Galoisian obstruction to the integrability of general dynamical systems

The Galoisian approach to study the integrability of classical Hamiltonian systems, the so-called Morales–Ramis theory, has been proved to be useful and powerful by many applications. Here, two analogous forms of the Morales–Ramis theory for general dynamical systems both in vector field and mapping forms are given. Galois groups of the corresponding variational equations are studied, and some necessary conditions of the system to possess a certain number of integrals are presented. Several applications are given at last to illustrate our results.     

Analytic methods for obstruction to integrability in discrete dynamical systems

A unique analytic continuation result is proven for solutions of a relatively general class of difference equations by using techniques of generalized Borel summability. This continuation allows for Painlevé property methods to be extended to difference equations. It is shown that the Painlevé property (PP) induces, under relatively general assumptions, a dichotomy within first‐order difference equations: all equations with PP can be solved in closed form; on the contrary, absence of PP implies, under some further assumptions, that the local conserved quantities are strictly local in the sense that they develop singularity barriers on the boundary of some compact set. The technique produces analytic formulas to describe fractal sets originating in polynomial iterations. © 2004 Wiley Periodicals, Inc.     

On integrability in elementary functions of certain classes of nonconservative dynamical systems

The results of the presented work are due to the study of the applied problem of the rigid body motion in a resisting medium; see [210, 211], where complete lists of transcendental first integrals expressed through a finite combination of elementary functions were obtained. This circumstance allowed the author to perform a complete analysis of all phase trajectories and highlight those properties of them which exhibit the roughness and preserve for systems of a more general form. The complete integrability of those systems is related to symmetries of a latent type. Therefore, it is of interest to study sufficiently wide classes of dynamical systems having analogous latent symmetries. As is known, the concept of integrability is sufficiently broad and undeterminate in general. In its construction, it is necessary to take into account in what sense it is understood (it is meant that a certain criterion according to which one makes a conclusion that the structure of trajectories of the dynamical system considered is especially “attractive and simple”), in which function classes the first integrals are sought for, etc. (see also [1, 4, 14, 17, 20–22, 35, 40–42, 47, 83–85, 117, 120]).     

On numerical integration by the shift and application to Wiener space

Since the advantages of quasi-Monte Carlo methods vanish when the dimension of the basic space increases, the question arises whether there are better methods than the classical Monte Carlo in large or infinite-dimensional basic spaces. We study here the use of the shift operator with the pointwise ergodic theorem whose implementation is particularly interesting. After recalling the theoretical results on the speed of convergence in a form useful for applications, we give sufficient criteria for the law of iterated logarithm in several cases and, in particular, in situations involving the Wiener space.     

On integrability in the LIL for degenerateU-statistics

It is shown by example that square integrability of the kennelh of a completely degenerateU-statistic is not a necessary condition for the law of the iterated logarithm to hold. Necessary integrability conditions are given onh which are in a sense the strongest possible. Sufficient conditions for a degenerateU-statistic of order 2 to satisfy the LIL, weaker than square integrability of the kernel, are also given.Research partially supported by NSF Grant No. DMS-93-00725. Part of this author's work was done at the University of Bielefeld, supported by the Deutsche Forschungsgemeinshaft, Sonderforschungbereich 343, Diskrete Structuren in der Mathematik.Research partially supported by the Army Research Office and the National Security Agency.     

Global periodicity and complete integrability of discrete dynamical systems

Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F ^p (x)=x for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.     

On properties of infimal topology of a map space

We study properties of the infimal topology τ_inf which is the infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. One of the main results of the research consists in obtaining necessary and sufficient condition for the topology τ_inf to have the Fréchet–Urysohn property. We also establish necessary and sufficient conditions for coincidence of the topology τinf and a topology of uniform convergence τ_μ (“attaining” the infimum). We prove that for this coincidence it is sufficient for the topology τ_inf to satisfy the first axiom of countability.     

Symplectic topology of integrable dynamical systems. Rough topological classification of classical cases of integrability in the dynamics of a heavy rigid body

Physical and mechanical systems with four-dimensional phase space are considered. The classification of nondegenerate integral systems is studied. A “physical zone,’ i.e., the systems connected with real physical applications, is determined. Bibliography: 27 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1996, pp. 104–183.     

On the symplectic phase space of KdV

We prove that the Birkhoff map for KdV constructed on can be interpolated between and . In particular, the symplectic phase space can be described in terms of Birkhoff coordinates.

     

On the Hamiltonian property of linear dynamical systems in Hilbert space

Mathematical Notes - Conditions for the operator differential equation $$dot x = Ax$$ possessing a quadratic first integral (1/2)(Bx, x) to be Hamiltonian are obtained. In the finite-dimensional...     

On an extended phase space

Summary This paper presents a generalization of the dynamical concept phase space. The generalization involves a relative increase in dimensionality and employs a basic function which contains higher order derivatives. To Enrico Bompiani on his scientific Jubile.     

On the Sidon-Telyakovskii integrability class for cosine series

Various reformulations of the classical Sidon integrability condition for cosine series are considered. Two apparent generalizations are shown to be equivalent to Sidon's condition. As an offshoot of these results it is shown that if the cosine Fourier coefficients satisfy $\text{n¦Δa}{}_{\mathrm{n}}\text{¦ = o(1) (n → ∞)}$, then ∥S_n(f) ?f ∥ = o(1) (n → ∞) is equivalent to a_nlgn = o(1) (n → ∞).