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NON-ALGEBRAIC LIMIT CYCLES(Topics Around Chaotic Dynamical Systems)

NON-ALGEBRAIC LIMIT CYCLES(Topics Around Chaotic Dynamical Systems)
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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
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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
Qian, Xinjie, Shen, Yang, Yang, Jiazhong
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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
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Polynomial Vector Fields with Prescribed Algebraic Limit Cycles

Geometriae Dedicata, 2001
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Planar Polynomial Systems with Non-Algebraic Limit Cycles

AIP Conference Proceedings, 2009
In this paper, we study the existence of the non‐algebraic limit cycles of the systems dxdt = Pn(x,y)+xRm(x,y) dydt = Qn(x,y)+yRm(x,y) where Pn(x,y), Qn(x,y) and Rm(x,y) are homogeneous polynomials of degrees n, n and m respectively with ...
Khalil I. T. Al-Dosary, George Maroulis, Theodore E. Simos   +2 more
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Hilbert’s 16th Problem for Algebraic Limit Cycles

2016
In this chapter we state Hilbert’s 16th problem restricted to algebraic limit cycles. Namely, consider the set Σ’ n of all real polynomial vector fields \( \chi = \left( {P,\,Q} \right)\) of degree n having real irreducible \( \left( {{\rm on}\, \mathbb{R}\left[ {x,\,y} \right]} \right)\) invariant algebraic curves.
Jaume Llibre, Rafael Ramírez
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NON-ALGEBRAIC LIMIT CYCLES FOR PARAMETRIZED PLANAR POLYNOMIAL SYSTEMS

International Journal of Mathematics, 2007
In this paper, we determine conditions for planar systems of the form [Formula: see text] where a, b and c are real constants, to possess non-algebraic limit cycles. This is done as an application of a former theorem gives description of the existence of the non-algebraic limit cycles of the family of systems: [Formula: see text] where Pn(x,y), Qn(x,y)
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Algebraic limit cycles in polynomial systems of differential equations

Journal of Physics A: Mathematical and Theoretical, 2007
Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree
Jaume Llibre, Yulin Zhao
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Algebraic Limit Cycles

International Journal of Bifurcation and Chaos
In the qualitative theory of differential equations in the plane [Formula: see text], one of the most difficult objects to study is the existence of limit cycles. Here, we summarize some results and open problems on the algebraic limit cycles of the planar polynomial differential systems.
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