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On the Multiplicity of Algebraic Limit Cycles
Journal of Dynamics and Differential Equations, 2012The 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 \(y\).
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Invariant Algebraic Curves and Hyperelliptic Limit Cycles of Liénard Systems
Qualitative Theory of Dynamical Systems, 2021The 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, 2008For 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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An algebraic approach to the design of robust limit cycle controllers
Proceedings of the 2003 American Control Conference, 2003., 2004The 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.
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Coexistence of algebraic and nonalgebraic limit cycles in Kukles systems
Periodica Mathematica Hungarica, 2008zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Eduardo Sáez, Iván Szántó
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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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