共查询到20条相似文献,搜索用时 46 毫秒
1.
Using the averaging theory of first and second order we study the maximum number of limit cycles of generalized Linard differential systems{x = y + εh_l~1(x) + ε~2h_l~2(x),y=-x- ε(f_n~1(x)y~(2p+1) + g_m~1(x)) + ∈~2(f_n~2(x)y~(2p+1) + g_m~2(x)),which bifurcate from the periodic orbits of the linear center x = y,y=-x,where ε is a small parameter.The polynomials h_l~1 and h_l~2 have degree l;f_n~1and f_n~2 have degree n;and g_m~1,g_m~2 have degree m.p ∈ N and[·]denotes the integer part function. 相似文献
2.
Upper Bound of the Number of Zeros for Abelian Integrals in a Kind of Quadratic Reversible Centers of Genus One 下载免费PDF全文
Qiuli Yu Houmei He Yuangen Zhan Xiaochun Hong 《Journal of Nonlinear Modeling and Analysis》2024,6(1):218-227
By using the methods of Picard-Fuchs equation and Riccati equation, we study the upper bound of the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under polynomial perturbations of degree $n$. We obtain that the upper bound is $7[(n-3)/2]+5$ when $n\ge 5$, $8$ when $n=4$, $5$ when $n=3$, $4$ when $n=2$, and $0$ when $n=1$ or $n=0$, which linearly depends on $n$. 相似文献
3.
Gary Nicklason 《Journal of Applied Analysis & Computation》2015,5(3):284-300
We consider the polynomial system $\td{x}{t}=-y-ax^{s+3}y^{n-s-3}-bx^{s+1}y^{n-s-1},$\, $\td{y}{t}=x+cx^{s+2}y^{n-s-2} + dx^sy^{n-s}$ where $n \ge 3$ is an odd integer and $s=0, \dots, n-3$ is an even integer. We calculate the first three nonzero Lyapunov coefficients for the system and obtain a Gr\"obner basis for the ideal generated by them. Potential centre conditions for the system are obtained by setting the basis elements equal to zero and solving the resulting system. This gives five basic solutions and within this set we find two well known classes of centres and three new centre producing systems. One of the three is a variant of one of the other new systems, so we obtain two general independent systems which produce multiple centre conditions for each $n \ge 5.$ 相似文献
4.
设n,a,b,c是正整数,gcd(a,b,c)=1,a,b≥3,且丢番图方程a~x+b~y=c~z只有正整数解(x,y,z)=(1,1,1).证明了若(x,y,z)是丢番图方程(an)~x+(bn)~y=(cn)~z的正整数解且(x,y,z)≠(1,1,1),则yzz或xzy.还证明了当(a,b,c)=(3,5,8),(5,8,13),(8,13,21),(13,21,34)时,丢番图方程(an)~x+(bn)~y=(cn)~z只有正整数解(x,y,z)=(1,1,1). 相似文献
5.
Upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one 下载免费PDF全文
Xiaochun Hong Junliang Lu Yanjie Wang 《Journal of Applied Analysis & Computation》2018,8(6):1959-1970
In this paper, by using the method of Picard-Fuchs equation and Riccati equation, we study the upper bounds for the associated number of zeros of Abelian integrals for two classes of quadratic reversible centers of genus one under any polynomial perturbations of degree $n$, and obtain that their upper bounds are $3n-3$ ($n\geq 2$) and $18\left[\frac{n}{2}\right]+3\left[\frac{n-1}{2}\right]$ ($n\geq 4$) respectively, both of the two upper bounds linearly depend on $n$. 相似文献
6.
A linear estimation to the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one 下载免费PDF全文
Lijun Hong Xiaochun Hong Junliang Lu 《Journal of Applied Analysis & Computation》2020,10(4):1534-1544
In this paper, using the method of Picard-Fuchs equation and Riccati equation, we consider the number of zeros for Abelian integrals in a kind of quadratic reversible centers of genus one under arbitrary polynomial perturbations of degree $n$, and obtain that the upper bound of the number is $2\left[{(n+1)}/{2}\right]+$ $\left[{n}/{2}\right]+2$ ($n\geq 1$), which linearly depends on $n$. 相似文献
7.
On the Number of Limit Cycles in Small Perturbations of a Piecewise
Linear Hamiltonian System with a Heteroclinic Loop 下载免费PDF全文
In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived. 相似文献
8.
Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton systems 下载免费PDF全文
In this paper, we consider the bifurcation of limit cycles for system $\dot{x}=-y(x^2+a^2)^m,~\dot{y}=x(x^2+a^2)^m$ under perturbations of polynomials with degree n, where $a\neq0$, $m\in \mathbb{N}$. By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from periodic orbits of the center of the unperturbed system. Particularly, if $m=2, n=5$, the sharp bound is 5. 相似文献
9.
Amel Boulfoul Amar Makhlouf Nawal Mellahi 《Journal of Applied Analysis & Computation》2019,9(3):864-883
In this paper, we consider the limit cycles of a class of polynomial differential systems of the form $\dot{x}=-y, \hspace{0.2cm} \dot{y}=x-f(x)-g(x)y-h(x)y^{2}-l(x)y^{3},$ where $f(x)=\epsilon f_{1}(x)+\epsilon^{2}f_{2}(x),$ $g(x)=\epsilon g_{1}(x)+\epsilon^{2}g_{2}(x),$ $h(x)=\epsilon h_{1}(x)+\epsilon^{2}h_{2}(x)$ and $l(x)=\epsilon l_{1}(x)+\epsilon^{2}l_{2}(x)$ where $f_{k}(x),$ $g_{k}(x),$ $h_{k}(x)$ and $l_{k}(x)$ have degree $n_{1},$ $n_{2},$ $n_{3}$ and $n_{4},$ respectively for each $k=1,2,$ and $\varepsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=-y,$ $\dot{y}=x$ using the averaging theory of first and second order. 相似文献
10.
Acta Applicandae Mathematicae - This paper study the type of integrability of differential systems with separable variables $\dot{x}=h\left (x\right )f\left (y\right )$ , $\dot{y}= g\left (y\right... 相似文献
11.
Ye Yanqian 《数学年刊B辑(英文版)》1996,17(2):167-174
For the quadratic system: x=-y δx lx2 ny2, y=x(1 ax-y) under conditions -10 the author draws in the (a, ()) parameter plane the global bifurcationdiagram of trajectories around O(0,0). Notice that when na2 l < 0 the system has one saddleN(0,1/n) and three anti-saddles. 相似文献
12.
一类具功能反应的食饵——捕食者系统定性分析 总被引:5,自引:0,他引:5
研究一类具功能反应的食饵-捕食者系统:x=xg(x)-y(?)(x),y=y(-d+e(?)(x).在g(x)=α-bxm,(?)(x)=cxθ及m+θ=1,m=1/n,n>2为正整数情形下,分析了该系统的平衡点性态,并得到了系统在正平衡点外围的极限环的不存在性、存在性与唯一性的相关条件. 相似文献
13.
ZHOU Yi 《数学年刊A辑(中文版)》2003,(3):293-302
This paper considers the following
Cauchy problem for semilinear wave equations in $n$ space
dimensions
$$\align
\square\p &=F(\partial\p ),\\p (0,x)&=f(x),\quad \partial_t\p (0,x)=g(x),
\endalign$$
where $\square =\partial_t^2-\triangle$ is the wave operator, $F$ is
quadratic in $\partial\p$ with
$\partial =(\partial_t,\partial_{x_1},\cdots ,\partial_{x_n})$.
The minimal value of $s$ is determined such that the above
Cauchy problem is locally well-posed in $H^s$. It turns out that
for the general equation $s$ must satisfy
$$s>\max\Big(\frac{n}{2}, \frac{n+5}{4}\Big).$$
This is due to Ponce and Sideris (when $n=3$) and Tataru (when $n\ge
5$). The purpose of this paper is to supplement with a proof in the
case $n=2,4$. 相似文献
14.
ZHOU Yi 《数学年刊B辑(英文版)》2003,24(3):293-302
This paper considers the following Cauchy problem for semilinear wave equations in n space dimensions □φ=F(δφ),φ(0,x)=f(x),δtφ(0,x)=g(x),whte □=δt^2-△ is the wave operator,F is quadratic in δεφ with δ=(δt,δx1,…,δxn).The minimal value of s is determined such that the above Cauchy problem is locally wellposed in H^s.It turns out that for the general equation s must satisfy s>max(n/2,n+5/4).This is due to Ponce and Sideris (when n=3)and Tataru (when n≥5).The purpose of this paper is to supplement with a proof in the case n=2,4. 相似文献
15.
Henryk Zoladek 《Transactions of the American Mathematical Society》1998,350(4):1681-1701
Odani has shown that if then after deleting some trivial cases the polynomial system does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of and . We give also a simple presentation of Yablonsky's example of a quartic limit cycle in a quadratic system.
16.
Huang Yuanshi 《数学年刊B辑(英文版)》1985,6(1):15-26
First the author considers the system (1)$\frac{dx}{dt}=f(t,x,y,\varepsilon),\varepsilon\frac{dy}{dt}=g(t,x,y,\varepsilon)$ and its degenerate system (2)$\frac{dx}{dt}=f(t,x, y, 0), g(f, x, y, 0) =0$. In both noncritical and critical cases, sufficient conditions are established for the existence of almost periodic solutions of system (1) near the given solutions of system (2). The main method of proof is that, by performing suitable transformation, the author establishes exponential dichotomies, and then applies the theory of integral manifolds. Secondly, for the autonomous system (3)
$\frac{dx}{dt}=f(x,y,\varepsilon),\varepsilon\frac{dy}{dt}=g(x,y,\varepsilon)$, analogous results are obtained by performing the generalized normal coordinate transformation. 相似文献
17.
The notion of Hilbert number from polynomial differential systems in the plane of degree $n$ can be extended to the differential equations of the form \[\dfrac{dr}{d\theta}=\dfrac{a(\theta)}{\displaystyle \sum_{j=0}^{n}a_{j}(\theta)r^{j}} \eqno(*)\] defined in the region of the cylinder $(\tt,r)\in \Ss^1\times \R$ where the denominator of $(*)$ does not vanish. Here $a, a_0, a_1, \ldots, a_n$ are analytic $2\pi$--periodic functions, and the Hilbert number $\HHH(n)$ is the supremum of the number of limit cycles that any differential equation $(*)$ on the cylinder of degree $n$ in the variable $r$ can have. We prove that $\HHH(n)= \infty$ for all $n\ge 1$. 相似文献
18.
Xingwu Chen Valery G. Romanovski Weinian Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2008
In 2002 X. Jarque and J. Villadelprat proved that no center in a planar polynomial Hamiltonian system of degree 4 is isochronous and raised a question: Is there a planar polynomial Hamiltonian system of even degree which has an isochronous center? In this paper we give a criterion for non-isochronicity of the center at the origin of planar polynomial Hamiltonian systems. Moreover, the orders of weak centers are determined. Our results answer a weak version of the question, proving that there is no planar polynomial Hamiltonian system with only even degree nonlinearities having an isochronous center at the origin. 相似文献
19.
20.
In this paper we investigate the integrability problem for the two-dimensional Lotka-Volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials, that is, we consider systems of the form $\dot{x}=x(1-a_{30}x^{3}-a_{21} x^{2} y-a_{12}x y^{2} -a_{03}y^{3})$ , $\dot{y}=-y(1-b_{30}x^{3}-b_{21} x^{2} y-b_{12}x y^{2}-b_{03} y^{3})$ . Conditions for the integrability of this system are found. From them the center conditions for corresponding real system can be derived. The study relays on making use of algorithms of computational algebra based on the Groebner basis theory. To simplify laborious manipulations with polynomial modular arithmetics is involved. 相似文献