On the 16th Hilbert problem for algebraic limit cycles

For a polynomial planar vector field of degree n?2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1+(n−1)(n−2)/2 when n is even, and (n−1)(n−2)/2 when n is odd. Furthermore, these upper bounds are reached.     

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401-1409] Llibre, Ramírez and Sadovskaia 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: Is1+(m−1)(m−2)/2the 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 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.     

Relationships between limit cycles and algebraic invariant curves for quadratic systems

Algebraic limit cycles for quadratic systems started to be studied in 1958. Up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2, 4, 5 and 6. Here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems. These results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles. In particular, it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system.     

Symbolic computation of limit cycles associated with Hilbert’s 16th problem

This paper is concerned with the practical complexity of the symbolic computation of limit cycles associated with Hilbert’s 16th problem. In particular, in determining the number of small-amplitude limit cycles of a non-linear dynamical system, one often faces computing the focus values of Hopf-type critical points and solving lengthy coupled polynomial equations. These computations must be carried out through symbolic computation with the aid of a computer algebra system such as Maple or Mathematica, and thus usually gives rise to very large algebraic expressions. In this paper, efficient computations for the focus values and polynomial equations are discussed, showing how to deal with the complexity in the computation of non-linear dynamical systems.     

Weak infinitesimal Hilbert’s 16th problem

The following weak infinitestimal Hilbert’s 16th problem is solved. Given a real polynomial H in two variables, denote by M(H, m) the maximal number possessing the following property: for any generic set {γ _i } of at most M(H,m) compact connected components of the level lines H = c _i of the polynomial H, there exists a form θ = P dx + Q dy with polynomials P and Q of degrees no greater than m such that the integral ∫ _H=c θ has nonmultiple zeros on the connected components {γ _i }. An upper bound for the number M(H,m) in terms of the degree n of the polynomial H is found; this estimate is sharp for almost every polynomial H of degree n. A multidimensional version of this result is proved. The relation between the weak infinitesimal Hilbert’s 16th problem and the following question is discussed: How many limit cycles can a polynomial vector field of degree n have if it is close to a Hamiltonian vector field?     

On the realization problem of real algebraic plane curves as Hessian curves

In this paper we study some problems in Hessian topology. We prove that certain real plane curves satisfy the requirements of the Hessian curve of a differential function. The real plane curves we consider are those with k outer ovals and also those which only have one nest of depth k, with \({k \in \mathbb{N}}\) .     

The infinitesimal 16th Hilbert problem in the quadratic case

Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let X _H =H _y -H _x be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of X _H in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles. Oblatum 23-III-2000 & 19-VI-2000?Published online: 11 October 2000     

Universal algebraic equivalences between tautological cycles on Jacobians of curves

We present a collection of algebraic equivalences between tautological cycles on the Jacobian J of a curve, i.e., cycles in the subring of the Chow ring of J generated by the classes of certain standard subvarieties of J. These equivalences are universal in the sense that they hold for all curves of given genus. We show also that they are compatible with the action of the Fourier transform on tautological cycles and compute this action explicitly. Supported in part by NSF grant DMS-0302215.     

Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH ⁱ (X) is finitely generated for i ≤ 4.     

Non-nested configuration of algebraic limit cycles in quadratic systems

This work deals with algebraic limit cycles of planar polynomial differential systems of degree two. More concretely, we show among other facts that a quadratic vector field cannot possess two non-nested algebraic limit cycles contained in different irreducible invariant algebraic curves.     

On pseudopoints of algebraic curves

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least four were defined in a previous paper of the authors. The present paper studies bundles which compute these Clifford indices. We show that under certain conditions on the curve all such bundles and their Serre duals are generated.     

Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

On the construction of bifurcation curves related to limit cycles of multiplicity three for planar vector fields

We consider planar vector fields f(x,y,λ)$f(x,y,\lambda )$ depending on a three-dimensional parameter vector λ$\lambda$. We assume f(0,0,λ)≡0$f(0,0,\lambda )\equiv 0$ and that there exists a parameter value λ=_λ0$\lambda ={\lambda }_0$ connected with the Andronov–Hopf bifurcation of a limit cycle of multiplicity three from the origin. We describe an algorithm to continue the corresponding local Andronov–Hopf bifurcation curve in the parameter space which is based on the continuation of a periodic orbit to some augmented vector field and the construction of a Poincaré function to another augmented system.     

The homotopy limit problem for two-primary algebraic K-theory

We solve the homotopy limit problem for two-primary algebraic K-theory of fields, that is, the Quillen-Lichtenbaum conjecture at the prime 2.     

On approximation of smooth submanifolds by nonsingular real algebraic subvarieties

Let M be a smooth submanifold of dimension m of a nonsingular real algebraic set X. If M can be approximated by nonsingular algebraic subsets of X, then the homology class in represented by M is algebraic. The converse, investigated in this paper, is true only in some exceptional cases.