Sufficient Conditions for Oscillation of the Liénard Equation

1980 ◽  
Vol 21 (1) ◽  
pp. 55-62
Author(s):  
Xiaojing Yang
Keyword(s):  
Sufficient Conditions ◽  
Liénard Equation ◽  
Lienard Equation

scholarly journals Stability analysis for the Lienard equation with discontinuous coefficients

2021 ◽  
Vol 31 (2) ◽  
pp. 226-240
Author(s):  
A.V. Platonov
Keyword(s):  
Sufficient Conditions ◽  
Discontinuous Coefficients ◽  
Liénard Equation ◽  
Auxiliary Result ◽  
Piecewise Constant ◽  
Nonlinear Mechanical ◽  
Nonlinear Mechanical System ◽  
Sign Definiteness ◽  
Break Points ◽  
Lienard Equation

A nonlinear mechanical system, whose dynamics is described by a vector ordinary differential equation of the Lienard type, is considered. It is assumed that the coefficients of the equation can switch from one set of constant values to another, and the total number of these sets is, in general, infinite. Thus, piecewise constant functions with infinite number of break points on the entire time axis, are used to set the coefficients of the equation. A method for constructing a discontinuous Lyapunov function is proposed, which is applied to obtain sufficient conditions of the asymptotic stability of the zero equilibrium position of the equation studied. The results found are generalized to the case of a nonstationary Lienard equation with discontinuous coefficients of a more general form. As an auxiliary result of the work, some methods for analyzing the question of sign-definiteness and approaches to obtaining estimates for algebraic expressions, that represent the sum of power-type terms with non-stationary coefficients, are developed. The key feature of the study is the absence of assumptions about the boundedness of these non-stationary coefficients or their separateness from zero. Some examples are given to illustrate the established results.

scholarly journals Monotonicity of a Quadratic Lienard Equation

2020 ◽  
Vol Volume 14 ◽  
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Liénard Equation ◽  
Period Function ◽  
Necessary And Sufficient ◽  
Lienard Equation

We study the period function of the quadratic Lienard equation of a certain type in order to give necessary and sufficient conditions for monotonicity and isochronicty of the period function. We apply this result to identify the region of monotonicity of the period function of particular cases.

On the Liénard system which has no periodic solutions

1993 ◽  
Vol 113 (2) ◽  
pp. 413-422 ◽  
Author(s):  
Jitsuro Sugie ◽  
Toshiaki Yoneyama
Keyword(s):  
Periodic Solutions ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Liénard Equation ◽  
Liénard System ◽  
Lienard System ◽  
Generalized Liénard Equation ◽  
Existence Of Periodic Solutions ◽  
Image Position ◽  
Lienard Equation

The problem of periodicity of solutions of the generalized Liénard equationhas attracted much attention. Many efforts have been made to give sufficient conditions to guarantee the existence and the uniqueness of periodic solutions (limit cycles) of (1·1). There are also some papers on the number of limit cycles of (1·1) (see, for example, [3, 5, 6, 13]). However, there are only a few results on non-existence of periodic solutions of (1·1).

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