On global attractors for a class of nonhyperbolic piecewise affine maps

Dynamical behavior of a class of nonhyperbolic discrete systems are considered. These systems are generated by iterating planar maps that are piecewise isometries, and they arise as mathematical models for signal processing, digital filters and modulator dynamics. Planar piecewise isometries may be discontinuous and/or non-invertible. First, the authors consider attraction caused by discontinuity in planar piecewise isometries. Namely, they have shown that the maximal invariant set can induce an invariant measure, and all the Lyapunov exponents are zero under this invariant measure. Second, they discuss various definitions of global attractors and their existence and uniqueness for discontinuous maps, and introduce a few examples in which the attractors are created due to discontinuity. Third, they study the relation between invariance and invertibility for various nonhyperbolic maps, and finally they investigate decomposability of global attractors for certain nonhyperbolic systems.     

A class of nonmixing dynamical systems with monotonic semigroup property

In order to illustrate the class of conservative dynamical systems for which a Boltzmann entropy can be obtained under finite coarse-graining [2], we consider dynamical systems defined by the shift transformation on K , where K is any finite set of integers. We give a class of non-Markovian invariant measures that verify the Chapman-Kolmogorov equation (equivalent to a Boltzmann entropy) for any positive stochastic matrix and that are ergodic but not weakly mixing.     

Existence of time evolution forv-dimensional statistical mechanics

Infinite systems of particles inv-dimensions are considered. The pair interaction is assumed to beC ², finite range and superstable. The existence of a time evolution which satisfies the infinite equations of motion in a set of full equilibrium measure is proved. This measure is proved to be invariant; so a dynamical system is obtained.     

Global Structure of Quaternion Polynomial Differential Equations

In this paper we mainly study the global structure of the quaternion Bernoulli equations \({\dot q=aq+bq^n}\) for \({q\in {\mathbb{H}}}\), the quaternion field and also some other form of cubic quaternion differential equations. By using the Liouvillian theorem of integrability and the topological characterization of 2–dimensional torus: orientable compact connected surface of genus one, we prove that the quaternion Bernoulli equations may have invariant tori, which possesses a full Lebesgue measure subset of \({{\mathbb{H}}}\). Moreover, if n = 2 all the invariant tori are full of periodic orbits; if n = 3 there are infinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits.     

The G-Scheme: A framework for multi-scale adaptive model reduction

The numerical solution of mathematical models for reaction systems in general, and reacting flows in particular, is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics. However, the dynamics can develop very-slow and very-fast time scales separated by a range of active time scales. An opportunity to reduce the complexity of the problem arises when the fast/active and slow/active time scales gaps becomes large. We propose a numerical technique consisting of an algorithmic framework, named the G-Scheme, to achieve multi-scale adaptive model reduction along-with the integration of the differential equations (DEs). The method is directly applicable to initial-value ODEs and (by using the method of lines) PDEs. We assume that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. The G-Scheme introduces locally a curvilinear frame of reference, defined by a set of orthonormal basis vectors with corresponding coordinates, attached to this decomposition. The evolution of the curvilinear coordinates associated with the active subspace is described by non-stiff DEs, whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem. Adjusting the active DEs dynamically during the time integration is the most significant feature of the G-Scheme, since the numerical integration is accomplished by solving a number of DEs typically much smaller than the dimension of the original problem, with corresponding saving in computational work. To demonstrate the effectiveness of the G-Scheme, we present results from illustrative as well as from relevant problems.     

Theory of tensor invariants of integrable Hamiltonian systems. I. Incompatible Poisson structures

This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM ⁿ. The central objects in this theory are supplementary invariant Poisson structuresP _c which are incompatable with the original Poisson structureP ₁ for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP ₁ andP ₂ in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP ₁ nor with respect toP ₂. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337.     

Spectrum of Lebesgue Measure Zero for Jacobi Matrices of Quasicrystals

We study one-dimensional random Jacobi operators corresponding to strictly ergodic dynamical systems. We characterize the spectrum of these operators via non-uniformity of the transfer matrices and vanishing of the Lyapunov exponent. For aperiodic, minimal subshifts satisfying the so-called Boshernitzan condition this gives that the spectrum is supported on a Cantor set with Lebesgue measure zero. This generalizes earlier results for Schrödinger operators.     

Escape dynamics in collinear atomic-like three mass point systems

The present paper studies the escape mechanism in collinear three point mass systems with small-range-repulsive/large-range-attractive pairwise interaction. Specifically, we focus on the asymptotic behaviour for systems with non-negative total energy.On the zero energy level set there are two distinct asymptotic states, called 1+1+1escape configurations, where all the three separations infinitely increase as t→∞. We show that 1+1+1 escapes are improbable by proving that the set of initial conditions leading to such asymptotic configurations has zero Lebesgue measure. When the outer mass points are of the same kind we deduce the existence of a heteroclinic orbit connecting the 1+1+1 escape configurations. We further prove that this orbit is stable under parameter perturbation.In the positive energies’ case, we show that the set of initial conditions leading to 1+1+1 escape configurations has positive Lebesgue measure.     

An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems

We present a computational method for determining the geometry of a class of three-dimensional invariant manifolds in non-autonomous (aperiodically time-dependent) dynamical systems. The presented approach can be also applied to analyse the geometry of 3D invariant manifolds in three-dimensional, time-dependent fluid flows. The invariance property of such manifolds requires that, at any fixed time, they are given by surfaces in R³. We focus on a class of manifolds whose instantaneous geometry is given by orientable surfaces embedded in R³. The presented technique can be employed, in particular, to compute codimension one (invariant) stable and unstable manifolds of hyperbolic trajectories in 3D non-autonomous dynamical systems which are crucial in the Lagrangian transport analysis. The same approach can also be used to determine evolution of an orientable ‘material surface’ in a fluid flow. These developments represent the first step towards a non-trivial 3D extension of the so-called lobe dynamics — a geometric, invariant-manifold-based framework which has been very successful in the analysis of Lagrangian transport in unsteady, two-dimensional fluid flows. In the developed algorithm, the instantaneous geometry of an invariant manifold is represented by an adaptively evolving triangular mesh with piecewise C² interpolating functions. The method employs an automatic mesh refinement which is coupled with adaptive vertex redistribution. A variant of the advancing front technique is used for remeshing, whenever necessary. Such an approach allows for computationally efficient determination of highly convoluted, evolving geometry of codimension one invariant manifolds in unsteady three-dimensional flows. We show that the developed method is capable of providing detailed information on the evolving Lagrangian flow structure in three dimensions over long periods of time, which is crucial for a meaningful 3D transport analysis.     

On the dimension of deterministic and random Cantor-like sets,symbolic dynamics,and the Eckmann-Ruelle Conjecture

In this paper we unify and extend many of the known results on the dimension of deterministic and random Cantor-like sets in ? ⁿ , and apply these results to study some problems in dynamical systems. In particular, we verify the Eckmann-Ruelle Conjecture for equilibrium measures for Hölder continuous conformal expanding maps and conformal Axiom A^# (topologically hyperbolic) homeomorphims. We also construct a Hölder continuous Axiom A^# homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere. this example shows that the non-conformal Hölder continuous version of the Eckmann-Ruelle Conjecture is false. The Cantor-like sets we consider are defined by geometric constructions of different types. The vast majority of geometric constructions studied in the literature are generated by a finite collection ofp maps which are either contractions or similarities and are modeled by the full shift onp symbols (or at most a subshift of finite type). In this paper we consider much more general classes of geometric constructions: the placement of the basic sets at each step of the construction can be arbitrary, and they need not be disjoint. Moreover, our constructions are modeled by arbitrary symbolic dynamical systems. The importance of this is to reveal the close and nontrivial relations between the statistical mechanics (and especially the absence of phase transitions) of the symbolic dynamical system underlying the geometric construction and the dimension of its limit set. This has not been previously observed since no phase transitions can occur for subshifts of finite type. We also consider nonstationary constructions, random constructions (determined by an arbitrary ergodic stationary distribution), and combinations of the above.     

Operator ideals and the statistical independence in quantum field theory

We consider analytic properties of a class of dynamical systems which are defined by the action of certain homomorphism groups on von Neumann algebras if restricted to subalgebras. In particular, the analyticity of nuclear maps in the nuclear norm is shown. Furthermore, the statistical independence will be derived from nuclearity conditions. These results give new insight in the statistical independence of commuting algebras.This paper is a result of a collaboration with H. J. Borchers.     

Gauge invariant unified field structures,quantizable dynamical systems,charge, and spin structures

Gauge invariant unified field structures on a manifold B are introduced. Necessary and sufficient conditions for their existence are studied. The connection with charge is studied; it is shown that such gauge invariant structures, e.g. quantizable dynamical systems, over simply connected manifolds B are completely classified by charge. Complex analytic gauge invariant unified field structures are studied. These structures over a complex analytic manifold B whose square is the canonical line bundle are in bijective correspondence with the spin structures on B. Finally, a class of homogeneous quantizable dynamical systems are shown not to carry spin structure.     

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

We give a global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces reduces to a parameter-dependent system of two or three ordinary differential equations, respectively. Here, we present a qualitative study of these systems’ global phase portrait, which uses techniques of dynamical systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.     

Die Klassifikation nach inneren Symmetrien beim quantenmechanischen Mehrkörperproblem

For systems ofn identical particles with harmonic two-body interaction a method is derived which allows for anyn the construction of a complete orthonormal set of functions that are translationally invariant and are classified according to energy, permutational symmetry,SU (3), angular momentum, and parity. Moshinsky's method for the determination of translationally invariant states with definite permutational symmetry for harmonic two-body interaction is briefly reviewed and is extended to additionalSU (3)-classification. His method, however, is seen to be restricted to the casesn=3,n=4.     

The Rossby wave extra invariant in the physical space

It was found out in 1991 that the Fourier space dynamics of Rossby waves possesses an extra positive-definite quadratic invariant, in addition to the energy and enstrophy. This invariant is similar to the adiabatic invariants in the theory of dynamical systems. For many years, it was unclear if this invariant—known only in the Fourier representation—is physically meaningful at all, and if it is, in what sense it is conserved. Does the extra conservation hold only for a class of solutions satisfying certain constraints (like the conservation in the Kadomtsev-Petviashvili equation)? The extra invariant is especially important because this invariant (provided it is meaningful) has been connected to the formation of zonal jets (like Jupiter’s stripes).In the present paper, we find an explicit expression of the extra invariant in the physical (or coordinate) space and show that the invariant is indeed physically meaningful for any fluid flow. In particular, no constraints are needed. The explicit form also enables us to note several properties of the extra invariant.     

Neumann-Bogolyubov-Rosochatius Oscillatory Dynamical Systems and their Integrability Via Dual Moment Maps. Part I

Abstract

The finite-dimensional invariant subspaces of the solutions of intergrable by Lax infinite-dimensional Benney-Kaup dynamical system are presented. These invariant subspaces carry the canonical symplectic structure, with relation to which the Neumann type dynamical systems are Hamiltonian and Liouville intergrable ones. For the Neumann-Bogolyubov and Neumann-Rosochatius dynamical systems, the Lax-type representations via the dual moment maps into some deformed loop algebras as well as the finite hierarchies of conservation laws are constructed.     

Theory of Tensor Invariants of Integrable Hamiltonian Systems. II. Theorem on Symmetries and Its Applications

The theorem on symmetries is proved that states that a Liouville-integrable Hamiltonian system is non-degene\-rate in Kolmogorov's sense and has compact invariant submanifolds if and only if the corresponding Lie algebra of symmetries is abelian. The theorem on symmetries has applications to the characterization problem, to the integrable hierarchies problem, to the necessary conditions for the strong dynamical compatibility problem, and to the problem on master symmetries. The invariant necessary conditions for the non-degenerate C-integrability in Kolmogorov's sense of a given dynamical system V are derived. It is proved that the C-integrable Hamiltonian system is non-degenerate in the iso-energetic sense if and only if the corresponding Lie algebra of the iso-energetic conformal symmetries is abelian. An extended concept of integrability of Hamiltonian systems on the symplectic manifolds M ⁿ , n= 2k, is introduced. The concept of integrability describes the Hamiltonian systems that have quasi-periodic dynamics on tori or on toroidal cylinders of an arbitrary dimension . This concept includes, as a particular case, all Hamiltonian systems that are integrable in Liouville's classical sense, for which . The A-B-C-cohomologies are introduced for dynamical systems on smooth manifolds. Received: 16 January 1996 / Accepted: 3 July 1996     

Pearson-walk visualization of the characteristic function of the invariant measure for 1-d chaos

The Pearson-walk visualization of one-dimensional (1-d) chaos, which has been proposed in a qualitative fashion by the same authors recently (Physica 134A (1985) 123) is treated quantitatively. Continuity of the Pearson image is deduced for a map f(x) whose kth iterate f^k(x) is continuous for any k. Then the Lyapunov exponent is used to describe the length of the Pearson image. The normalized Pearson-walk visualization is introduced to show that it can be related to the existence of the invariant measure. For a 1-d map that has a definite invariant measure it is shown that the characteristic function of the invariant measure is represented by a unique point in the normalized Pearson plane for a large iteration-number limit.     

Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions

We investigate one-dimensional discrete Schrödinger operators whose potentials are invariant under a substitution rule. The spectral properties of these operators can be obtained from the analysis of a dynamical system, called the trace map. We give a careful derivation of these maps in the general case and exhibit some specific properties. Under an additional, easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying Schrödinger operator is singular and supported on a set of zero Lebesgue measure. A condition allowing to exclude point spectrum is also given. The application of our theorems is explained on a series of examples.