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.     

On the number of limit cycles of planar quadratic vector fields

Doklady Mathematics -     

On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields

We prove that the number of limit cycles which bifurcate from a two-saddle loop of an analytic planar vector field X ₀ under an arbitrary finite-parameter analytic deformation X _λ , λ ∈ (? ^N , 0), is uniformly bounded with respect to λ.     

On the number of limit cycles which appear by perturbation of Hamiltonian two-saddle cycles of planar vector fields

We find an upper bound to the maximal number of limit cycles, which bifurcate from a hamiltonian two-saddle loop of an analytic vector field, under an analytic deformation.     

Bifurcations of limit cycles in equivariant quintic planar vector fields

In this paper, we obtain 23 limit cycles for a _Z3$Z_3$-equivariant near-Hamiltonian system of degree 5 which is the perturbation of a _Z6$Z_6$-equivariant quintic Hamiltonian system. The configuration of these limit cycles is new and different from the configuration obtained by H.S.Y. Chan, K.W. Chung and J. Li, where the unperturbed system is a _Z3$Z_3$-equivariant quintic Hamiltonian system. Our unperturbed system is different from the unperturbed systems studied by Y. Wu and M. Han. The limit cycles are obtained by Poincaré–Pontryagin theorem and Poincaré–Bendixson theorem.     

Planar polynomial vector fields having first integrals and algebraic limit cycles

With the help of Abel differential equations we obtain a new class of Darboux integrable planar polynomial differential systems, which have degenerate infinity. Moreover such integrable systems may have algebraic limit cycles. Also we present the explicit expressions of these algebraic limit cycles for quintic systems.     

On the number of limit cycles of a monodromic polynomial vector field on the plane

Limit cycles of monodromic polynomial vector fields are studied. These are fields on the real plane with the unique singular point 0 whose components are polynomials and whose phase portraits are diffeomorphic to a linear focus. The number of limit cycles of such a field is estimated from above in terms of the degrees of the polynomials, the maximum of the absolute values of their coefficients, and certain characteristics of the monodromy transformation, which can be called multipliers at zero and infinity. The estimate is based on the growth and zeros theorem for holomorphic functions.     

Bifurcation set and number of limit cycles in Z2-equivariant planar vector fields of degree 5

An Z₂-equivariant polynomial Hamiltonian system of degree 5 with two perturbation terms is considered in this paper. The phase plane (a, b) is divided into 15 different regions which give the bifurcation set of the system. Using the bifurcation theory of planar dynamical system and the method of detection function, we obtain the bifurcation set and the configurations of compound eyes of the system with 21 or 23 limit cycles.     

Upper bounds for Euclidean minima of algebraic number fields

The aim of this paper is to give new upper bounds for Euclidean minima of algebraic number fields. In particular, to show that Minkowski's conjecture holds for the maximal totally real subfields of cyclotomic fields of prime power conductor.     

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.     

Lower bounds for the number of limit cycles of trigonometric Abel equations

We consider the Abel equation , where A(t) and B(t) are trigonometric polynomials of degree n and m, respectively, and we give lower bounds for its number of isolated periodic orbits for some values of n and m. These lower bounds are obtained by two different methods: the study of the perturbations of some Abel equations having a continuum of periodic orbits and the Hopf-type bifurcation of periodic orbits from the solution x=0.     

On the limit cycles of a quintic planar vector field

This paper concerns the number and distributions of limit cycles in a Z_2-equivariant quintic planar vector field.25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation.It can be concluded that H(5)≥25=5~2, where H(5)is the Hilbert number for quintic polynomial systems.The results obtained are useful to study the weakened 16th Hilbert problem.     

Criteria on the existence of limit cycles in planar polynomial differential systems

We summarize known criteria for the non-existence, existence and on the number of limit cycles of autonomous real planar polynomial differential systems, and also provide new results. We give examples of systems which realize the maximum number of limit cycles provided by each criterion. In particular we consider the class of differential systems of the form $\stackrel{?}{x}=P_n(x,y)+P_m(x,y),\stackrel{?}{y}=Q_n(x,y)+Q_m(x,y),$ where $n,m$ are natural numbers with $m>n\ge 1$ and $(P_i,Q_i)$ for $i=n,m$, are quasi-homogeneous vector fields.     

Some criteria for the existence of limit cycles for quadratic vector fields

In this paper we consider real quadratic systems. We present new criteria for the existence and uniqueness of limit cycles for such systems by using Darbouxian particular solutions. Some results are based on the study of such systems in . We also generalize the well-know result of Bautin on the nonexistence of limit cycles for quadratic Lotka-Volterra systems.     

A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities

This paper is devoted to study the planar polynomial system:

$\dot{x}=ax?y+P_n(x,y),\dot{y}=x+ay+Q_n(x,y),$

where $a\in \mathbb{R}$ and $P_n,Q_n$ are homogeneous polynomials of degree $n\ge 2$. Denote $\psi (\theta )=\cos ?(\theta )?Q_n(\cos ?(\theta ),\sin ?(\theta ))?\sin ?(\theta )?P_n(\cos ?(\theta ),\sin ?(\theta ))$. We prove that the system has at most 1 limit cycle surrounding the origin provided $(n?1)a\psi (\theta )+\dot{\psi }(\theta )\ne 0$. Furthermore, this upper bound is sharp. This is maybe the first uniqueness criterion, which only depends on a (linear) condition of ψ, for the limit cycles of this kind of systems. We show by examples that in many cases, the criterion is applicable while the classical ones are invalid. The tool that we mainly use is a new estimate for the number of limit cycles of Abel equation with coefficients of indefinite signs. Employing this tool, we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin.     

Integral invariants and limit sets of planar vector fields

We show that if a planar system of differential equations admits an inverse integrating factor V defined in a neighborhood of a singular point with exactly one zero eigenvalue then V vanishes along any separatrix of the singular point. Additionally we prove that if K is a compact α- or ω-limit set that contains a regular point (or has an elliptic or parabolic sector if not), and if V is defined on a neighborhood of K, then V vanishes at at least one point of K (and on all of K if V is real analytic or Morse).