Darboux integrating factors: Inverse problems

We discuss planar polynomial vector fields with prescribed Darboux integrating factors, in a nondegenerate affine geometric setting. We establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems, and show that the space of vector fields with a given integrating factor, modulo a subspace of explicitly known “standard” vector fields, has finite dimension. For several classes of examples we determine this space explicitly.     

The nondegenerate center problem and the inverse integrating factor

In this paper we study some aspects of the nondegenerate center problem for analytic and, in particular, for polynomial vector fields. The relation between the existence of an inverse integrating factor and the center problem is studied. The relationship between the conditions for a center using the Poincaré formal series and the inverse integrating factor formal series for systems with a linear center perturbed by homogeneous polynomials is proved.     

Darboux theory of integrability in the sparse case

Darboux's theorem and Jouanolou's theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that we can get the same kind of results if we consider the size of a Newton polytope associated to the vector field. Furthermore, we show that in this context the bound is optimal.     

Cyclicity of several planar graphics and ensembles through three singular points without generic conditions

This paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle-node P₀ and two hyperbolic saddles P₁ and P₂ for the non-generic cases of r₁(0)=1, r₂(0)≠1 and r₁(0)≠1, r₂(0)=1, where r₁(0) and r₂(0) are the hyperbolicity ratio of the saddles P₁ and P₂, respectively. For the case of r₁(0)=1, r₂(0)≠1, we suppose that the connection from P₀ to P₂ and the connection from P₀ to P₁ keep unbroken. We prove that these graphics and ensembles are of finite cyclicity respectively. Moreover, the cyclicity is linearly dependent on the order of the neutral saddle P₁ if P₂ is contractive and r₂(0)∈Q. We also show that the nearer r₂(0) is close to 1, the more the limit cycles are bifurcated. For the case of r₁(0)≠1, r₂(0)=1, we obtain that these graphics and ensembles are of finite cyclicity respectively if P₁ is of finite order and the hp-connection from P₀ to P₂ keeps unbroken.     

Sufficient conditions for the existence of at least n or exactly n limit cycles for the Liénard differential systems

In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles.     

Polynomial first integrals of quadratic vector fields

We classify all quadratic polynomial differential systems having a polynomial first integral, and provide explicit normal forms for such systems and for their first integrals.     

Configurations of limit cycles and planar polynomial vector fields

We show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field. Moreover, the realization can be made by algebraic limit cycles, and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles.     

Limit cycles near hyperbolas in quadratic systems

In this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane. We study a strip of hyperbolas occurring in some quadratic systems. We deal with the cyclicity of the degenerate graphics DI2a from the programme, set up in [F. Dumortier, R. Roussarie, C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations 110 (1994) 86-133], to solve the finiteness part of Hilbert's 16th problem for quadratic systems. Techniques from geometric singular perturbation theory are combined with the use of the Bautin ideal. We also rely on the theory of Darboux integrability.     

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré-Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincaré-Liapunov method.     

Bifurcations of traveling wave solutions for Zhiber–Shabat equation

In this paper, we investigate the dynamical behavior of traveling wave solutions in the Zhiber–Shabat equation by using the bifurcation theory and the method of phase portraits analysis. As a result, we obtain the conditions under which smooth and non-smooth traveling wave solutions exist, and give some exact explicit solutions for some special cases.     

Planar quasi-homogeneous polynomial differential systems and their integrability

In this paper we study the quasi-homogeneous polynomial differential systems and provide an algorithm for obtaining all these systems with a given degree. Using this algorithm we obtain all quasi-homogeneous vector fields of degree 2 and 3.     

Centers for a 6-parameter family of polynomial vector fields of arbitrary degree

For all non-negative integers n₁,n₂,n₃,j₁,j₂ and j₃ with nk+jk>1 for k=1,2,3, (nk,jk)≠(nl,jl) if k≠l, j₃=n₃−1 and jk≠nk−1 for k=1,2, we study the center variety of the 6-parameter family of real planar polynomial vector given, in complex notation, by , where z=x+iy and A,B,C∈C\{0}.     

Note on a Uniqueness Theorem of Schiffer

AMS (MOS): 35R30, 76Q05

The uniqueness of the inverse scattering problem is examined. It is shown that the boundary conditions must be suitably restricted to secure uniqueness. For one class the far-field pattern for one incident field is sufficient for uniqueness and, for a wider class, information from a finite number of distinct incident waves suffices. In either case, the far-field also dictates the boundary condition on the scatterer.     

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface Σ=H⁻¹(1). If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.     

Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree

In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R² of degree d that in complex notation z=x+iy can be written as where j is either 0 or 1. If j=0 then d?5 is an odd integer and n is an even integer satisfying 2?n?(d+1)/2. If j=1 then d?3 is an integer and n is an integer with converse parity with d and satisfying 0<n?[(d+1)/3] where [⋅] denotes the integer part function. Furthermore λ∈R and A,B,C,D∈C. Note that if d=3 and j=0, we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) [17], N.I. Vulpe and K.S. Sibirskii (1988) [25], J. Llibre and C. Valls (2009) [15], and if d=2, j=1 and C=0, we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) [2], H. Zoladek (1994) [26]. So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations.     

Cup-length estimate for Lagrangian intersections

In this paper, we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (nontransversal) case.     

The Brake Orbits of Hamiltonian Systems on Positive-type Hypersurfaces

This paper deals with the brake orbits of Hamiltonian system on given energy hypersurfaces Σ = H ⁻¹(1). We introduce a class of contact type but not necessarily star-shaped hypersurfaces in ℝ²ⁿ and call them normalized positive-type hypersurfaces. By using of the critical point theory, we prove that if Σ is a partially symmetric normalized positive-type hypersurface, it must carries a brake orbit of (HS). Furthermore, we obtain some multiplicity results under certain pinching conditions. Our results include the earlier works on this subject given by P. Rabinowitz and A. Szulkin in star-shaped case. An example of partially symmetric normalized positive-type hypersurface in ℝ⁴ that is not star-shaped is also presented Partially supported by NNSF of China (10571085) and Science Foundation of Hohai University.     

Limit cycles in a general two-stroke oscillation

Conditions of the existence of limit cycles in a general two-stroke oscillation system are obtained. The general oscillation is a special kind of Liénard-type system with isoclines approaching both positive infinites or negative infinites as t→±∞$t\to \pm \infty$. The conditions for the uniqueness of the two-stroke oscillation are also proved. Some examples are used to illustrate our results.