𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Global bifurcations of limit and separatrix cycles in a generalized Liénard system

✍ Scribed by Valery A. Gaiko; Wim T. van Horssen

Publisher
Elsevier Science
Year
2004
Tongue
English
Weight
182 KB
Volume
59
Category
Article
ISSN
0362-546X

No coin nor oath required. For personal study only.

✦ Synopsis

We construct a canonical cubic dynamical system of Kukles type and carry out the global qualitative analysis of its special case corresponding to a generalized Liénard equation. In particular, we prove that the foci of such a Liénard system can be at most of second order, and that this particular system can have at least three limit cycles in the whole phase plane. Moreover, unlike all previous works on the Kukles-type systems, we study global bifurcations of limit and separatrix cycles, using arbitrary (including as large as possible) field-rotation parameters of our canonical system. As a result, we have obtained a classification of all possible types of separatrix cycles for this generalized Liénard system and also all possible distributions of its limit cycles.

📜 SIMILAR VOLUMES

The number of limit cycle bifurcation di
The number of limit cycle bifurcation diagrams for the generalized mixed Rayleigh–Liénard oscillator
✍ Q. Ding; A.Y.T. Leung 📂 Article 📅 2009 🏛 Elsevier Science 🌐 English ⚖ 237 KB

This paper investigates the generalized mixed Rayleigh-Lie ´nard oscillator with highly nonlinear terms. Not restrict to the number of limit cycles, this analysis considers mainly the number of limit cycle bifurcation diagrams of the system. First, the singularity theory approach is applied to the f

A Note on “Uniqueness of Limit Cycles in
A Note on “Uniqueness of Limit Cycles in a Liénard-Type System”
✍ Robert E. Kooij; Sun Jianhua 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 255 KB

proposed a theorem to guarantee the uniqueness of limit cycles for the generalized Lienard system Ž . Ž . Ž . dxrdt s h y y F x , dyrdt s yg x . We will give a counterexample to their Ž . theorem. It will be shown that their theorem is valid only if F x is monotone on certain intervals. For this ca

Separatrix Cycles and Multiple Limit Cyc
Separatrix Cycles and Multiple Limit Cycles in a Class of Quadratic Systems
✍ A. Zegeling 📂 Article 📅 1994 🏛 Elsevier Science 🌐 English ⚖ 994 KB

In this paper a class of quadratic systems is studied. By quadratic systems we mean autonomous quadratic vector fields in the plane. The class under consideration is class \(\mathrm{II}_{n=0}\) in the Chinese classification of quadratic systems. Bifurcation sets \(\delta=\delta^{*}(l, m)(m>2, l>0)\)

A New Criterion for Controlling the Numb
A New Criterion for Controlling the Number of Limit Cycles of Some Generalized Liénard Equations
✍ Armengol Gasull; Hector Giacomini 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 180 KB

We consider a class of planar differential equations which include the Lie´nard differential equations. By applying the Bendixson-Dulac Criterion for '-connected sets we reduce the study of the number of limit cycles for such equations to the condition that a certain function of just one variable do