Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComplete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincare rank 1
108
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this, paper, we give a complete system of analytic invariants for the unfoldings of nonresonant linear differential systems with an irregular singularity of Poincare rank 1 at the origin over a fixed neighborhood $D_r$. The unfolding parameter $epsilon $ is taken in a sector S pointed at the origin of opening larger than $2 pi$ in the complex plane, thus covering a whole neighborhood of the origin. For each parameter value in S, we cover $D_r$ with two sectors and, over each sector, we construct a well chosen basis of solutions of the unfolded linear differential systems. This basis is used to find the analytic invariants linked to the monodromy of the chosen basis around the singular points. The analytic invariants give a complete geometric interpretation to the well-known Stokes matrices at $epsilon =0$: this includes the link (existing at least for the generic cases) between the divergence of the solutions at $epsilon =0$ and the presence of logarithmic terms in the solutions for resonance values of the unfolding parameter. Finally, we give a realization theorem for a given complete system of analytic invariants satisfying a necessary and sufficient condition, thus identifying the set of modules.
rate research
Read More
Let $[A]: Y=AY$ with $Ain mathrm{M}_n (k)$ be a differential linear system. We say that a matrix $Rin {cal M}_{n}(bar{k})$ is a {em reduced form} of $[A]$ if $Rin mathfrak{g}(bar{k})$ and there exists $Pin GL_n (bar{k})$ such that $R=P^{-1}(AP-P)in mathfrak{g}(bar{k})$. Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendants. In this article, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of hamiltonian systems. We use this to give an effective form of the Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.
We study singularly perturbed linear systems of rank two of ordinary differential equations of the form $varepsilon xpartial_x psi (x, varepsilon) + A (x, varepsilon) psi (x, varepsilon) = 0$, with a regular singularity at $x = 0$, and with a fixed asymptotic regularity in the perturbation parameter $varepsilon$ of Gevrey type in a fixed sector. We show that such systems can be put into an upper-triangular form by means of holomorphic gauge transformations which are also Gevrey in the perturbation parameter $varepsilon$ in the same sector. We use this result to construct a family in $varepsilon$ of Levelt filtrations which specialise to the usual Levelt filtration for every fixed nonzero value of $varepsilon$; this family of filtrations recovers in the $varepsilon to 0$ limit the eigen-decomposition for the $varepsilon$-leading-order of the matrix $A (x, varepsilon)$, and also recovers in the $x to 0$ limit the eigen-decomposition of the residue matrix $A (0, varepsilon)$.
We consider discrete-time dynamical systems with a linear relaxation dynamics that are driven by deterministic chaotic forces. By perturbative expansion in a small time scale parameter, we derive from the Perron-Frobenius equation the corrections to ordinary Fokker-Planck equations in leading order of the time scale separation parameter. We present analytic solutions to the equations for the example of driving forces generated by N-th order Chebychev maps. The leading order corrections are universal for N larger or equal to 4 but different for N=2 and N=3. We also study diffusively coupled Chebychev maps as driving forces, where strong correlations may prevent convergence to Gaussian limit behavior.
In this paper, I have proved that for a class of polynomial differential systems of degree n+1 ( where n is an arbitrary positive integer) the composition conjecture is true. I give the sufficient and necessary conditions for these differential systems to have a center at origin point by using a different method from the previous references. By this I can obtain all the focal values of these systems for an arbitrary n and their expressions are succinct and beautiful. I believe that the idea and method of this article can be used to solve the center-focus problem of more high-order polynomial differential systems.
The paper continues the authors study of the linearizability problem for nonlinear control systems. In the recent work [K. Sklyar, Systems Control Lett. 134 (2019), 104572], conditions on mappability of a nonlinear control system to a preassigned linear system with analytic matrices were obtained. In the present paper we solve more general problem on linearizability conditions without indicating a target linear system. To this end, we give a description of invariants for linear non-autonomous single-input controllable systems with analytic matrices, which allow classifying such systems up to transformations of coordinates. This study leads to one problem from the theory of linear ordinary differential equations with meromorphic coefficients. As a result, we obtain a criterion for mappability of nonlinear control systems to linear control systems with analytic matrices.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
