The 16th Hilbert problem on algebraic limit cycles
Copyright policy )The 16th Hilbert problem on algebraic limit cycles
For real planar polynomial differential systems there appeared a simple version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree $m$?} In [J. Differential Equations, 248(2010), 1401--1409] Llibre, Ramírez and Sadovskia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: {\it Is $1+(m-1)(m-2)/2$ the maximal number of algebraic limit cycles that a polynomial vector field of degree $m$ can have?} In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre {\it et al}\,'s as a special one. For the polynomial vector fields having only non--dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.
16. Journal Differential Equations, 2011
- Shanghai Jiao Tong University China (People's Republic of)
- Department of Mathematics
holomorphic singular foliations, Algebraic limit cycles, Simple version of the 16th Hilbert problem, simple version of the 16th Hilbert problem, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Dynamical Systems (math.DS), algebraic limit cycles, polynomial differential systems, Polynomial differential systems, Mathematics - Classical Analysis and ODEs, Bifurcations of limit cycles and periodic orbits in dynamical systems, 34C05, 34C07, 37G15, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Holomorphic singular foliations, Mathematics - Dynamical Systems, Analysis
holomorphic singular foliations, Algebraic limit cycles, Simple version of the 16th Hilbert problem, simple version of the 16th Hilbert problem, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Dynamical Systems (math.DS), algebraic limit cycles, polynomial differential systems, Polynomial differential systems, Mathematics - Classical Analysis and ODEs, Bifurcations of limit cycles and periodic orbits in dynamical systems, 34C05, 34C07, 37G15, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Holomorphic singular foliations, Mathematics - Dynamical Systems, Analysis
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