Narrow Results

In this paper, the periodic solutions of the equation of Friedmann–Robertson–Walker cosmology with a cosmological constant are investigated. Using variable transformation, the original second-order ordinary differential equation is converted to a planar dynamical system with cosmic time t. Numerical simulations indicate that period function T(h) of this dynamical system is monotonically increasing. However, a new planar dynamical system could be deduced by using conformal time variable $\tau$. We prove that the new planar dynamical system has two isochronous centers under certain parameter conditions by using Picard–Fuchs equation. Explicitly, we find that there exist two families of periodic solutions with equal period for the new planar dynamical system which is derived from the Friedmann–Robertson–Walker model.

An upper bound Z(2,n)≤[n/2]+6[(n-1)/2]+5 is derived for the number of zeros of Abelian integrals Ĩ(h)=∮_γ(h)g(x,y)dx-f(x,y)dy on the open interval (0,1/6), where γ(h) is an oval lying on the algebraic curve H_λ=(1/3)x³+(1/2)x²+λy³-y²=h, f(x,y), g(x,y) are polynomials of x and y, and n=max{deg f(x,y), deg g(x,y)}. The proof exploits the expansion of near λ=0.

In this paper, we investigate the number of isolated zeros of the Abelian integrals for a kind of non-Hamiltonian integrable systems with one center and two invariant straight lines and with other orbits formed by quartics. It is proved that the exact upper bound of the number of isolated zeros of the Abelian integrals under cubic polynomial perturbations is equal to two.

In this paper, we provide a complete study of the zeros of Abelian integrals obtained by integrating the 1-form (α + βx + x²)ydx over the compact level curves of the hyperelliptic Hamiltonian . Such a family of compact level curves is bounded by a polycycle passing through a nilpotent cusp and a hyperbolic saddle of this hyperelliptic Hamiltonian system, which is not the exceptional family of ovals proposed by Gavrilov and Iliev. It is shown that the least upper bound for the number of zeros of the related hyperelliptic Abelian integral is two, and this least upper bound can be achieved for some values of parameters (α, β). This implies that the Abelian integral still has Chebyshev property for this nonexceptional family of ovals. Moreover, we derive the asymptotic expansion of Abelian integrals near a polycycle passing through a nilpotent cusp and a hyperbolic saddle in a general case.

This paper is concerned with the bifurcation of limit cycles from the period annulus of a quadratic reversible system. The outer boundary of the period annulus contains a degenerate critical point. The exact upper bound of the number of limit cycles is given. Our result shows that the conjecture on the cyclicity of (r4) system is correct.

In this paper, we consider Liénard systems of the form

In this paper, we consider the planar system ẋ = -yF(x, y) + εP(x, y), ẏ = xF(x, y) + εQ(x, y), where the set {F(x, y) = 0} consists of m nonzero points (a_i, b_i)(i = 1, …, m) with multiple multiplicities, P(x, y) and Q(x, y) are arbitrary real polynomials. We study the number of limit cycles bifurcating from the periodic annulus surrounding the origin by using Abelian integrals and residue integration.

In this study, we determine the associated number of zeros for Abelian integrals in four classes of quadratic reversible centers of genus one. Based on the results of [Li et al., 2002b],, we prove that the upper bounds of the associated number of zeros for Abelian integrals with orbits formed by conics, cubics, quartics, and sextics, under polynomial perturbations of arbitrary degree $n$, depend linearly on $n$.

We report a new result on the traveling wave solutions of a biological invasion model with density-dependent migrations and Allee effect. It has been shown in the literature that such a model can exhibit one periodic wave solution by using Hopf bifurcation theory. In this paper, global bifurcation theory is applied to prove that there exists maximal one periodic solution which can be reached in a large feasible parameter regime. The basic idea used in our technique is to examine the monotonicity of the ratio of related Abelian integrals. Especially, the existence condition for the solution near a homoclinic loop is obtained.

This paper investigates the planar differential systems $ẋ=-yG(x,y)$, $ẏ=xG(x,y)$ under the perturbations of polynomials of $(x,y)$ with degree $n$, where $G(x,y)={(x-a)}^{m_1}{(y-b)}^{m_2}$ with $ab\ne 0$ and $m_1,m_2\in \mathbb{N}$. The upper bound for the number of zeros of Abelian integral from the period annulus around the center is obtained.

In this paper, we consider the number of zeros of Abelian integral for the system $ẋ=-y(a{x}^2+bx+1)+𝜀P_n(x,y)$, $ẏ=x(a{x}^2+bx+1)+𝜀Q_n(x,y)$, where $a\ne 0$, $b^2-4a>0$, and $P_n,Q_n$ are arbitrary polynomials of degree $n$. We obtain that $H(n)=2[\frac{n+1}{2}]-1$ if $b=0$ and $2[\frac{n+1}{2}]+[\frac{n}{2}]\le H(n)\le 3[\frac{n+1}{2}]$ if $b\ne 0$, where $H(n)$ is the maximum number of limit cycles bifurcating from the period annulus up to the first order in $𝜀$. So, the bounds for $b=0$ or $b\ne 0$, $n=2k$, $k\in \mathbb{N}$ are exact.

This paper is concerned with the quadratic perturbations from one parameter family of generic reversible quadratic vector fields having a simple center and an invariant straight line. It is shown that the system can generate at least two limit cycles. As the parameter is rational, we propose a procedure for finding the upper bound to cyclicity of period annulus based on the Chebyshev criterion for Abelian integrals together with one rationalizing transformation. To illustrate our approach, we determine the cyclicity of three representative reversible systems. Our results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev [1998].

In this paper, we study a class of $Z_p$-equivariant planar polynomial differential systems $ż=(a+\mathrm{\text{i}})z+(b+\mathrm{\text{i}})z|z{|}^{2(p-2)}+cz|z{|}^2-\frac{5\mathrm{\text{i}}}{2}{\overline{z}}^{p-1}$. It is shown that for any $4\le p\in \mathbb{N}$ there is a differential system of the above type having at least $2p$ limit cycles. This is proved by estimating the number of zeros of the first-order Melnikov function.

In this paper, we study the maximum number of limit cycles for the unfolding of codimension-3 planar singularities with nilpotent linear parts. In [J. Math. Anal. Appl.499 (2017)], the authors proved that when parameter $a\in (-1,1)\backslash \{0,-2/3\}$ is rational, the corresponding problem could be transformed to solving semi-algebraic systems. At the same time, it is pointed out that when $a=-2/3$, the logarithmic function will appear according to the method, which makes it impossible to solve the problem. In this paper, we use some techniques to avoid the occurrence of logarithmic function, and get the corresponding system to produce at most two limit cycles.

In this paper, we study the bound on the number of isolated zeros of Abelian integrals associated with the seventh-degree hyperbolic Hamiltonian system. When the unperturbed system has a unique period annulus bounded by a heteroclinic loop, a series of results on the upper bound of the number of zeros of the related Abelian integrals are obtained. However, there are still some open problems left about the exact bound. Furthermore, we give some results for the system when the parameters in the Hamiltonian system are fixed to certain values. The main tools involve algebraic symbolic computations such as counting the zeros of semi-algebraic systems.

In this paper, the perturbation of a class of cubic Hamiltonian systems is considered. Using Abelian integral, we prove that there exists a neighborhood of the center where the system has at most two limit cycles for arbitrary polynomial perturbation of degree three or four, and at most four limit cycles for arbitrary polynomial perturbation of degree five or six, respectively, which can be reached.

In this paper, a perturbed quintic BBM equation with weak backward diffusion and dissipation effects is investigated. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree six, we prove the existence of isolated periodic wave solutions with certain wave speed in an open interval. It is also shown that isolated periodic wave solutions persist for any energy parameter $h$ in an open interval under small perturbation. Furthermore, we prove that the wave speed $c(h)$ of periodic wave is strictly monotonically increasing with respect to $h$ by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained. Our analysis is mainly based on Melnikov theory, Chebyshev criteria, and symbolic computation, which may be useful for other problems.

We analyze the dynamics of a class of $Z_4$-equivariant Hamiltonian systems of the form $ż=(a+b|z{|}^2)zi+{\overline{z}}^{3}i$, where $z$ is complex, the time $t$ is real, while $a$ and $b$ are real parameters. The topological phase portraits with at least one center are given. The finite generators of Abelian integral $I(h)={\oint }_{{\mathrm{\Gamma }}_h}g(x,y)dx-f(x,y)dy$ are obtained, where ${\mathrm{\Gamma }}_h$ is a family of closed ovals defined by $H(x,y)=-\frac{a}{2}{y}^2-\frac{a}{2}{x}^2-\frac{1+b}{4}{x}^4-\frac{1+b}{4}{y}^4-\frac{b-3}{2}{x}^2{y}^2=h,$$h\in \mathrm{\Sigma }$, $\mathrm{\Sigma }$ is the open interval on which ${\mathrm{\Gamma }}_h$ is defined, $f(x,y)$, $g(x,y)$ are real polynomials in $x$ and $y$ with degree $n$. We give an estimation of the number of isolated zeros of the corresponding Abelian integral by using its algebraic structure. We show that for the given polynomials $f(x,y)$ and $g(x,y)$ in $x$ and $y$ with degree $n$, the number of the limit cycles of the perturbed $Z_4$-equivariant Hamiltonian system does not exceed $83n-3$ (taking into account the multiplicity).

This paper focuses on the persistence of solitary waves and periodic waves of a singularly perturbed generalized Drinfel’d–Sokolov system. Geometric singular perturbation theory is first employed to reduce the higher-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we are then able to find some sufficient conditions about the wave speed to guarantee the existence of homoclinic orbits and periodic orbits, which indicates the existence of solitary waves and periodic waves. Furthermore, we find the lower and upper bounds of the limit wave speed.

For the quadratic reversible systems of genus one, all of their periodic orbits are higher-order algebraic curves. When they are perturbed by polynomials of degree $n$, the numbers of zeros of their Abelian integrals will change and we study the upper bounds of these numbers by using the methods of Riccati equation and Picard–Fuchs equation. We consider both the highest and lowest degrees of polynomials, and more importantly, we consider the law of polynomials and the range of values for their variables. Consequently, some laws of the polynomials are discovered and many upper bounds are obtained, and these upper bounds are sharper than the results obtained by other techniques.