
As is known, the problems for the differential equations with continuously changing order of the derivatives are not considered completely. In this paper we consider the initial and boundary value problems for this type of linear ordinary…
We describe the first term of the $\Lambda_{k-1}\mathcal{C}$--spectral sequence (see math.DG/0610917) of the diffiety (E,C), E being the infinite prolongation of an l-normal system of partial differential equations, and C the Cartan…