scholarly journals ON THE MAXIMUM NUMBER OF PERIOD ANNULI FOR SECOND ORDER CONSERVATIVE EQUATIONS

2021 ◽  
Vol 26 (4) ◽  
pp. 612-630
Author(s):  
Armands Gritsans ◽  
Inara Yermachenko
Keyword(s):  
Differential Equation ◽  
Finite Number ◽  
Potential Function ◽  
Critical Points ◽  
Upper Bound ◽  
Morse Function ◽  
Second Order

We consider a second order scalar conservative differential equation whose potential function is a Morse function with a finite number of critical points and is unbounded at infinity. We give an upper bound for the number of nonglobal nontrivial period annuli of the equation and prove that the upper bound obtained is sharp. We use tree theory in our considerations.

On a criterion for oscillatory solutions of a linear differential equation of the second order

1940 ◽  
Vol 36 (3) ◽  
pp. 283-287 ◽  
Author(s):  
J. C. P. Miller
Keyword(s):  
Differential Equation ◽  
Finite Number ◽  
Linear Differential Equation ◽  
Second Order ◽  
Oscillatory Solutions ◽  
Number Of Zeros ◽  
Independent Variable ◽  
Linear Differential

This paper gives a criterion for determining whether real, non-zero solutions of a linear differential equation of the second order have an infinite or a finite number of zeros, or, in short, are oscillatory or non-oscillatory, as the independent variable tends to infinity.

scholarly journals Stability of viscous flow between rotating cylinders. III. Integration of a sixth order linear equation

1946 ◽  
Vol 187 (1011) ◽  
pp. 492-504 ◽  
Keyword(s):  
Differential Equation ◽  
Viscous Flow ◽  
Critical Points ◽  
Second Order ◽  
Sixth Order ◽  
Large Parameter ◽  
Rotating Cylinders ◽  
Linear Differential ◽  
High Reynolds ◽  
Second Order Equations

The aim of this paper is to derive the asymptotic integrals, and their transformations through the critical points, of a certain linear differential equation of the sixth order containing a large parameter. This particular equation is of importance in connexion with the question of stability of viscous flow between rotating cylinders. Since, however, similar equations occur in all questions of stability of viscous flow, a development of proper methods of solution of such equations is of very great importance for problems of viscous flow at high Reynolds numbers. The method of finding asymptotic integrals of linear differential equations containing a large parameter is well known; it was developed by Horn (1899), Sehlesinger (1907), Birkhoff (1908) and Fowler & Lock (1922). The main difficulty of the problem consists in the following. The coefficients of the differential equation are expressions like λΦ(x) , where λ is a large parameter, and Φ(x) is a slowly varying function of the independent variable; the function Φ(x) usually vanishes within the range of x under consideration, with the result that the asymptotic expansions become infinite at such critical points, lose their validity round these points and change their form in passing through such points. The main problem of integration consists, thus, in finding the transformations of the asymptotic integrals in passing through critical points. This problem was considered by Jeffreys (1924, 1942), Kramers (1926) and Goldstein (1928, 1932) for certain second-order equations. Langer (1931), using a different method, discussed several cases of second-order equations; a summary of methods used and results obtained was also given by Langer (1934). A case of a fourth-order equation was solved by Meksyn (in Press).

scholarly journals Asymptotic distribution of singularities of solutions of Matrix-Riccati differential equations

1992 ◽  
Vol 34 (1) ◽  
pp. 112-131
Author(s):  
Gerhard Jank
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Number ◽  
Periodic Solutions ◽  
Critical Points ◽  
Riccati Differential Equation ◽  
Riccati Differential Equations ◽  
The Matrix ◽  
Matrix Riccati Differential Equations

AbstractIn the present paper, we make use of the method of asymptotic integration to get estimates on those regions in the complex plane where singularities and critical points of solutions of the Matrix-Riccati differential equation with polynomial co-efficients may appear. The result is that most of these points lie around a finite number of permanent critical directions. These permanent directions are defined by the coefficients of the differential equation. The number of singularities outside certain domains around the permanent critical directions, in a circle of radius r, is of growth O(log r). Applications of the results to periodic solutions and to the determination of critical points are given.

scholarly journals On the Zeroes and the Critical Points of a Solution of a Second Order Half-Linear Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Pedro Almenar ◽  
Lucas Jódar
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Critical Point ◽  
Lower Bounds ◽  
Critical Points ◽  
Upper Bounds ◽  
Second Order ◽  
Lyapunov Inequalities ◽  
Piecewise Continuous ◽  
Linear Differential

This paper presents two methods to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second-order half-linear differential equation(p(x)Φ(y'))'+q(x)Φ(y)=0, withp(x)andq(x)piecewise continuous andp(x)>0,Φ(t)=|t|r-2tandrbeing real such thatr>1. It also compares between them in several examples. Lower bounds (i.e., Lyapunov inequalities) for such a distance are also provided and compared with other methods.

scholarly journals The Distribution of Zeroes and Critical Points of Solutions of a Second Order Half-Linear Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Pedro Almenar ◽  
Lucas Jódar
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Critical Point ◽  
Critical Points ◽  
Upper Bounds ◽  
Second Order ◽  
Distribution Of Zeroes ◽  
Linear Differential

This paper reuses an idea first devised by Kwong to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second order half-linear differential equation(p(x)Φ(y'))'+q(x)Φ(y)=0, withp(x),q(x)>0,Φ(t)=|t|r-2t, andrreal such thatr>1. It also compares it with other methods developed by the authors.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

scholarly journals Oscillatory behavior of a second order nonlinear advanced differential equation with mixed neutral terms

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hongwei Shi ◽  
Yuzhen Bai
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

AbstractIn this paper, we present several new oscillation criteria for a second order nonlinear differential equation with mixed neutral terms of the form $$ \bigl(r(t) \bigl(z'(t)\bigr)^{\alpha }\bigr)'+q(t)x^{\beta } \bigl(\sigma (t)\bigr)=0,\quad t\geq t_{0}, $$(r(t)(z′(t))α)′+q(t)xβ(σ(t))=0,t≥t0, where $z(t)=x(t)+p_{1}(t)x(\tau (t))+p_{2}(t)x(\lambda (t))$z(t)=x(t)+p1(t)x(τ(t))+p2(t)x(λ(t)) and α, β are ratios of two positive odd integers. Our results improve and complement some well-known results which were published recently in the literature. Two examples are given to illustrate the efficiency of our results.

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