arXiv 1208.46972012-08-23math.CAmath.DS

Cooperative irreducible systems of ordinary differential equations with first integral

Janusz Mierczyński

英文摘要：

This note considers cooperative irreducible systems of ordinary differential equations admitting a $C^{1}$ first integral with positive gradient. We prove that all forward or backward nonwandering points are equilibria. We obtain also some results on the global phase portrait of such systems. The main tool of proof is a canonically defined Finsler structure with respect to which the derivative skew-product dynamical system is contractive.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Instability in linear cooperative systems of ordinary differential equations

It is well known that, contrary to the autonomous case, the stability/instability of solutions of nonautonomous linear ordinary differential equations $x' = A(t) x$ is in no relation to the sign of the real parts of the eigenvalues of $A(t)$. In particular, the real parts of all eigenvalues can be negative and bounded away from zero, nonetheless there is a solution of magnitude growing to infinity. In this paper we present a method of constructing examples of such systems when the matrices $A(t)$ have positive off-diagonal entries (strongly cooperative systems). We illustrate those examples bo…

2017-08-24math.CA

Lower estimates of top Lyapunov exponent for cooperative random systems of linear ODEs

For cooperative random linear systems of ordinary differential equations a method is presented of obtaining lower estimates of the top Lyapunov exponent. The proofs are based on applying some polynomial Lyapunov-like function. Known estimates for the dominant eigenvalue of a nonnegative matrix due to G. Frobenius and L. Yu. Kolotilina are shown to be specializations of our results.

2017-08-18math.CAmath.DS

A simple proof of monotonicity for linear cooperative systems of ODEs

We present a simple proof of monotonicity for cooperative systems of linear ordinary differential equations, without having recourse to approximation by strongly cooperative systems.

2013-04-24math.CA

On the Study of the General Group Classification of Systems of Linear Second-Order Ordinary Differential Equations

In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results and subsequent Theorem arising from this particular study are discussed here. This paper considers the study of irreducible systems of second-order ordinary differential equations.

2014-01-31math.CA

Some Results on Ordinary Differential Operators with Periodic Coefficients

For a general ordinary differential operator $\mathcal{L}$ with periodic coefficients we prove that the characteristic polynomial of the Floquet matrix is irreducible over the field of meromorphic functions. We also consider a multipoint eigenvalue problem and show that its eigenspaces are spanned by pure or generalized Floquet solutions. Finally, at the end of the paper we mention some relevant conjectures and open questions.

2015-01-15math.CAmath.CVmath.SP

Rational Solutions of First Order Algebraic Ordinary Differential Equations

Let $f(t,y,y')=\sum_{i=0}^n a_i(t,y)y'^i=0$ be an irreducible first order ordinary differential equation with polynomial coefficients. Eremenko in 1998 proved that there exists a constant $C$ such that every rational solution of $f(t,y,y')=0$ is of degree not greater than $C$. Examples show that this degree bound $C$ depends not only on the degrees of $f$ in $t,y,y'$ but also on the coefficients of $f$ viewed as the polynomial in $t,y,y'$. In this paper, we show that if $f$ satisfies $deg(f,y)<deg(f,y')$ or $\max_{i=0}^n \{deg(a_i,y)-2(n-i)\}>0 $ then the degree bound $C$ only depends on the d…

2022-01-27cs.SCmath.CA

更多相关论文收起更多论文

一个Kac-Moody根系与线性常微分方程

A Kac-Moody root system and linear ordinary differential equations

本文证明黎曼球面上不可约线性微分方程可通过可逆变换化为平凡方程或$E_8$-基本谱型Fuchs系统，谱型由原方程唯一确定，并基于星形Kac-Moody根系建立对应关系。

2026-10-01math.CA

Estimates for principal Lyapunov exponents: A survey

This is a survey of known results on estimating the principal Lyapunov exponent of a time-dependent linear differential equation possessing some monotonicity properties. Equations considered are mainly strongly cooperative systems of ordinary differential equations and parabolic partial differential equations of second order. The estimates are given either in terms of the principal (dominant) eigenvalue of some derived time-independent equation or in terms of the parameters of the equation itself. Extensions to other differential equations are considered. Possible directions of further researc…

2014-11-15math.APmath.CAmath.DS