Topic
Hilbert's sixteenth problem
About: Hilbert's sixteenth problem is a research topic. Over the lifetime, 34 publications have been published within this topic receiving 1259 citations.
Papers
TL;DR: The progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 15th problem and bifurcations of planar vector fields is discussed.
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 Zq–equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.
532 citations
Book•
30 Sep 2003
TL;DR: The study of limit cycles is an interesting and very difficult problem of the qualitative theory of differential equations as mentioned in this paper, which was origi- nated at the end of the nineteenth century in the works of two geniuses of the world science: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare.
Abstract: On the 8th of August 1900 outstanding German mathematician David Hilbert delivered a talk "Mathematical problems" at the Second Interna- tional Congress of Mathematicians in Paris. The talk covered practically all directions of mathematical thought of that time and contained a list of 23 problems which determined the further development of mathema- tics in many respects (1, 119]. Hilbert's Sixteenth Problem (the second part) was stated as follows: Problem. To find the maximum number and to determine the relative position of limit cycles of the equation dy Qn(X, y) -= dx Pn(x, y)' where Pn and Qn are polynomials of real variables x, y with real coeffi- cients and not greater than n degree. The study of limit cycles is an interesting and very difficult problem of the qualitative theory of differential equations. This theory was origi- nated at the end of the nineteenth century in the works of two geniuses of the world science: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare. A. M. Lyapunov set forth and solved completely in the very wide class of cases a special problem of the qualitative theory: the problem of motion stability (154]. In turn, H. Poincare stated a general problem of the qualitative analysis which was formulated as follows: not integrating the differential equation and using only the properties of its right-hand sides, to give as more as possi- ble complete information on the qualitative behaviour of integral curves defined by this equation (176].
82 citations
56 citations
49 citations
TL;DR: For planar vector fields with nodal invariant algebraic curves, the Llibre et al. as mentioned in this paper conjecture was shown to hold for all polynomial vector fields.
45 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2016 | 1 |
| 2014 | 2 |
| 2013 | 1 |
| 2012 | 2 |
| 2011 | 3 |
| 2010 | 2 |