共查询到20条相似文献,搜索用时 570 毫秒
1.
V. N. Laptinskii 《Differential Equations》2008,44(2):213-218
We obtain constructive conditions for the unique solvability of the singular problem dx/dt = f(t, x), x ∞ = 0, where f ∈ C (0,1)([0, ∞) × ? n , ? n ). 相似文献
2.
Isochronicity Problem of Higher-Order Singular Point for Polynomial Differential Systems 总被引:1,自引:0,他引:1
Due to the difficulty, the isochronicity problems with respect to higher-order singular point (or degenerate singular point)
of polynomial differential systems are far from being solved. The calculation of period constants is an effective way to find
necessary conditions for isochronicity. In this paper, by means of a homeomorphic transformation, higher-order singular point
is transferred into the origin. At the same time, a new recursive algorithm to compute period constants at the origin of the
transformed system is deduced which is easy to realize with the computer algebraic system such as MATHEMATICA or MAPLE. Finally,
to illustrate the effectiveness of our algorithm, the pseudo-isochronous center conditions of higher-order singular point
for a class of septic system are investigated. Our work is new in terms of research about the isochronicity problem of higher-order
singular point and consists of the existing results related to the origin as a special case when it is an elementary singular
point. 相似文献
3.
A. Raouf Chouikha 《Journal of Mathematical Analysis and Applications》2007,331(1):358-376
In this work we study Eq. (E) with a center at 0 and investigate conditions of its isochronicity. When f and g are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of (E) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems (LD,F). We also classify a 5-parameters family of reversible cubic systems with isochronous centers. 相似文献
4.
This paper deals with the relation between isochronicity and first integral for a class of reversible systems: , , which associates to the first integral of the form H(x,y)=F(x)y2+G(x). Two necessary and sufficient conditions are given to characterize isochronicity for these systems. Moreover, we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n. 相似文献
5.
Wing-Sum Cheung Patricia J.Y. Wong 《Journal of Mathematical Analysis and Applications》2007,329(2):851-869
We consider a system of focal boundary value problems where the nonlinearities may be singular in the independent variable and may also be singular in the dependent arguments. Using Schauder fixed point theorem, we establish criteria such that the system of boundary value problems has at least one fixed-sign solution. 相似文献
6.
Our purpose in this paper is to delineate precisely the extent to which one can make explicit calculations involving the most basic linear feedback systems. Our results center around the Galois theory of the “root-locus” equation p(s) + kq(s) = 0 and the Lie symmetries associated with the related differential equation p(D)x + k(t)q(D)x = 0, D = d/dt. We show that the Galois theory leads to a more refined classification, but that these theories are related in a substantial way. Considerable insight into this is obtained through the study of the monodromy group associated with algebraic curve defined by p(s) + kq(s) = 0. 相似文献
7.
Olof J. Staffans 《Journal of Differential Equations》1983,50(2):183-217
The linear autonomous, neutral system of functional differential equations , (1) x(t) = ?(t) (t ? 0), in a fading memory space is studied. Here μ and ν are matrix-valued measures supported on [0, ∞), finite with respect to a weight function, and are Cn-valued, continuous or locaily integrable functions, bounded with respect to a fading memory norm. Conditions which imply that solutions of (1) can be decomposed into a stable part and an unstable part are given. These conditions are of frequency domain type. The usual assumption that the singular part of μ vanishes is not needed. The results can be used to decompose the semigroup generated by (1) into a stable part and an unstable part. 相似文献
8.
D. I. Martynyuk V. Ya. Danilov A. N. Stanzhitskii 《Ukrainian Mathematical Journal》1997,49(2):247-252
We prove a theorem on the existence of periodic solutions of a system of differential equations with random right-hand sides and small parameter of the form dx/dt=εX(t, x, ξ(t)) in a neighborhood of the equilibrium state of the averaged deterministic system dx/dt=εX 0(t). 相似文献
9.
M. Sabatini 《Journal of Differential Equations》2004,196(1):151-168
We study the period function T of a center O of the title's equation. A sufficient condition for the monotonicity of T, or for the isochronicity of O, is given. Such a condition is also necessary, when f and g are odd and analytic. In this case a characterization of isochronous centers is given. Some classes of plane systems equivalent to such equation are considered, including some Kukles’ systems. 相似文献
10.
Qin-long Wang Yi-rong Liu 《应用数学学报(英文版)》2007,23(3):451-466
In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given. 相似文献
11.
Floyd B Hanson 《Journal of Mathematical Analysis and Applications》1983,96(2):405-419
The singular functional differential equation x(1 ? x)A(x)y′(x) + by(h(x)) ? by(x) = ?bg(x), x in (0, 1), is studied for initial data y = 0 on x ? a, y continuous on (a, 1) and y(1?) bounded. The singularity at x = 0+ is removable for a certain class of delayed arguments, h(x). The final end point at x = 1? is the most important singularity because it results in a genuine singular boundary value problem. A formal solution is constructed and is shown to be unique and bounded when g(x) is bounded. A singular decomposition transforms the problem into two nonsingular initial value problems. Singular FDEs of this type arise in the study of the persistence of populations undergoing large random fluctuations when modeled by compound Poisson processes superimposed on logistic-type growth. 相似文献
12.
Qingliu Yao 《Applications of Mathematics》2013,58(1):93-110
We consider the classical nonlinear fourth-order two-point boundary value problem . In this problem, the nonlinear term h(t)f(t, u(t), u′(t), u″(t)) contains the first and second derivatives of the unknown function, and the function h(t)f(t, x, y, z) may be singular at t = 0, t = 1 and at x = 0, y = 0, z = 0. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals. 相似文献
13.
We obtain criteria for the strong isochronicity of a center with the indication of the maximum order of strong isochronicity. 相似文献
14.
In this paper, we consider the second order Hamiltonian system $\left\{ \begin{gathered} u''(t) + A(t)u(t) + \nabla H(t,u(t)) = 0,t \in R, \hfill \\ u(0) = u(T),u'(0) = u'(T),T > 0. \hfill \\ \end{gathered} \right.$ Here, we assume 0 lies in a gap of σ(B) (the spectrum of B:= ?d 2/dt 2 ?A(t)). We find nontrivial and ground state T-periodic solutions for the second order Hamiltonian system under conditions weaker than those previously assumed; also, our proof is much more direct. 相似文献
15.
L. E. Payne G. A. Philippin S. Vernier Piro 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,61(6):999-1007
We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived. 相似文献
16.
We consider a linear system of PDEs of the form 1 $$\begin{aligned} & \begin{aligned} u_{tt} - c\Delta u_t - \Delta u &= 0 \quad\text{in } \varOmega\times (0,T)\\ u_{tt} + \partial_n (u+cu_t) - \Delta_\varGamma(c \alpha u_t + u)& = 0 \quad\text{on } \varGamma_1 \times(0,T)\\ u &= 0 \quad\text{on } \varGamma_0 \times(0,T) \end{aligned} \\ &\quad (u(0),u_t(0),u|_{\varGamma_1}(0),u_t|_{\varGamma_1}(0)) \in {\mathcal{H}} \end{aligned}$$ on a bounded domain Ω with boundary Γ=Γ 1∪Γ 0. We show that the system generates a strongly continuous semigroup T(t) which is analytic for α>0 and of Gevrey class for α=0. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form ∥(d/dt)T(t)∥?|t|?1 for α>0, ∥(d/dt)T(t)∥?|t|?2 for α=0. Moreover, when α=0 we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to 1/2 power of the generator. 相似文献
17.
We consider the problem of boundary control by displacement at one boundary point x = 0 for a process described by the Klein-Gordon-Fock equation with a variable coefficient on a finite interval 0 ≤ x ≤ l with the Dirichlet condition u(l, t) = 0 at the other boundary point. For the critical time interval T = 2l, we show that there exists a unique boundary function u(0, t) = µ(t) bringing the system from an arbitrary initial state into an arbitrary terminal state. 相似文献
18.
研究一类五次系统无穷远点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换将系统无穷远点转化成原点,然后求出该原点的前8个奇点量,从而导出无穷远点成为中心和最高阶细焦点的条件,在此基础上给出了五次多项式系统在无穷远点分支出8个极限环的实例.同时通过一种最新算法求出无穷远点为中心时的周期常数,得到了拟等时中心的必要条件,并利用一些有效途径一一证明了条件的充分性. 相似文献
19.
We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that an orbifoldO(α/β) has a uniqueD 2-covering by an orbifold? n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only if? n(α/β) is hyperbolic. As the volumes of the orbifolds? n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels). 相似文献