HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS

DOI10.1142/S0218127403006352zbMath1063.34026MaRDI QIDQ4653787

Ji-Bin Li

Publication date: 8 March 2005

Published in: International Journal of Bifurcation and Chaos (Search for Journal in Brave)

zbMATH Keywords

limit cyclebifurcation of vector fieldsReal algebraic curveconfiguration of limit cyclesHilbert number \(H(n)\)scheme of ovals

Mathematics Subject Classification ID

Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Bifurcation theory for ordinary differential equations (34C23) 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) Topology of real algebraic varieties (14P25) Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) (34C08) Research exposition (monographs, survey articles) pertaining to ordinary differential equations (34-02)

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• Hilbert's sixteenth problem
• On hereditarily indecomposable Banach spaces
• The shape of limit cycles that bifurcate from non-Hamiltonian centers
• Link theory and new restrictions for \(M\)-curves of degree nine
• Ergodicity of limit cycles in quadratic systems
• Existence of at most 1, 2, or 3 zeros of a Melnikov function and limit cycles
• Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment
• Perturbations from an elliptic Hamiltonian of degree four. II: Cuspidal loop
• The Hilbert-Arnold problem and an estimate of the cyclicity of polycycles of the plane and in space
• Perturbations from an elliptic Hamiltonian of degree four. I: Saddle loop and two saddle cycles
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• Further considerations on the number of limit cycles of vector fields of the form \(X(v)=Av+f(v)Bv\)
• Computing integrability conditions for a cubic differential system
• Bifurcation at infinity in polynomial vector fields
• Distribution of ovals of the real plane of algebraic curves, of involutions of four-dimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms
• On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields
• M-curves of degree 10
• Limit cycles of quadratic systems in the plane
• A new class of quadratic systems
• Bifurcation of limit cycles from quadratic isochrones
• REDUCE and the bifurcation of limit cycles
• Configurations of fans and nests of limit cycles for polynomial vector fields in the plane
• Algebraic and topological classification of the homogeneous cubic vector fields in the plane
• A codimension two bifurcation with a third order Picard-Fuchs equation
• Estimate of the number of zeros of an Abelian integral depending on a parameter and limit cycles
• Real analytic varieties with the finiteness property and complex abelian integrals
• On a class of complete polynomial vector fields in the plane
• Keeping track of limit cycles
• Limit cycles of vector fields of the form \(X(v)=Av+f(v)Bv\)
• A note on: Abelian integrals and bifurcation theory
• Bifurcations of certain family of planar vector fields tangent to axes
• The Chebyshev property of elliptic integrals
• Elliptic integrals and their nonoscillation
• Limit cycles of polynomial systems with homogeneous non-linearities
• Bifurcation formulae derived from center manifold theory
• Ljapunov approach to multiple Hopf bifurcation
• Global bifurcations in a disturbed Hamiltonian vector field approaching a 3:1 resonant Poincaré map. I
• Zeroes of complete elliptic integrals for 1:2 resonance
• Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system
• Global bifurcations in a perturbed cubic system with \(Z_ 2\)-symmetry
• Singularities of vector fields on the plane
• Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields
• Computation of normal forms
• Cubic systems and Abel equations
• Petrov modules and zeros of abelian integrals
• Center problem for several differential equations via Cherkas' method
• Abelian integrals related to Morse polynomials and perturbations of plane Hamiltonian vector fields
• Link theory and oval arrangements of real algebraic curves
• Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians
• Rokhlin's inequalities for real algebraic curves and an extremal property
• Cyclicity 1 and 2 conditions for a 2-polycycle of integrable systems on the plane
• Existence of limit cycles in a multiply-connected region
• Melnikov functions and perturbation of a planar Hamiltonian system
• On sufficient conditions for certain two-dimensional systems to have at most two limit cycles
• Abelian integrals in non-symmetric perturbation of symmetric Hamiltonian vector field
• Quadratic systems with center and their perturbations
• Hilbert's 16th problem for quadratic vector fields
• Separatrix cycles and mutiple limit cycles in a class of quadratic systems
• Degenerate and non-trivial hyperbolic polycycles with two vertices
• Weak focus, limit cycles, and bifurcations for bounded quadratic systems
• The first derivative of the period function of a plane vector field
• Quadratic Liénard equations with quadratic damping
• Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops
• Algebraic properties of the Lyapunov and period constants
• Perturbations of quadratic centers
• Integrability of a linear center perturbed by a fifth degree homogeneous polynomial
• Cyclicity of planar homoclinic loops and quadratic integrable systems
• On the shape of limit cycles that bifurcate from Hamiltonian centers
• On Hopf cyclicity of planar systems
• Abelian integrals for cubic vector fields
• Mathematical problems for the next century
• Linear estimation of the number of zeros of abelian integral for some cubic isochronous centers
• Saddle quantities and applications
• A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4
• Hilbert's sixteenth problem and global bifurcations of limit cycles.
• Linear estimate of the number of zeros of Abelian integrals for a class of integrable non-Hamiltonian systems.
• A note on a result of G. S. Petrov about the weakened 16th Hilbert problem
• Bifurcation of limit cycles from quadratic centers
• Multiple limit cycle bifurcation surfaces and global families of multiple limit cycles
• On cyclicity of triangles and segments in quadratic systems
• Simple exponential estimate for the number of real zeros of complete abelian integrals
• Cyclicity of a family of vector fields
• Double exponential estimate for the number of zeros of complete Abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group
• Perturbations of quadratic Hamiltonian systems with symmetry
• Abelian integrals of quadratic Hamiltonian vector fields with an invariant straight line
• The cyclicity of the period annulus of the quadratic Hamiltonian triangle
• Integrability of a linear center perturbed by a fourth degree homogeneous polynomial
• Phase portraits of planar semi-homogeneous vector fields. II
• On the limit cycles available from polynomial perturbations of the Bogdanov-Takens Hamiltonian
• The cyclicity of the period annulus of the quadratic Hamiltonian systems with non-Morsean point
• A survey of quadratic systems
• The cusp of order n
• On the topology of real plane algebraic curves
• The number of limit cycles of polynomial deformations of a Hamiltonian vector field
• On the conditions of Kukles for the existence of a Centre
• Comparaison de la Méthode des Constantes de Lyapunov et de la Bifurcation de Hopf^*
• ON THE VERSALITY OF A FAMILY OF SYMMETRIC VECTOR FIELDS IN THE PLANE
• The number of small-amplitude limit cycles of Liénard equations
• REAL PLANE CURVES OF DEGREES 7 AND 8: NEW PROHIBITIONS
• Limit Cycles for a Class of Abel Equations
• Bounded Polynomial Vector Fields
• The number of limit cycles of certain polynomial differential equations
• Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3
• Progress in the topology of real algebraic varieties over the last six years
• On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields
• Quadratic systems with a degenerate critical point
• Condition for generating limit cycles by bifurcation of the loop of a saddle-point separatrix
• Generalized Hopf Bifurcation and Its Dual Generalized Homoclinic Bifurcation
• The Saddle-Node Separatrix-Loop Bifurcation
• Small-amplitude limit cycles of certain Liénard systems
• Limit cycles of a class of polynomial systems