Results 221 to 230 of about 155,898 (247)

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   +4 more sources

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.
J. Llibre
semanticscholar   +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.
Xiang Zhang, Jaume Llibre
exaly   +2 more sources

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.
J. Llibre, R. Ramírez
semanticscholar   +2 more sources

Coexistence of algebraic and non-algebraic limit cycles, explicitly given, using Riccati equations

Nonlinearity, 2006
We give a family of planar polynomial differential systems whose limit cycles can be explicitly described using polar coordinates. Moreover, we characterize the multiplicity of each one of the limit cycles whenever they exist. The given family of planar polynomial differential systems can have at most two limit cycles, counted with multiplicity.As an ...
Jaume Gine, Maite Grau
exaly   +2 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

Classification of Invariant Algebraic Curves and Nonexistence of Algebraic Limit Cycles in Quadratic Systems from Family (I) of the Chinese Classification

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2020
We give the complete classification of irreducible invariant algebraic curves in quadratic systems from family (I) of the Chinese classification, that is, of differential system x′ = y,y′ = δy − x ...
M. Demina, C. Valls
semanticscholar   +1 more source

Limit Cycles from Perturbed Center on the Invariant Algebraic Surface of Unified Lorenz-Type System

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2023
For a three-dimensional chaotic system, little seems to be known about the perturbation of invariant algebraic surface and the center on this surface. This question is very interesting and worth investigating. This paper is devoted to analyzing the limit
Yuming Chen, Qigui Yang
semanticscholar   +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
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Eduardo Sáez, Iván Szántó
openaire   +2 more sources