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

2003/01/01 by Jibin Li, JIBIN LI · 9 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Quantum chaos and dynamical systems #Advanced Topics in Algebra

paper · doi:10.1142/s0218127403006352

Abstract

The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert's 16th problem? Section 2: The first part of Hilbert's 16th problem. Section 3: The second part of Hilbert's 16th problem: introduction. Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert's 16th problem. Section 7: Global and local bifurcations of Z q –equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.

Citations

• Differential Equations Defined by the Sum of two Quasi-Homogeneous Vector Fields

Cited by

• Hilbert's 16th problem. I. When differential systems meet variational methods
• Algorithmic Averaging for Studying Periodic Orbits of Planar Differential Systems
• Monotonous Period Function for Equivariant Differential Equations with Homogeneous Nonlinearities
• Four limit cycles from perturbing quadratic integrable systems by quadratic polynomials
• A Chebyshev criterion for at most two non-zero limit cycles in Abel equations
• Limit cycles of piecewise differential equations on the cylinder
• A new approach for the study of limit cycles
• The maximum number of centers of planar polynomial Hamiltonian vector fields
• Hilbert's 16th problem. When variational principles meet differential\n systems

Related

• A note on a recent attempt to solve the second part of Hilbert's 16th Problem
• A counterexample to a multidimensional version of the weakened Hilbert's 16-th problem
• A restricted version of the Hilbert's 16th problem for quadratic vector fields
• Limit cycles of planar vector fields: Hilbert's 16th problem and o-minimality
• Hilbert's 16th problem on a period annulus and Nash space of arcs