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Limit cycles and invariant algebraic curves

Communications on Pure and Applied Analysis
We give lower bounds in terms of~$n,$ for the number of limit cycles of polynomial vector fields of degree~$n,$ having any prescribed invariant algebraic curve. By applying them when the ovals of this curve are also algebraic limit cycles we obtain a new recurrent property for the Hilbert numbers. Finally, we apply our results to two important families
Paulo Santana, Armengol Gasull
exaly   +4 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   +5 more sources

Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

Qualitative Theory of Dynamical Systems, 2023
Abstract For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve.
Armengol Gasull, Hector J. Giacomini
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

Relationships between limit cycles and algebraic invariant curves for quadratic systems [PDF]

Journal of Differential Equations, 2006
Algebraic limit cycles for quadratic systems started to be studied in 1958. Up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2, 4, 5 and 6.
Jaume Llibre, Grzegorz Swirszcz
exaly   +2 more sources

Counterexample to a Conjecture on the Algebraic Limit Cycles of Polynomial Vector Fields [PDF]

Geometriae Dedicata, 2005
In Geometriae Dedicata 79 (2000), 101{108, Rudolf Winkel conjectured: For a given algebraic curve f = 0 of degree m > 4 there is in general no polynomial vector ¯eld of degree less than 2m ¡ 1 leaving invariant f = 0 and having exactly the ovals of f = 0 as limit cycles. Here we show that this conjecture is not true.
Chara Pantazi, Jaume Llibre
exaly   +3 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