Uniqueness of limit cycles for polynomial first-order differential equations

We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE in terms of {i_k}, {j_k}, {n_k}. Our main result characterizes, under some additional hypotheses, the exponents {i_k}, {j_k}, {n_k}, such that for any choice of the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form x^′=y+xR(x,y), y^′=−x+yR(x,y), where . Concretely, when the set has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents {i_k}, {j_k} such that the origin of the rigid system is a center for any choice of and also when there are no limit cycles surrounding the origin for any choice of .     

一类E13系统极限环的惟一性

研究一类E13系统x=y,y=-x+δy+nx2+mxy+ly2+bxy2,求出奇点O的焦点量W0=δ,W1=m(n+l),W2=-mnb.证明了W0=W1=W2=0时O为中心.其次证明了W0=0,W1W2≥0时系统无极限环;W0=0,W1W2＜0时系统至多有一个极限环.最后讨论了n=0,b＞0的情况.证明了存在δ0,0＜δ0≤-l/m,当0＜δ＜δ0时系统存在惟一极限环,δ=δ0时系统存在无穷远分界线环,δ≤0或δ＞δ0时系统无闭轨与奇闭轨.     

Ⅲ类二次系统极限环的惟一性(Ⅰ)

证明了一般的Ⅲ类二次系统当参数a很小时极限环的大范围惟一性,对于一般的参数值a,在适当的条件下也证明了极限环的大范围惟一性.文中也给出了极限环随参数d变化时产生和消失的过程.     

Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response

The goal of this paper is to establish the uniqueness of limit cycles of the predator-prey systems with Beddington-DeAngelis functional response. Through a change of variables, the predator-prey system can be transformed into a better studied Gause-type predator-prey system. As a result, the uniqueness of limit cycles can be solved.     

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.     

The predator-prey model with two limit cycles

This paper investigates the predator-prey system:x=k_1(x-ax)-k(x)y,y=(-k_3 βk(x)ywit.k(x)=k_2x, x≤x, k_2x, x>τ,where α, β, τ; k_1, k_2, k_3 are positive constants.The main results are as follows(i) In case k_3-βk_2τ≥0 system (1) has no limit cycle.(ii) In case k_3-βk_2τ<0, k_1 k_3-βk_2τ>0, and for O<α<<1, system (1) at least has two limitcycles.     

Uniqueness of limit cycles in predator-prey system: the role of weight functions

We consider a Gause type model of interactions between predator and prey populations. Using the ideas of Cheng and Liou we give a sufficient condition for uniqueness of the limit cycle, which is more general than their condition. That is, we include a kind of weight function in the condition. It was motivated by a result due to Hwang, where the prey isocline plays a role of weight function. Moreover, we show that the interval where the condition from Hwang's result is to be fulfilled can be narrowed.     

Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system  = (0, 1, 2, . . .) of L2H(D), the space of holomorphic and absolutely square integrable functions in the bounded domain D of Cn, we construct a holomorphic imbedding ι : D →■(n, ∞), the complex infinite dimensional Grassmann manifold of rank n. It is known that in ■(n, ∞) there are n closed (μ, μ)-forms τμ (μ = 1, . . . , n) which are invariant under the holomorphic isometric automorphism of ■(n, ∞) and generate algebraically all the harmonic differential forms of ...     

Foliation without limit cycles

The following theorem is proved for a closed manifold M with an oriented foliated structure of codimension 1 without limit cycles, supplemented by a foliation of one-dimensional normals: if every normal in M intersects every leaf, the same is true of the induced foliation on M (a universal covering of M).Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 181–191, February, 1971     

Bifurcation of limit cycles near equivariant compound cycles

In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.     

On limit classes of bounded perturbations

We obtain sufficient conditions for a given class of piecewise continuous bounded perturbations to be a limit class, i.e., to admit the computation of the least upper bound for the exponents of linear differential systems with perturbations in that class by a formula similar to well-known formulas for the central and exponential exponents.     

On invariant subspaces for polynomially bounded operators

We discuss the invariant subspace problem of polynomially bounded operators on a Banach space and obtain an invariant subspace theorem for polynomially bounded operators. At the same time, we state two open problems, which are relative propositions of this invariant subspace theorem. By means of the two relative propositions (if they are true), together with the result of this paper and the result of C.Ambrozie and V.Müller (2004) one can obtain an important conclusion that every polynomially bounded operator on a Banach space whose spectrum contains the unit circle has a nontrivial invariant closed subspace. This conclusion can generalize remarkably the famous result that every contraction on a Hilbert space whose spectrum contains the unit circle has a nontrivial invariant closed subspace (1988 and 1997).     

Uniqueness and existence for bounded boundary value problems

The existence and uniqueness of solutions for the boundary value problems with general linear point evaluation boundary conditions is established. We assume that f is bounded and that there is uniqueness on a homogeneous problem and on the linear variational problems.     

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.     

Uniqueness of bounded solutions for some abstract differential equations

Summary For some differential equations in Hilbert space we prove uniqueness of bounded solutions in the Stepanoff norm.

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Riassunto Si dimostra l’unicità delle soluzioniS ²-limitate per alcune equazioni differenziali astratte.

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This research is supported by a grant of the N.R.C. of Canada.