scholarly journals On the Exponents of Exponential Dichotomies

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 651
Author(s):  
Flaviano Battelli ◽  
Michal Fečkan
Keyword(s):  
Differential Equations ◽  
Exponential Dichotomy ◽  
Nonlinear Differential Equations ◽  
Linear Differential Equations ◽  
Constructive Method ◽  
Bifurcation Method ◽  
Exponential Dichotomies ◽  
The Difference ◽  
Linear Differential

An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. This roughness result is crucial in developing a Melnikov bifurcation method for either discontinuous or implicit perturbed nonlinear differential equations.

Entire solutions of certain type of non-linear differential equations

2020 ◽  
Vol 70 (1) ◽  
pp. 87-94
Author(s):  
Bo Xue
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Nonlinear Differential Equations ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Linear Differential Equations ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential

AbstractUtilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| > |α2| > |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

The Analysis of the Systems Stability of Linear Differential Equations Based on the Transformation of Difference Schemes

2019 ◽  
Vol 20 (9) ◽  
pp. 542-549 ◽  
Author(s):  
S. G. Bulanov
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computer Analysis ◽  
Linear Differential Equations ◽  
System Stability ◽  
The Difference ◽  
Linear Differential ◽  
The Stability

The approach to the analysis of Lyapunov systems stability of linear ordinary differential equations based on multiplicative transformations of difference schemes of numerical integration is presented. As a result of transformations, the stability criteria in the form of necessary and sufficient conditions are formed. The criteria are invariant with respect to the right side of the system and do not require its transformation with respect to the difference scheme, the length of the gap and the step of the solution. A distinctive feature of the criteria is that they do not use the methods of the qualitative theory of differential equations. In particular, for the case of systems with a constant matrix of the coefficients it is not necessary to construct a characteristic polynomial and estimate the values of the characteristic numbers. When analyzing the system stability with variable matrix coefficients, it is not necessary to calculate the characteristic indicators. The varieties of criteria in an additive form are obtained, the stability analysis based on them being equivalent to the stability assessment based on the criteria in a multiplicative form. Under the conditions of a linear system stability (asymptotic stability) of differential equations, the criteria of the systems stability (asymptotic stability) of linear differential equations with a nonlinear additive are obtained. For the systems of nonlinear ordinary differential equations the scheme of stability analysis based on linearization is presented, which is directly related to the solution under study. The scheme is constructed under the assumption that the solution stability of the system of a general form is equivalent to the stability of the linearized system in a sufficiently small neighborhood of the perturbation of the initial data. The matrix form of the criteria allows implementing them in the form of a cyclic program. The computer analysis is performed in real time and allows coming to an unambiguous conclusion about the nature of the system stability under study. On the basis of a numerical experiment, the acceptable range of the step variation of the difference method and the interval length of the difference solution within the boundaries of the reliability of the stability analysis is established. The approach based on the computer analysis of the systems stability of linear differential equations is rendered. Computer testing has shown the feasibility of using this approach in practice.

scholarly journals Efficient quantum algorithm for dissipative nonlinear differential equations

2021 ◽  
Vol 118 (35) ◽  
pp. e2026805118
Author(s):  
Jin-Peng Liu ◽  
Herman Øie Kolden ◽  
Hari K. Krovi ◽  
Nuno F. Loureiro ◽  
Konstantina Trivisa ◽  
...  
Keyword(s):  
Differential Equations ◽  
Quantum Algorithms ◽  
Nonlinear Differential Equations ◽  
Quantum Algorithm ◽  
Euler Method ◽  
Linear Differential Equations ◽  
Dimensional System ◽  
Worst Case ◽  
Potential Applications ◽  
Linear Differential

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T2q poly(log⁡T,log⁡n,log⁡1/ϵ)/ϵ, where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R≥2. Finally, we discuss potential applications, showing that the R<1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.

Sturmian comparison theory for half-linear and nonlinear differential equations via Picone identity

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1185-1194
Author(s):  
Abdullah Özbekler
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Linear Differential Equations ◽  
Comparison Theorems ◽  
Comparison Theory ◽  
Second Order Differential Equations ◽  
Nonlinear Version ◽  
Linear And Nonlinear ◽  
Linear Differential ◽  
Picone Identity

In this paper, Sturmian comparison theory is developed for the pair of second order differential equations; first of which is the nonlinear differential equations of the form (m(t)??(y'))' + Xn,i=1 qi(t)??i(y)=0(1) and the second is the half-linear differential equations (k(t)??(x'))'+ p(t)??(x) = 0 (2) where ?*(s) = |s|*-1s and ?1 >...> ?m > ? > ?m+1 > ... > ?n > 0. Under the assumption that the solution of Eq. (2) has two consecutive zeros, we obtain Sturm-Picone type and Leighton type comparison theorems for Eq. (1) by employing the new nonlinear version of Picone?s formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for Eq. (1). Examples are given to illustrate the relevance of the results.

scholarly journals On the Iterated Exponent of Convergence of Solutions of Linear Differential Equations with Entire and Meromorphic Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Rabab Bouabdelli ◽  
Benharrat Belaïdi
Keyword(s):  
Differential Equations ◽  
Meromorphic Functions ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Exponent Of Convergence ◽  
Order Linear ◽  
Convergence Of Solutions ◽  
Small Functions ◽  
The Difference ◽  
Linear Differential

We investigate the zeros of the difference of the derivative of solutions of the higher-order linear differential equationsf(k)+Ak-1(z)f(k-1)+⋯+A1(z)f′+A0(z)f=0and small functions, whereA0(z),…,Ak-1(z)are entire or meromorphic functions of finite iteratedporder.

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