Multiplicity of limit cycles and analytic m-solutions for planar differential systems

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f₀(x,y), defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f₀(x,y) being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f₀(x,y) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.     

Abelian integrals and limit cycles

The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation.     

The number of Hamiltonian decompositions of regular graphs

A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph K _n on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended by Kühn and Osthus. They proved that every r-regular n-vertex graph Γ with even degree r = cn for some fixed c > 1/2 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating H(Γ), the number of such decompositions of Γ. Our main result is that H(Γ) = r ^(1+o(1))nr/2. In particular, the number of Hamiltonian decompositions of K _n is \({n^{\left( {1 + o\left( 1 \right)} \right){n^2}/2}}\).     

On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle

In this paper, we make a complete study on small perturbations of Hamiltonian vector field with a hyper-elliptic Hamiltonian of degree five, which is a Liénard system of the form x^′=y, y^′=Q₁(x)+εyQ₂(x) with Q₁ and Q₂ polynomials of degree respectively 4 and 3. It is shown that this system can undergo degenerated Hopf bifurcation and Poincaré bifurcation, which emerges at most three limit cycles in the plane for sufficiently small positive ε. And the limit cycles can encompass only an equilibrium inside, i.e. the configuration (3,0) of limit cycles can appear for some values of parameters, where (3,0) stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria.     

The generating graph of some monolithic groups

For a finite group G let Γ(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. We look for conditions on the positive integer m that ensure that Γ(G) contains a Hamiltonian cycle when G=S?Cm is the wreath product of a finite simple group S and a cyclic group of order m.     

Construction d’entrelacs hyperpoliques et arithmétiques

In this Note, we continue the study started in [4] about arithmetic hyperbolic links L such that S³\L is homeomorphic to H³/Γ, where Γ is not conjugate to a subgroup of any Bianchi group. One describes the steps of the construction, different from the construction in [4], which permits us to determine the first known examples in M₂(Q (i√39)) and M₂(Q (i√6 respectively.     

Hamiltonian decomposition of lexicographic product

In this paper we prove the conjecture of J.-C. Bermond (Ann. Discrete Math.36 (1978), 21–28): If two graphs are decomposable into Hamiltonian cycles, then their lexicographic product is decomposable, too.     

Analysis on bifurcations of multiple limit cycles for a parametrically and externally excited mechanical system

Research on the bifurcations of the multiple limit cycles for a parametrically and externally excited mechanical system is presented in this paper. The original mechanical system is first transformed to the averaged equation in the Cartesian form, which is in the form of a Z₂-symmetric perturbed polynomial Hamiltonian system of degree 5. Then, using the bifurcation theory of planar dynamical system and the method of detection function, the bifurcations of the multiple limit cycles of the system are investigated and the configurations of compound eyes are also obtained.     

Deformations of plane graphs

It is proven that, if Γ₀ and Γ₁ are isomorphic strictly convex graphs such that their outer polygons correspond to each other and have the same orientations, then Γ₀ can be continuously deformed into Γ₁ such that, at each stage, the graph under consideration is convex. This extends a result of Cairns (Ann of Math.45 (2) (1944), 207–217; Amer. Math. Monthly51 (1944), 247–252) and proves a conjecture of Grünbaum and Shepard (“Proceedings, 8th British Combinatorial Conf.”, 1981). This result is applied to prove an analogous conjecture by Grünbaum and Shepard on deformations of straight graphs in general and it is shown how the proof method also can be used to verify a conjecture of Robinson (“Proceedings, 8th British Combinatorial Conf.”, 1981) on deformations of rectanguloid curves.     

Γ-convergence of nonlinear functionals in thin reticulated structuresΓ-convergence des fonctionelles non linéaires dans des structures réticulées de faible épaisseur

We study the Γ-convergence of nonlinear functionals considered in nonperiodic 2D lattice-like structures. The Γ-limit functional is obtained in the explicit form. To cite this article: L. Pankratov, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 315–320.     

Incompressible nonlinearly elastic thin membranes

Nonlinearly elastic thin membrane models are derived for hyperelastic incompressible materials using Γ-convergence arguments. We obtain an integral representation of the limit two-dimensional energy owing to a result of singular functionals relaxation due to Ben Belgacem [ESAIM Control Optim. Calc. Var. 5 (2000) 71–85 (electronic)]. To cite this article: K. Trabelsi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Finite extensions and unipotent shadows of affine crystallographic groupsExtensions finies et ombres unipotentes des groupes affines cristallographiques

Let Γ be a virtually polycyclic group so that the Fitting subgroup is torsion-free and contains its centralizer. We prove that an effective extension of Γ by a finite group μ is isomorphic to an affine crystallographic group if and only if there exists a fixed point for the action of μ on the deformation space of affine crystallographic actions of Γ. We associate to Γ a finitely generated torsion-free nilpotent group Θ which is called the unipotent shadow of Γ, and we relate the deformation space of Γ to the deformation space of Θ. As an application, we show that Γ is isomorphic to an affine crystallographic group if, e.g., Θ has nilpotency class ?3, or if the polycylic rank of Γ is ?5, and also in some other cases. To cite this article: O. Baues, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 785–788.     

On the number of Hamiltonian cycles in tournaments

The main results assert that the minimum number of Hamiltonian bypasses in a strong tournament of order n and the minimum number of Hamiltonian cycles in a 2-connected tournament of order n increase exponentially with n. Furthermore, the number of Hamiltonian cycles in a tournament increases at least exponentially with the minimum outdegree of the tournament. Finally, for each k?1 there are infinitely many tournaments with precisely k Hamiltonian cycles.     

Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H(6) ⩾ 35 = 6² − 1, where H(6) is the Hilbert number for sixth-degree polynomial systems.     

Minimum degree conditions for large subgraphs

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (such as Turán's theorem [Turán, P., On an extremal problem in graph theory (in Hungarian), Matematiko Fizicki Lapok 48 (1941), 436–452]) or on finding spanning subgraphs (such as Dirac's theorem [Dirac, G.A., Some theorems on abstract graphs, Proc. London Math. Soc. s3-2 (1952), 69–81] or more recently work of Komlós, Sárközy and Szemerédi [Komlós, J., G. N. Sárközy and E. Szemerédi, On the square of a Hamiltonian cycle in dense graphs, Random Struct. Algorithms 9 (1996), 193-211; Komlós, J., G. N. Sárközy and E. Szemerédi, Proof of the Seymour Conjecture for large graphs, Ann. Comb. 2 (1998), 43–60] towards a proof of the Pósa-Seymour conjecture). Only a few results give conditions to obtain some intermediate-sized subgraph. We contend that this neglect is unjustified. To support our contention we focus on the illustrative case of minimum degree conditions which guarantee squared-cycles of various lengths, but also offer results, conjectures and comments on other powers of paths and cycles, generalisations thereof, and hypergraph variants.     

Limit points badly approximable by horoballs

For a proper, geodesic, Gromov hyperbolic metric space X, a discrete subgroup of isometries Γ whose limit set is uniformly perfect, and a disjoint collection of horoballs {H _j }, we show that the set of limit points badly approximable by {H _j } is absolutely winning in the limit set Λ(Γ). As an application, we deduce that for a geometrically finite Kleinian group acting on ${\mathbb{H}^{n+1}}$ , the limit points badly approximable by parabolics, denoted BA(Γ), is absolutely winning, generalizing previous results of Dani and McMullen. As a consequence of winning, we show that BA(Γ) has dimension equal to the critical exponent of the group. Since BA(Γ) can alternatively be described as the limit points representing bounded geodesics in the quotient ${\mathbb{H}^{n+1}/\Gamma}$ , we recapture a result originally due to Bishop and Jones.     

Hamiltonian cycles in cubic Cayley graphs: the 〈2,4k,3〉 case

It was proved by Glover and Maru?i? (J. Eur. Math. Soc. 9:775–787, 2007), that cubic Cayley graphs arising from groups G=〈a,x∣a ²=x ^s =(ax)³=1,…〉 having a (2,s,3)-presentation, that is, from groups generated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4, and have a Hamiltonian path when |G| is congruent to 0 modulo 4. In this article the existence of a Hamiltonian cycle is proved when apart from |G| also s is congruent to 0 modulo 4, thus leaving |G| congruent to 0 modulo 4 with s either odd or congruent to 2 modulo 4 as the only remaining cases to be dealt with in order to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs.     

Real-analytic,volume-preserving actions of lattices on 4-manifoldsActions analytiques réelles,conservant le volume,de réseaux sur les variétés de dimension 4

We prove that if Γ is a lattice of $\text{Q}$-rank at least 7 in a simple linear Lie group, then any real-analytic, volume-preserving action of Γ on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action. To cite this article: B. Farb, P.B. Shalen, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1011–1014.     

Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms

A class of cubic Hamiltonion system with the higher-order perturbed term of degree n=5, 7, 9, 11, 13 is investigated. We find that there exist at least 13 limit cycles with the distribution C¹₉⊃2[C²₃⊃2C²₂] (let C_m^k denote a nest of limit cycles which encloses m singular points, and the symbol `⊂' is used to show the enclosing relations between limit cycles, while the sign `+' is used to divide limit cycles enclosing different critical points. Denote simply C_m^k+C_m^k=2C_m^k, etc.) in the Hamiltonian system under the perturbed term of degree 7, and give the complete bifurcation diagrams and classification of the phase portraits by using bifurcation theory and qualitative method and numerical simulations. These results in this paper are useful for the study of the weaken Hilbert 16th problem.     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.