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This paper is the second of a two-part series in which we review the properties of
the rotation number for a random family of linear non-autonomous Hamiltonian systems. In
Part I, we defined the rotation number for such a family and discussed its basic properties. Here
we define and study a complex quantity - the Floquet coefficient w - for such a family. The
rotation number is the imaginary part of w.
We derive a basic trace formula satisfied by w,
and give applications to Atkinson-type spectral problems. In particular we use w to discuss
the convergence properties of the Weyl M-functions,
the Kotani theory, and the gap-labelling phenomenon for these problems. 相似文献
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R. Fabbri R. Johnson C. Nú?ez 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2003,40(2):484-502
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Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes. 相似文献
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《Journal of Applied Mathematics and Mechanics》2002,66(3):347-355
A new algorithm is proposed for reducing non-autonomous Hamiltonian systems to normal Birkhoff form. The criterion for the normal form is the condition that the vector fields of the perturbed and unperturbed parts of the system should commute. The invariant character of the criterion enables the system to be normalized in a unified way, without first simplifying the unperturbed part and without distinguishing between resonance and non-resonance, or autonomous and non-autonomous, cases. The whole algorithm reduces to a one-dimensional recurrence formula. The result is obtained by using the Campbell-Hausdorff formula for the ring of asymptotic forms, as well as the solution of a homologicla equation in the form of a quadrature. Three examples are considered to illustrate the various special features of the new algorithm. One of the examples is of interest for nuclear magnetic resonance theory. 相似文献
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The aim of this paper is to develop the Floquet theory for linear implicit difference systems (LIDS). It is proved that any index-1 LIDS can be transformed into its Kronecker normal form. Then the Floquet theorem on the representation of the fundamental matrix of index-1 periodic LIDS has been established. As an immediate consequence, the Lyapunov reduction theorem is proved. Some applications of the obtained results are discussed. 相似文献
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Russell A. Johnson 《Annali di Matematica Pura ed Applicata》1987,147(1):211-248
Sunto Si definisce un esponente di Floquet per certe equazioni differenziali lineari nonperiodiche, la parte immaginaria del quale rappresenta una «rotazione» delle soluzioni di dette equazioni. Inoltre si discute la relazione fra l'esponente di Floquet e le funzioni m di Weyl-Kodaira, e fra la rotazione e certi problemi spettrali.
The author would like to thank Dr. RichardCushman for stimulating conversations on the subjects considered in this paper. 相似文献
The author would like to thank Dr. RichardCushman for stimulating conversations on the subjects considered in this paper. 相似文献
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Nurbek AizmahinTianqing An 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(14):4862-4867
The purpose of this paper is to study the existence of periodic solutions for a class of non-autonomous second-order Hamiltonian systems. Some new existence theorems are obtained by using the least action principle and the saddle point theorem. 相似文献
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The purpose of this paper is to study the existence of periodic solutions for a class of non-autonomous second order Hamiltonian system. Some new existence theorems are obtained by the least action principle. 相似文献
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n independent adiabatic invariants in involution are found for a slowly varying Hamiltonian system of order 2n × 2n. The Hamiltonian system considered is , where A(t) is a 2n × 2n real matrix with distinct, pure imaginary eigen values for each t? [?∞, ∞], and , for all j > 0. The adiabatic invariants Is(u, t), s = 1,…, n are expressed in terms of the eigen vectors of A(t). Approximate solutions for the system to arbitrary order of ? are obtained uniformly for t? [?∞, ∞]. 相似文献
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Chun LiZeng-Qi Ou Chun-Lei Tang 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(4):2262-2272
Some existence theorems are obtained for periodic and subharmonic solutions of a class of non-autonomous Hamiltonian systems. Our technical approach is based on a version of the Local Linking Theorem, and the Generalized Mountain Pass Theorem. 相似文献
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Ziheng Zhang 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):4125-4130
In this paper we consider the existence of homoclinic solutions for the following second-order non-autonomous Hamiltonian system:
(HS) 相似文献
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By using mountain pass theorem and local link theorem, some existence theorems are obtained for periodic solutions of second order non-autonomous Hamiltonian systems under local superquadratic condition and other suitable conditions. 相似文献
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Zhaowen Zheng 《Mathematische Nachrichten》2008,281(11):1664-1671
In this paper some new oscillation criteria of interval type for linear matrix Hamiltonian systems are established which are different from most known ones in the sense that they are based on the information only on a sequence of subinterval of [t0, ∞) rather than on the whole half line. Our results are shaper than previous results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献