Phase portraits for quadratic homogeneous polynomial vector fields on $$ \mathbb{S}^2 $$

Let X be a homogeneous polynomial vector field of degree 2 on $ \mathbb{S}^2 $ \mathbb{S}^2 . We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on $ \mathbb{S}^2 $ \mathbb{S}^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 ^th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on $ \mathbb{S}^2 $ \mathbb{S}^2 of degree n.     

Limit cycles, invariant meridians and parallels for polynomial vector fields on the torus

We study the polynomial vector fields of arbitrary degree in R³ having the 2-dimensional torus invariant by their flow. We characterize all the possible configurations of invariant meridians and parallels that these vector fields can exhibit. Furthermore we analyze when these invariant either meridians or parallels can be limit cycles.     

Periodic orbits for a class of C 1 three-dimensional systems

We studyC ¹ perturbations of a reversible polynomial differential system of degree 4 in. We introduce the concept of strongly reversible vector field. If the perturbation is strongly reversible, the dynamics of the perturbed system does not change. For non-strongly reversible perturbations we prove the existence of an arbitrary number of symmetric periodic orbits. Additionally, we provide a polynomial vector field of degree 4 in with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied. The first two authors are partially supported by a MCYT grant number MTM2005-06098-C02-01, and by a CICYT grant number 2005SGR 00550. The second author is partially supported by a FAPESP-BRAZIL grant 10246-2. All authors are also supported by the joint project CAPES-MECD grant HBP2003-0017.     

The behavior of the Laplace transform of the invariant measure on the hypersphere of high dimension

We consider the sequence of the hyperspheres M _n , i.e., the homogeneous transitive spaces of the Cartan subgroup of the group and study the normalized limit of the corresponding sequence of invariant measures m _n on those spaces. In the case of compact groups and homogeneous spaces, for example, for the classical pairs (SO(n), S ^n-1), n = 1, 2, … , the limit of the corresponding measures is the classical infinite-dimensional Gaussian measure; this is the well-known Maxwell-Poincaré lemma. Simultaneously the Gaussian measure is a unique (up to a scalar) invariant measure with respect to the action of the infinite orthogonal group O(∞). This coincidence implies the asymptotic equivalence between grand and small canonical ensembles for the series of the pairs (SO(n), S ^n-1). Our main result shows that the situation for noncompact groups, for example for the case , is completely different: the limit of the measures m _n does not exist in the literal sense, and we show that only a normalized logarithmic limit of the Laplace transforms of those measures does exist. At the same time, there exists a measure which is invariant with respect to a continuous analogue of the Cartan subgroup of the group GL(∞), the so-called infinite-dimensional Lebesgue measure (see [7]). This difference is an evidence for non-equivalence between the grand and small canonical ensembles in the noncompact case. To my friend Dima Arnold     

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.     

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.     

Note on a paper of J. Llibre and G. Rodríguez concerning algebraic limit cycles

In a recent paper of Llibre and Rodríguez (J. Differential Equations 198 (2004) 374-380) it is proved that every configuration of cycles in the plane is realizable (up to homeomorphism) by a polynomial vector field of degree at most 2(n+r)-1, where n is the number of cycles and r the number of primary cycles (a cycle C is primary if there are no other cycles contained in the bounded region limited by C). In this letter we prove the same theorem by using an easier construction but with a greater polynomial bound (the vector field we construct has degree at most 4n-1). By using the same technique we also construct R³ polynomial vector fields realizing (up to homeomorphism) any configuration of limit cycles which can be linked and knotted in R³. This answers a question of R. Sverdlove.     

A characterization of spherical distributions

It is shown that when the random vector X in Rⁿ has a mean and when the conditional expectation E(u′X|v′X) = 0 for all vectors u, v Rⁿ which satisfy u′v = 0, then the distribution of X is orthogonally invariant. A version of this characterization is also established when X does not have a mean vector.     

Geometric invariants of link cobordism

A geometric notion of a “derivative” is defined for 2-component links ofS ⁿ inS ⁿ⁺² and used to construct a sequenceβ ⁱ ,i=1,2,... of abelian concordance invariants which vanish for boundary links. Forn>1, these generalize the only heretofore known invariant, the Sato-Levine invariant. Forn=1, these invariants are additive under any band-sum and consequently provide new information about which 1-links are concordant to boundary links. Examples are given of concordance classes successfully distinguished by theβ ⁱ but not by their , Murasugi 2-height, Sato-Levine invariant or Alexander polynomial. Supported in part by a grant from the National Science Foundation.     

On processes which cannot be distinguished by finite observation

A function J defined on a family C of stationary processes is finitely observable if there is a sequence of functions s _n such that s _n (x ₁,…, x _n ) → J(X) in probability for every process X=(x _n ) ∈ C. Recently, Ornstein and Weiss proved the striking result that if C is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant defined on C is entropy [8]. We sharpen this in several ways. Our main result is that if X → Y is a zero-entropy extension of finite entropy ergodic systems and C is the family of processes arising from generating partitions of X and Y, then every finitely observable function on C is constant. This implies Ornstein and Weiss’ result, and extends it to many other families of processes, e.g., it follows that there are no nontrivial finitely observable isomorphism invariants for processes arising from the class of Kronecker systems, the class of mild mixing zero entropy systems, or the class of strong mixing zero entropy systems. It also follows that for the class of processes arising from irrational rotations, every finitely observable isomorphism invariant must be constant for rotations belonging to a set of full Lebesgue measure. This research was supported by the Israel Science Foundation (grant No. 1333/04)     

Lipschitzian retracts and curves as invariant sets

By an invariant set in a metric space we mean a non-empty compact set K such that K = ⋃ _i=1 ⁿ T _i (K) for some contractions T ₁, …, T _n of the space. We prove that, under not too restrictive conditions, the union of finitely many invariant sets is an invariant set. Hence we establish collage theorems for non-affine invariant sets in terms of Lipschitzian retracts. We show that any rectifiable curve is an invariant set though there is a simple arc which is not an invariant set.      

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.     

Conjectures on the Quotient Ring by Diagonal Invariants

We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring in two sets of variables by the ideal generated by all S _n invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x ₁, ..., x _n} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory.     

Upper bounds for the number of orbital topological types of planar polynomial vector fields modulo limit cycles

The purpose of this paper is to find an upper bound for the number of orbital topological types of nth-degree polynomial fields in the plane. An obstacle to obtaining such a bound is related to the unsolved second part of the Hilbert 16th problem. This obstacle is avoided by introducing the notion of equivalence modulo limit cycles. Earlier, the author obtained a lower bound of the form $2^{cn^2 } $ . In the present paper, an upper bound of the same form but with a different constant is found. Moreover, for each planar polynomial vector field with finitely many singular points, a marked planar graph is constructed that represents a complete orbital topological invariant of this field.     

Strong S-domains

S-domains and strong S-rings are studied extensively with special emphasis on integral and polynomial ring extensions. The main theorem of this paper is that for a Prüfer domain R, the polynomial ring R[X₁,…X_n] in finitely many indeterminates is a strong S-domain. We also prove that any Prüfer υ-multiplication domain is an S-domain.     

For a fixed Turaev shadow Jones-Vassiliev invariants depend polynomially on the gleams

We use Turaev's technique of shadows and gleams to parametrize the set of all knots in S ³ with the same Hopf projection. We show that the Vassiliev invariants arising from the Jones polynomial J _t (K) are polynomials in the gleams, i.e., for , the n-th order Vassiliev invariant u _n , defined by , is a polynomial of degree 2n in the gleams. Received: April 30, 1996     

Best Sobolev Constants and Manifolds with Positive Scalar Curvature Metrics

We study the Yamabe invariant of manifolds which admit metrics of positive scalar curvature. Analysing `best Sobolev constants'we give a technique to find positive lower bounds for the invariant.We apply these ideas to show that for any compact Riemannian manifold (N ⁿ ,g) of positive scalarcurvature there is a positive constant K =K(N, g), which depends only on (N, g), such that for any compact manifold M ^m , the Yamabe invariantof M ^m × N ⁿ is no less than K times the invariant ofS ^{n + m} . We will find some estimates for the constant K in the case N =S ⁿ .     

Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies

We illustrate and discuss the computer-assisted study (approximation and visualization) of two-dimensional (un)stable manifolds of steady states and saddle-type limit cycles for flows in R ⁿ . Our investigation highlights a number of computational issues arising in this task, along with our solutions and “quick-fixes” for some of these problems. Two examples illustrative of both successes and shortcomings of our current approach are presented. Representative “snapshots” demonstrate the dependence of two-dimensional invariant manifolds on a bifurcation parameter as well as their interactions. Such approximation and visualization studies are a necessary component of the computer-assisted study and understanding of global bifurcations. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the limit cycles available from polynomial perturbations of the Bogdanov-Takens Hamiltonian

The displacement map related to small polynomial perturbations of the planar Hamiltonian systemdH=0 is studied in the elliptic caseH=1/2y ²+1/2x ²−1/3x ³. An estimate of the number of isolated zeros for each of the successive Melnikov functionsM _k(h),k=1, 2,…is given in terms of the orderk and the maximal degreen of the perturbation. This sets up an upper bound to the number of limit cycles emerging from the periodic orbits of the Hamiltonian system under polynomial perturbations. Research partially supported by grant MM810/98 from the NSF of Bulgaria and MURST, Italy.