Hilbert's sixteenth problem for polynomial Liénard equations

DOI10.1007/s12346-012-0068-yzbMath1272.34002OpenAlexW2063451147MaRDI QIDQ1936475

Magdalena Caubergh

Publication date: 5 February 2013

Published in: Qualitative Theory of Dynamical Systems (Search for Journal in Brave)

Full work available at URL: http://ddd.uab.cat/record/228095

zbMATH Keywords

compactificationplanar vector fieldperiodic orbitLiénard equationlarge amplitude limit cycle

Mathematics Subject Classification ID

Transformation and reduction of ordinary differential equations and systems, normal forms (34C20) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Bifurcations of limit cycles and periodic orbits in dynamical systems (37G15) Research exposition (monographs, survey articles) pertaining to ordinary differential equations (34-02)

Related Items (12)

Normal forms near a symmetric planar saddle connection ⋮ Hopf bifurcation for a \(\varphi \)-Laplacian Liénard system ⋮ On the integrability of Liénard systems with a strong saddle ⋮ Bifurcations of small limit cycles in Liénard systems with cubic restoring terms ⋮ Limit cycle enumeration in random vector fields ⋮ Nilpotent center conditions in cubic switching polynomial Liénard systems by higher-order analysis ⋮ Center Conditions and Bifurcation of Limit Cycles Created from a Class of Second-Order ODEs ⋮ Multiple limit cycles of some strongly nonlinear Liénard-van der Pol oscillator ⋮ Maximum Number of Small Limit Cycles in Some Rational Liénard Systems with Cubic Restoring Terms ⋮ The number of limit cycles for regularized piecewise polynomial systems is unbounded ⋮ Slow divergence integrals in classical Liénard equations near centers ⋮ Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields

Cites Work

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• An upper estimate for the number of limit cycles of even-degree Liénard equations in the focus case
• Classical Liénard equations of degree \(n\geqslant 6\) can have \([\frac{n-1}{2}+2\) limit cycles]
• Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems
• A cubic system with thirteen limit cycles
• Birth of canard cycles
• Polynomial Liénard equations near infinity
• Quadratic Liénard equations with quadratic damping
• Bifurcation of planar vector fields and Hilbert's sixteenth problem
• Some lower bounds for \(H(n)\) in Hilbert's 16th problem
• Mathematical problems for the next century
• Perturbation from an elliptic Hamiltonian of degree four. IV: Figure eight-loop.
• Hopf-Takens bifurcations and centres
• Limit cycle bifurcations of some Liénard systems
• Putting a boundary to the space of Liénard equations
• Generalized Liénard equations, cyclicity and Hopf-Takens bifurcations
• Hilbert's 16th problem for classical Liénard equations of even degree
• Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation
• Cubic Lienard equations with linear damping
• More limit cycles than expected in Liénard equations
• ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD
• Limit cycles of the generalized polynomial Liénard differential equations
• Small-amplitude limit cycles of certain Liénard systems
• On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations
• Centennial History of Hilbert's 16th Problem
• HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
• Polynomial systems: a lower bound for the Hilbert numbers
• BIFURCATIONS OF LIMIT CYCLES IN A Z₂-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELD OF DEGREE 7

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