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1.
In this paper, the conditions under which there exits a uniformly hyperbolic invariant set for the map fa(x) = ag(x) are studied, where a is a real parameter, and g(x) is a monic real-coefficient polynomial. It is shown that for certain parameter regions, the map has a uniformly hyperbolic invariant set on which it is topologically conjugate to the one-sided subshift of finite type for A, where ∣a∣ is sufficiently large, A is an eventually positive transition matrix, and g has at least two different real zeros or only one real zero. Further, it is proved that there exists an invariant set on which the map is topologically semiconjugate to the one-sided subshift of finite type for a particular irreducible transition matrix under certain conditions, and one type of these maps is not hyperbolic on the invariant set.  相似文献   

2.
We use a topological technique based on covering relations to prove the existence of chaotic orbits for certain Hamiltonian systems with several degrees of freedom. This paper relies on an earlier work of Zgliczyński and Gidea.  相似文献   

3.
4.
In this paper, we present a new technique of proving the existence of an infinite number of symmetric periodic solutions. This technique combines the covering relations method introduced by Zgliczyński (J. Differential Equations 171 (2001) 173; Methods in Nonlinear Anal. 8 (1996) 169; Nonlinearity 10 (1997) 243) with fixed set iteration method described in Lamb (Reversing symmetries in dynamical systems, Ph.D. Thesis, Universiteit van Amsterdam, 1994). As an example we present a computer-assisted proof of symbolic dynamics in the Kuramoto-Sivashinsky equations. Moreover, we prove the existence of an infinite number of symmetric and an infinite number of nonsymmetric periodic solutions.  相似文献   

5.
We study affine operators on a unitary or Euclidean space U up to topological conjugacy. An affine operator is a map f:UU of the form f(x)=Ax+b, in which A:UU is a linear operator and bU. Two affine operators f and g are said to be topologically conjugate if g=h-1fh for some homeomorphism h:UU.If an affine operator f(x)=Ax+b has a fixed point, then f is topologically conjugate to its linear part A. The problem of classifying linear operators up to topological conjugacy was studied by Kuiper and Robbin [Topological classification of linear endomorphisms, Invent. Math. 19 (2) (1973) 83-106] and other authors.Let f:UU be an affine operator without fixed point. We prove that f is topologically conjugate to an affine operator g:UU such that U is an orthogonal direct sum of g-invariant subspaces V and W,
the restriction gV of g to V is an affine operator that in some orthonormal basis of V has the form
(x1,x2,…,xn)?(x1+1,x2,…,xn-1,εxn)  相似文献   

6.
It is well known that the congruence lattice ConA of an algebra A is uniquely determined by the unary polynomial operations of A (see e.g. [K. Denecke, S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, CRC Press, Boca Raton, London, New York, Washington DC, 2002 [2]]). Let A be a finite algebra with |A|=n. If Imf=A or |Imf|=1 for every unary polynomial operation f of A, then A is called a permutation algebra. Permutation algebras play an important role in tame congruence theory [D. Hobby, R. McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, Providence, Rhode Island, 1988 [3]]. If f:AA is not a permutation then AImf and there is a least natural number λ(f) with Imfλ(f)=Imfλ(f)+1. We consider unary operations with λ(f)=n-1 for n?2 and λ(f)=n-2 for n?3 and look for equivalence relations on A which are invariant with respect to such unary operations. As application we show that every finite group which has a unary polynomial operation with one of these properties is simple or has only normal subgroups of index 2.  相似文献   

7.
Using rigorous numerical methods, we validate a part of the bifurcation diagram for a Poincaré map of the Rössler system (Rössler in Phys. Lett. A 57(5):397–398, 1976)—the existence of two period-doubling bifurcations and the existence of a branch of period two points connecting them. Our approach is based on the Lyapunov–Schmidt reduction and uses the C r -Lohner algorithm (Wilczak and Zgliczyński, available at http://www.ii.uj.edu.pl/~wilczak) to obtain rigorous bounds for the Rössler system.  相似文献   

8.
We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is constructed from certain basis vectors of an invariant subspace of A, such as eigenvectors, Jordan vectors, or Schur vectors. We show that most of the eigenvalues of the low-rank perturbed matrix stayed unchanged from the eigenvalues of A; the perturbation can only change the eigenvalues of A that are related to the invariant subspace. Existing results mostly studied using eigenvectors with full column rank for perturbations, we generalize the results to more general settings. Applications of our results to a few interesting problems including the Google’s second eigenvalue problem are presented.  相似文献   

9.
We consider elliptic operators A on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of A through an observation, with an exponential cost. Following the strategy of Lebeau and Robbiano (1995) [25], we deduce the construction of a control for the non-selfadjoint parabolic problem tu+Au=Bg. In particular, the L2 norm of the control that achieves the extinction of the lower modes of A is estimated. Examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of A.  相似文献   

10.
In [4], following [2], we have defined an invariant Δ(f)∈Q, for any f∈Mg,1, the mapping class group of a compact, connected, oriented surface with connected boundary, genus g. For f∈Tg,1 (a certain subgroup of the Torelli group), we have shown in [4], using Casson surgery formula, that Δ(f) co??ncides with the Casson invariant [1] of the homology sphere Mf, obtained by gluing two handlebodies along f. The purpose of this Note is to prove directly (i.e., without reference to Casson) that Δ(f), for f∈Tg,1, depends only on Mf. The surgery formula, which is a difficult point in Casson version, follows almost immediatly from the definition of Δ(f). To cite this article: B. Perron, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 199–204.  相似文献   

11.
We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L1-setting.  相似文献   

12.
Let G=〈f〉 be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g≥2. Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some xS, x is fixed by fd but not by any smaller power of f. For an arbitrary subset A of proper divisors of N, there is always an associated action and, if gA denotes the minimal genus for such an action, an algorithm is obtained here to determine gA. Furthermore, a set Amax is determined for which gA is maximal.  相似文献   

13.
We consider nonautonomous equations v=A(t)v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t). This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ(t)≠t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v=A(t)v+f(t,v). We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations.  相似文献   

14.
Let , where I is a proper closed subinterval of R. Our main goal in this paper is to describe all f,gFI such that the action of the semigroup generated by f and g is topologically transitive on I. We also prove that this action is never weakly topologically mixing. Finally, we describe all pairs of affine functions on R whose generated semigroup action is weakly topologically mixing.  相似文献   

15.
Recently, Fang and Ming [Y.L. Fang, Q.H. Ming, Embedded pair of extended Runge-Kutta-Nyström type methods for perturbed oscillators, Appl. Math. Modelling 34 (2010) 2665-2675] constructed an embedded pair of extended Runge-Kutta-Nyström type methods for perturbed oscillators based on the order conditions of extended Runge-Kutta-Nyström type methods proposed by Yang et al. [H.L. Yang, X.Y. Wu, X. You, Y.L. Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Commun. 180 (2009) 1777-1794]. The authors applied their embedded pair to one-dimensional and two-dimensional problems in numerical experiments. However, the extended Runge-Kutta-Nyström type methods by Yang et al. are designed for one-dimensional perturbed oscillators or systems of perturbed oscillators with a diagonal and positive semi-definite matrix M and a function f(y). For multidimensional perturbed oscillators y″ + My = f(y) with M ∈ Rm×m, a symmetric positive semi-definite matrix, the order conditions of the extended RKN-type methods must be reanalyzed. In this paper, the order conditions for the multidimensional perturbed oscillators are stated and accordingly Fang et al.’s ERKN method of order five for systems of perturbed oscillators is reconsidered. The numerical experiments of the fifth order ERKN method for multidimensional perturbed oscillators are accompanied in comparison with some existing well-known methods in the scientific literature.  相似文献   

16.
In locale theory, weak triquotient assignments on a map f:X?Y can be represented as the points of the double power locale of f relative to the topos of sheaves over Y. A categorical proof of this representation theorem is given based on a categorical account of the Sierpiński locale.  相似文献   

17.
We study the quasisymmetric geometry of the Julia sets of McMullen maps fλ(z) = zm + λ/z?, where λ ∈ ? {0} and ? and m are positive integers satisfying 1/?+1/m < 1. If the free critical points of fλ are escaped to the infinity, we prove that the Julia set Jλ of fλ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet (which is also standard in some sense). If the free critical points are not escaped, we give a suffcient condition on λ such that Jλ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.  相似文献   

18.
Let G be a σ-compact and locally compact group. If f?L(G) let Uf be the closed subspace of L(G) generated by the left translations of f. Conditions are given which ensure that each function in Uf may be expanded in an essentially unique way as an absolutely convergent series of translations of f. In this case Uf contains subspaces which are isometrically isomorphic to l1. If G is metrizable and nondiscrete there is a continuum Γ in L(G) such that, for each f?Γ, Uf contains no non-zero continuous function, and for f, g?Γ with fg, UfUg = {0}. If G is non-compact, metrizable, and non-discrete there is a continuum Γ of bounded continuous functions on G such that, for each f?Γ, Uf contains no non-zero left uniformly continuous function, and for f, g?Γ with fg, UfUg = {0}. The subspaces Uf above are translation invariant but are not convolution invariant.  相似文献   

19.
20.
Given a set A and a function A: AA, we study the set of all functions g: AA that are continuous for all topologies for which f continuous. We prove that in a sense to be made precise in the text, for any essentially infinitary function f, any non-constant such g equals f n , for some n∈ ?. We also prove a similar result for the clone of n-ary functions from A n A.  相似文献   

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