Results 101 to 110 of about 3,260 (131)

Uniqueness of Algebraic Limit Cycles for Quadratic Systems

Journal of Mathematical Analysis and Applications, 2001
All known quadratic systems (QS) having an algebraic limit cycle are contained in five families. The degree of the algebraic curve containing the limit cycle is 4 in four of these families and 2 for the other one. Furthermore, it is also known that if there is another QS having an algebraic limit cycle, it should have at least degree 5. The main result
Jaume Llibre, Hector Giacomini
exaly   +3 more sources

Rational Lyapunov Functions and Stable Algebraic Limit Cycles [PDF]

IEEE Transactions on Automatic Control, 2014
The main goal of this technical note is to show that the class of systems described by a planar differential equation having a rational proper Lyapunov function has asymptotically stable sets which are either locally asymptotically stable equilibrium points, stable algebraic limit cycles or asymptotically stable algebraic graphics. The use of the Zubov
Emmanuel Moulay
exaly   +4 more sources

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Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems

Qualitative Theory of Dynamical Systems, 2021
The paper under review studies Liénard systems of the form \[ \dot x=y, \quad \dot y=-f_m(x)y-g_n(x) \] with the focus on the following two aspects: the existence of invariant algebraic curves and hyperelliptic limit cycles of the systems. The functions \(f_m(x)\) and \(g_n(x)\) involved are real polynomials of degree \(m\) and \(n\), respectively. One
Jiazhong Yang
exaly   +3 more sources

On the Multiplicity of Algebraic Limit Cycles

Journal of Dynamics and Differential Equations, 2012
The present paper is devoted to the problem of determining the multiplicity of the unit circle as a periodic orbit of the planar differential system \[ \dot{x}=-y+f(x, y)a(x, y), \;\dot{y}=x+f(x, y)b(x, y), \] where \(f(x, y)=x^2+y^2-1\) and \(a\), \(b\) are real polynomials of the variables \(x\) and \(y\).
García, Belén, Giné, Jaume, Grau, Maite, Pérez del Río, Jesús S.   +3 more
openaire   +1 more source

Explicit non-algebraic limit cycles for polynomial systems

Journal of Computational and Applied Mathematics, 2007
We consider a system of the form x'=P_n(x,y)+xR_m(x,y), y'=Q_n(x,y)+yR_m(x,y), where P_n(x,y), Q_n(x,y) and R_m(x,y) are homogeneous polynomials of degrees n, n and m, respectively, with n<=m. We prove that this system has at most one limit cycle and that when it exists it can be explicitly found.
Armengol Gasull, Joan Torregrosa, H Giacomini   +2 more
exaly   +4 more sources

The Limit Cycle of the van der Pol Equation Is Not Algebraic

Journal of Differential Equations, 1995
It is shown that no solution curve of the so-called Liénard equation \((y'' + f(y)y' + g(y) = 0)\) lies in an algebraic curve if \(f\) and \(g\) are nonzero polynomials, if \(\deg f \geq \deg g\) and if \(f/g\) is not constant. The proof is based on a curious but elementary lemma that gives a necessary and sufficient condition for a solution curve of a
exaly   +3 more sources

Algebraic limit cycles of degree 4 for quadratic systems

Journal of Differential Equations, 2004
This interesting paper contains basic concepts on algebraic curves and some results on polynomial differential systems having invariant algebraic curves. The main result is the proof that quadratic systems have exactly four different families of algebraic limit cycles of degree four.
Jaume Llibre
exaly   +2 more sources

On the algebraic limit cycles of Liénard systems

Nonlinearity, 2008
For the Lienard systems with fm and gn polynomials of degree m and n, respectively, we present explicit systems having algebraic limit cycles in the cases m ≥ 2 and n ≥ 2m + 1 and m ≥ 3 and n = 2m. Also we prove that the Lienard system for m = 3 and n = 5 has no hyperelliptic limit cycles.
Jaume Llibre, Xiang Zhang
openaire   +1 more source

An algebraic approach to the design of robust limit cycle controllers

Proceedings of the 2003 American Control Conference, 2003., 2004
The design of robust limit cycle controllers introduced here can be used for autonomous systems with separable single-input-single-output nonlinearities and unavoidable limit cycles. The objective is to design a controller to secure specified oscillation amplitude and frequency.
Neusa Maria Franco de Oliveira, Karl Heinz Kienitz, Eduardo A. Misawa   +2 more
openaire   +1 more source

Coexistence of algebraic and nonalgebraic limit cycles in Kukles systems

Periodica Mathematica Hungarica, 2008
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Eduardo Sáez, Iván Szántó
openaire   +2 more sources