The period function of Liénard systems

2013 ◽  
Vol 143 (1) ◽  
pp. 205-221 ◽  
Author(s):  
Li-Jun Yang ◽  
Xian-Wu Zeng
Keyword(s):  
Critical Point ◽  
Sufficient Conditions ◽  
Period Function ◽  
Liénard Systems

The period function of Liénard systems ẋ = −y + F(x), ẏ = g(x) associated with a centre is considered and sufficient conditions are presented for the period function to be strictly and globally monotone increasing and to have at least one critical point. Examples of applications and remarks on related work are also given.

scholarly journals Local and global extrema for functions of several variables

1980 ◽  
Vol 29 (3) ◽  
pp. 362-368 ◽  
Author(s):  
Bruce Calvert ◽  
M. K. Vamanamurthy
Keyword(s):  
Critical Point ◽  
Local Minimum ◽  
Global Minimum ◽  
Sufficient Conditions ◽  
Subject Classification ◽  
General Function ◽  
Several Variables ◽  
Open Question

AbstractLet p: R2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is < 5, but not necessarily if the degree is ≥ 5. It is an open question what the result is for cubics and quartics in more variables, except cubics in three variables. Other sufficient conditions for a global minimum of a general function are given.1980 Mathematics subject classification (Amer. Math. Soc.): 26 B 99, 26 C 99.

Characterization of isochronous foci for planar analytic differential systems

2005 ◽  
Vol 135 (5) ◽  
pp. 985-998 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau
Keyword(s):  
Critical Point ◽  
Analytic Functions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Image Position ◽  
Necessary And Sufficient

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

scholarly journals On the breakup of air bubbles in a Hele-Shaw cell

2010 ◽  
Vol 22 (2) ◽  
pp. 125-149 ◽  
Author(s):  
VLADIMIR ENTOV ◽  
PAVEL ETINGOF
Keyword(s):  
Critical Point ◽  
Sufficient Conditions ◽  
Small Bubble ◽  
Air Bubbles ◽  
Air Bubble ◽  
Degenerate Critical Point ◽  
Hele Shaw Cell ◽  
Big Bubble

We study the problem of breakup of an air bubble in a Hele-Shaw cell. In particular, we propose some sufficient conditions of breakup of the bubble, and ways to find the contraction points of its parts. We also study regulated contraction of a pair of bubbles (in which the rates of air extraction from the bubbles are controlled) and study various asymptotic questions (such as the asymptotics of contraction of a bubble to a degenerate critical point, and asymptotics of contraction of a small bubble in the presence of a big bubble)

Small-amplitude limit cycles of certain Liénard systems

1988 ◽  
Vol 418 (1854) ◽  
pp. 199-208 ◽  
Keyword(s):  
Critical Point ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Liénard Systems

The paper is concerned with the number of limit cycles of systems of the form ẋ = y – F ( x ), ẏ = –g( x ), where F and g are polynomials. For several classes of such systems, the maximum number of limit cycles that can bifurcate out of a critical point under perturbation of the coefficients in F and g is obtained (in terms of the degree of F and g ).

scholarly journals A New Method for Research on the Center-Focus Problem of Differential Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Zhengxin Zhou
Keyword(s):  
Critical Point ◽  
Sufficient Conditions ◽  
New Method ◽  
Qualitative Behavior ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Planar Polynomial Differential Systems

We will introduce Mironenko’s method to discuss the Poincaré center-focus problem, and compare the methods of Lyapunov and Mironenko. We apply the Mironenko method to discuss the qualitative behavior of solutions of some planar polynomial differential systems and derive the sufficient conditions for a critical point to be a center.

Some applications of Hausdorff dimension inequalities for ordinary differential equations

1986 ◽  
Vol 104 (3-4) ◽  
pp. 235-259 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Hausdorff Dimension ◽  
Differential Equations ◽  
Finite Number ◽  
Ordinary Differential Equations ◽  
Critical Point ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Invariant Sets ◽  
Lorenz Equation ◽  
Independent Variable

SynopsisUpper bounds are obtained for the Hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable. From these are derived sufficient conditions for dissipative analytic n-dimensional ω-periodic differential equations to have only a finite number of ω-periodic solutions. For autonomous equations the same conditions ensure that each bounded semi-orbit converges to a critical point. These results yield some information about the Lorenz equation and the forced Duffing equation.

scholarly journals Existence of periodic solutions with prescribed minimal period of a 2nth-order discrete system

2019 ◽  
Vol 17 (1) ◽  
pp. 1392-1399
Author(s):  
Xia Liu ◽  
Tao Zhou ◽  
Haiping Shi
Keyword(s):  
Critical Point ◽  
Periodic Solutions ◽  
Critical Point Theory ◽  
Discrete System ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Minimal Period ◽  
Existence Of Periodic Solutions ◽  
First Time

Abstract In this paper, we concern with a 2nth-order discrete system. Using the critical point theory, we establish various sets of sufficient conditions for the existence of periodic solutions with prescribed minimal period. To the best of our knowledge, this is the first time to discuss the periodic solutions with prescribed minimal period for a 2nth-order discrete system.

scholarly journals Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yu Guo ◽  
Xiao-Bao Shu ◽  
Qianbao Yin
Keyword(s):  
Differential Equations ◽  
Critical Point ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Energy Functional ◽  
Dirichlet Boundary Conditions ◽  
Mountain Pass Lemma ◽  
First Order

<p style='text-indent:20px;'>In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian random impulsive differential equation. Finally, an example is presented to illustrate the feasibility and effectiveness of our results.</p>

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