Recent contributions and some open questions in the qualitative behaviour of certain generalized Liénard equations [ Back ]

Online seminar
Gabriele Villari
Università degli Studi di Firenze
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The aim of this talk is to present some recent results, together with some open questions, concerning the phase-portrait of certain generalized Liénard equations. At first I will briefly discuss some joint work with Jean Mawhin, Fabio Zanolin and Timoteo Carletti for the relativistic Liénard equation $$\frac{d}{dt}\frac{\dot{x}}{\sqrt{1-x^{2}}}+f(x)\dot{x}+g(x)=0$$ as well as for the case with prescribed curvature $$\frac{d}{dt}\frac{\dot{x}}{\sqrt{1+x^{2}}}+\lambda f(x)\dot{x}+g(x)=0.$$ In this framework, a generalization in which a function $f(x,\dot{x})$ takes the role of the function $f(x)$ will be also presented. However, the main part of the talk will concentrate on a very recent joint result with Fabio Zanolin, concerning the generalized Liénard system $$\dot{x}=y-F(x,y)$$ $$\dot{y}=-g(x)$$ and focusing on the case in which $$F(x,y)=\lambda B(y)A(x).$$

PDF 1: Periodic solutions of some autonomous Liénard equations with relativistic acceleration

PDF 2: Existence and non-existence of limit cycles for Liénard prescribed curvature equations

PDF 3: Existence of limit cycles for some generalisation of the Liénard equations: the relativistic and the prescribed curvature cases

PDF 4: On the qualitative behavior of a class of generalized Liénard planar systems