scholarly journals On the Period Function of Liénard Systems

1999 ◽  
Vol 152 (2) ◽  
pp. 467-487 ◽  
Author(s):  
M. Sabatini
Keyword(s):  
Period Function ◽  
Liénard Systems

scholarly journals On the critical periods of Liénard systems with cubic restoring forces

2004 ◽  
Vol 2004 (61) ◽  
pp. 3259-3274 ◽  
Author(s):  
Zhengdong Du
Keyword(s):  
Taylor Series ◽  
Type I ◽  
Type Ii ◽  
Critical Periods ◽  
Period Function ◽  
Weak Center ◽  
Local Bifurcations ◽  
Liénard Systems ◽  
Lienard System ◽  
Restoring Forces

We study local bifurcations of critical periods in the neighborhood of a nondegenerate center of a Liénard system of the formx˙=−y+F(x),y˙=g(x), whereF(x)andg(x)are polynomials such thatdeg(g(x))≤3,g(0)=0, andg′(0)=1,F(0)=F′(0)=0and the system always has a center at(0,0). The set of coefficients ofF(x)andg(x)is split into two strata denoted bySIandSIIand(0,0)is called weak center of type I and type II, respectively. By using a similar method implemented in previous works which is based on the analysis of the coefficients of the Taylor series of the period function, we show that for a weak center of type I, at most[(1/2)deg(F(x))]−1local critical periods can bifurcate and the maximum number can be reached. For a weak center of type II, the maximum number of local critical periods that can bifurcate is at least[(1/4)deg(F(x))].

scholarly journals Simultaneous Bifurcation of Limit Cycles and Critical Periods

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Regilene D. S. Oliveira ◽  
Iván Sánchez-Sánchez ◽  
Joan Torregrosa
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Critical Periods ◽  
Bifurcation Of Limit Cycles ◽  
Period Function ◽  
Polynomial Differential Equations ◽  
Return Map ◽  
Lie Bracket ◽  
Liénard Systems ◽  
Quadratic Family

AbstractThe present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

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 On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

scholarly journals CK1α Collaborates with DOUBLETIME to Regulate PERIOD Function in the Drosophila Circadian Clock

2018 ◽  
Vol 38 (50) ◽  
pp. 10631-10643 ◽  
Author(s):  
Vu H. Lam ◽  
Ying H. Li ◽  
Xianhui Liu ◽  
Katherine A. Murphy ◽  
Jonathan S. Diehl ◽  
...  
Keyword(s):  
Circadian Clock ◽  
Period Function

scholarly journals Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Special Form ◽  
Bifurcation Of Limit Cycles ◽  
Liénard Systems

We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.

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