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1.
In this work we give a unified proof for the existence, under appropriate gap conditions, of the standard invariant manifolds for a Ck map of a Banach space near a fixed point where k ≥ 1 is an integer.  相似文献   

2.
The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.  相似文献   

3.
In this paper, a result on the persistence of lower dimensional invariant tori in Cd reversible systems is obtained under some conditions. The theorem is proved for any d which is larger than some constants.  相似文献   

4.
Knowing a probability density (ideally, an invariant density) for the trajectories of a dynamical system allows many significant estimates to be made, from the well-known dynamical invariants such as Lyapunov exponents and mutual information to conditional probabilities which are potentially more suitable for prediction than the single number produced by most predictors. Densities on typical attractors have properties, such as singularity with respect to Lebesgue measure, which make standard density estimators less useful than one would hope. In this paper we present a new method of estimating densities which can smooth in a way that tends to preserve fractal structure down to some level, and that also maintains invariance. We demonstrate with applications to real and artificial data.  相似文献   

5.
Invariant tori in nonlinear oscillations   总被引:2,自引:0,他引:2  
The boundedness of all the solutions for semilinear Duffing equationx″ + ω2 x + φ(x) =p(t), ω ∈ ℝ+ℕ is proved, wherep (t) is a smooth 2π-periodic function and the perturbation ⌽(x) is bounded.  相似文献   

6.
讨论一类三维自治系统的闭轨在周期扰动下的分支问题.利用Poincare映射与积分流形定理,得到扰动系统存在次调和解和不变环面的条件,以及次调和解的鞍结点分支.  相似文献   

7.
    
A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation method. The results obtained are extensions of Chicone‘s for the three dimensional non-Hamiltonian systems.  相似文献   

8.
朱德明  白玉真 《数学学报》2002,45(5):959-968
本文给出了关于哈密顿系统低维环面的一个推广的KAM定理,它适用于同时存在法向频率和双曲法向分量的情况.其证明基于尤建功的一个定理的光滑性表述及法向双曲不变流形理论的应用.文中还给出了另外两种情况下的推广.  相似文献   

9.
广义指数二分性与微分方程的不变流形   总被引:4,自引:0,他引:4  
张伟年 《数学进展》1993,22(1):1-45
用 Hadamar和 Bogoliubov的方法,在常微分方程、泛函微分方程以及半线性抛物型方程所能满足的条件下对Banach或Hilbert空间上的非自治抽象微分方程建立了不变流形理论。首先,对相应的线性方程提出了“广义指数二分性”概念并讨论了它和线性方程谱的关系,然后,我们给出了不变流形存在性结论以及强稳定(不稳定)流形、弱稳定(不稳定)流形和弱双曲流形的分类。进而,我们对这些不变流形给出了C~k光滑性、周期性、概周期性和吸斥性的结论。  相似文献   

10.
In this paper we investigate the nearly small twist mappings with intersection property. With a certain non-degenerate condition, we proved that the most of invariant tori of the original small twist mappings will survive afer small perturtations. The persisted invariant tori are close to the unperturbed ones when the perturbation are small. The orbits reduced by those mappings are quasi-periodic in the invariant tori with the frequences closing to the original ones.  相似文献   

11.
Invariant tori for asymptotically linear impact oscillators   总被引:1,自引:0,他引:1  
The existence of invariant tori and quasi-periodic solutions for asymptotically linear impact oscillators is proved by using the successor map and some generalized versions of the Moser's twist theorem.  相似文献   

12.
In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub manifolds.  相似文献   

13.
14.
本文简要介绍了无穷维动力系统研究中的一些重要问题。  相似文献   

15.
In this paper we mainly concern the persistence of invariant tori in generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, system under consideration can be odd dimensional. Under the Riissmann type non-degenerate condition, we proved that the majority of the lower-dimension invariant tori of the integrable systems in generalized Hamiltonian system are persistent under small perturbation. The surviving lower-dimensional tori might be elliptic, hyperbolic, or of mixed type.  相似文献   

16.
This paper analyses the existence of invariant manifolds ofperiodic orbits for a specific piecewise linear three-dimensionalsystem with two zones, whose linear parts share a pair of imaginaryeigenvalues. This degenerate situation is obtained from thelack of controllability. The analysis proceeds by its reductionto a periodic one-dimensional equation for which some resultsof the Ambrosetti–Prodi type are given.  相似文献   

17.
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.  相似文献   

18.
We consider one parameter families of vector fields depending on a parameter ? such that for ?=0 the system becomes a rotation of R2×Rn around {0}×Rn and such that for ?>0 the origin is a hyperbolic singular point of saddle type with, say, attraction in the rotation plane and expansion in the complementary space. We look for a local subcenter invariant manifold extending the stable manifolds to ?=0. Afterwards the analogous case for maps is considered. In contrast with the previous case the arithmetic properties of the angle of rotation play an important role.  相似文献   

19.
20.
Invariant sets of dynamical systems are considered. The localization inside some compact set and isolating neighborhoods are constructed. Bibliography: 6 titles. Illustrations: 3 figures.  相似文献   

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