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1.
The authors study the bifurcation problems of rough heteroclinic loop connecting three saddle points for the case β1 > 1, β2 > 1, β3 < 1 and β1β2β3 < 1. The existence, number, coexistence and incoexistence of 2-point-loop, 1-homoclinic orbit and 1-periodic orbit are studied. Meanwhile, the bifurcation surfaces and existence regions are given.  相似文献   

2.
The homoclinic bifurcations under resonant conditions are considered in the ho- moclinic manifold consisting of a series of homoclinic orbits for the fourth-dimensional system.The existence,coexistence and uniqueness of 1-homoclinic orbit,1-periodic orbit and 2-fold 1-periodic orbit are obtained under resonant condition,the correspon- ding bifurcation surfaces and existing regions are also given.  相似文献   

3.
The paper studies a codimension-4 resonant homoclinic bifurcation with one orbit flip and two inclination flips, where the resonance takes place in the tangent direction of homoclinic orbit.Local active coordinate system is introduced to construct the Poincar′e returning map, and also the associated successor functions. We prove the existence of the saddle-node bifurcation, the perioddoubling bifurcation and the homoclinic-doubling bifurcation, and also locate the corresponding 1-periodic orbit, 1-homoclinic orbit, double periodic orbits and some 2n-homoclinic orbits.  相似文献   

4.
Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1 -periodic orbit and three-fold 1 -periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.  相似文献   

5.
Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.  相似文献   

6.
In this work, bifurcation analysis near double homoclinic loops with Ws inclination ?ip of Γ1 and nonresonant eigenvalues is presented in a four-dimensional system. We establish a Poincar´e map by constructing local active coordinates approach in some tubular neighborhood of unperturbed double homoclinic loops. Through studying the bifurcation equations, we obtain the condition that the original double homoclinic loops are persistent, and get the existence or the nonexistence regions of the large 1-homoclinic orbit and the large 1-periodic orbit. At last, an analytical example is given to illustrate our main results.  相似文献   

7.
The homoclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near the homoclinic orbit. This homoclinic orbit is non-principal in the meanings that its positive semi-orbit takes orbit flip and its unstable foliation takes inclination flip. The existence, nonexistence, uniqueness and coexistence of the 1-homoclinic orbit and the 1-periodic orbit are studied. The existence of the twofold periodic orbit and three-fold periodic orbit are also obtained.  相似文献   

8.
《分析论及其应用》2015,(3):307-320
In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].  相似文献   

9.
陈玉明  黄立宏 《东北数学》2003,19(3):213-223
Under some minor technical hypotheses, for each T larger than a certain rS > 0, Krisztin, Walther and Wu showed the existence of a periodic orbit for the positive feedback delay differential equation x(t) = -rμx(t) + rf(x(t-1)), where r and μ are positive constants and f : R → R satisfies f(0) = 0 and f' > 0. Combining this with a unique result of Krisztin and Walther, we know that this periodic orbit is the one branched out from 0 through Hopf bifurcation. Using the normal form theory for delay differential equations, we show the same result under the condition that f ∈ C3(R,R) is such that f'(0) = 0 and f'(0) < 0, which is weaker than those of Krisztin and Walther.  相似文献   

10.
Consider a hyperbolic system of conservation laws (1) where is a smooth nonlinear mapping. ▽f has two real and distinct eigenvalues λ_1 and λ_2(λ_1<λ_2), one of which is genuinely nonlinear, i. e. (2) the other one of which is linearly degenerate in sense of Lax, i, e,  相似文献   

11.
Let {Y i ;−∞<i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically dominated by a random variable X. Let {a i ;−∞<i<∞} be an absolutely summable sequence of real numbers and set V i =∑ k=−∞ a i+k Y i ,i≥1. In this paper, we derive that if and E|X| μ log  ρ |X|<0, for some μ (0<μ<2, μ≠1) and ρ>0 then for all ε>0. This work was partially supported by the Korean Research Foundation Grant funded by the Korean Government (KRF-2006-353-C00006, KRF-2006-251-C00026).  相似文献   

12.
Bifurcations of Rough Heteroclinic Loops with Three Saddle Points   总被引:5,自引:0,他引:5  
In this paper, we study the bifurcation problems of rough heteroclinic loops connecting three saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition, the existence, uniqueness, and incoexistence of the 1-heteroclinic loop with three or two saddle points, 1-homoclinic orbit and 1-periodic orbit near Γ are obtained. Meanwhile, the bifurcation surfaces and existence regions are also given. Moreover, the above bifurcation results are extended to the case for heteroclinic loop with l saddle points. Received January 4, 2001, Accepted July 3, 2001.  相似文献   

13.
Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X i) i 0/∞ over ℂ. Assume that theX i's are chosen from a finite set {D 0,D 1...,D t-1(ℂ), withP(X i=Dj)>0, and that the monoid generated byD 0, D1,…, Dq−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN ≠0(F(x)n,r) of nonzero coefficients inF(x) n.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN ≠0(F(x)n,r) ≈n α for “almost” everyn. Supported in part by FWF Project P16004-N05  相似文献   

14.
We obtain a generalization of the complete Perron effect whereby the characteristic exponents of all solutions change their sign from negative for the linear approximation system to positive for a nonlinear system with perturbations of higher-order smallness [Differ. Uravn., 2010, vol. 46, no. 10, pp. 1388–1402]. Namely, for arbitrary parameters λ 1λ 2 < 0 and m > 1 and for arbitrary intervals [b i , d i ) ⊂ [λ i ,+∞), i = 1, 2, with boundaries d 1b 2, we prove the existence of (i) a two-dimensional linear differential system with bounded coefficient matrix A(t) infinitely differentiable on the half-line t ≥ 1 and with characteristic exponents λ 1(A) = λ 1λ 2(A) = λ 2 < 0; (ii) a perturbation f(t, y) of smallness order m > 1 infinitely differentiable with respect to time t > 1 and continuously differentiable with respect to y 1 and y 2, y = (y 1, y 2) ∈ R 2 such that all nontrivial solutions y(t, c), cR 2, of the nonlinear system .y = A(t)y + f(t, y), yR 2, t ≥ 1, are infinitely extendible to the right and have characteristic exponents λ[y] ∈ [b 1, d 1) for c 2 = 0 and λ[y] ∈ [b 2, d 2) for c 2 ≠ 0.  相似文献   

15.
LetK be an algebraically closed field of characteristic zero. ForAK[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ i=1 n(λ) A iλ k μ whereA iλ are distinct primes. Let ϱλ(A) =n(λ) − 1 and let ρ(A) = Σλɛσ(A)ϱλ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.  相似文献   

16.
§ 1  HypothesesConsider the following system:z.=f(z) , (1 .1 )and its perturbed systemz.=f(z) +g(z,μ) (1 .2 )where z∈ Rm+n,μ∈ Rk,k≥ 3,0≤ |μ| 1 ,f,g∈ Cr,r≥ 4 ,g(z,0 ) =0 .For simplicity,we sup-pose thatf(p) =0 ,g(p,μ) =0 .Moreover,for(1 .1 ) we assume(H1 ) The stable manifold Wspand the unstable manifold Wupof z=p are m-dimension-al and n-dimensional,respectively.The linearization Df(p) atthe equilibrium z=p has realmultiple-2 eigenvaluesλ1 and -ρ1 ,such thatany remaining eige…  相似文献   

17.
18.
M. Sánchez  M. I. Sobrón 《TOP》1997,5(2):307-311
The easiest thecnique to reduce the classical multiple criteria decision problem into a reasonable single criterion decision problem is the weighting method. Po-Lung Yu (1985) gives a well known necessary condition fory 0 to be a Pareto optimal, namelyy 0 maximizes λty overY, for some λ ∈ p, such that λi≥0 for alli and some λj>0. In this brief note we generalize the necessary condition of Po-Lung Yu.  相似文献   

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