Nearly linear mappings and their applications

A simple 2-dimensional mapping is considered, both analytically and numerically, for which all nonlinear effects are of the same order as the perturbations and of the same origin. Properties of the stochastic instability are investigated, taking the beam-beam interaction in a storage ring as an important particular example of a dynamic system that can be modelled with such a mapping. The special case of time-dependent mappings is discussed. It is shown that low-frequency time dependence sharply decreases the critical perturbation strength for the stochastic transition.     

A Class of Integrable and Nonintegrable Mappings and their Dynamics

We analyse a class of mappings which by construction do not belong to the QRT family. We show that some of the members of this class have invariants of high degree. A new linearisable mapping is also identified. A mapping which possesses confined singularities while having nonzero algebraic entropy is presented. Its dynamics are studied in detail and shown to be related intimately to the Fibonacci recurrence.      

Classical Description of Intense Field Stabilization Dynamics of 1D Model Atom

Abstract In this paper classical nonlinear dynamics approaches are used to study the intense field stabilization (IFS) for ID model atom according to differential equation (DE) and map approximation (MA). The exact stability boundary and region of I-period fixed points in the parameter space of field strength and frequency are obtained, and the results of MA are in excellent agreement with those of DE. The electrons dynamical behavior in phase space is investigated and the IFS mainly occurring in high frequency regime is confirmed. The stabilization phenomenon connected with the stable fixed points is also examined. In particular it is shown that this phenomenon depends on tightly the KAM tori. It can be predicted that the stabilization probability is just the fraction of initial electrons confined by the KAM tori.     

Self-similarities in the frequency-amplitude space of a loss-modulated CO2 laser

We show the standard two-level continuous-time model of loss-modulated CO2 lasers to display the same regular network of self-similar stability islands known so far to be typically present only in discrete-time models based on mappings. Our results suggest that the two-parameter space of class B laser models and that of a certain class of discrete mappings could be isomorphic.     

Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors

We study numerically the complex domains of validity for KAM theory in generalized standard mappings. We compare methods based on Padé approximants and methods based on the study of periodic orbits.     

KAM theorems for the product of two involutions of different types

The properties of quasiperiodic motions in Hamiltonian, volume preserving, and reversible systems are summarized. KAM theorems concerning lower-dimensional invariant tori are announced for G-reversible mappings A such that the fixed point manifolds Fix(G) and Fix(AG) of the reversing involutions G and AG are of different dimensions. The case where the manifold Fix(G) itself consists of several connected components of different dimensions is also briefly discussed.     

A generalized iterative approach to curved-ray tomography

A generalized iterative approach to the tomographic reconstruction of strongly refracting fields is proposed. The mappings for existing iterative schemes are recognized to be special cases of a more general form, and this form is shown to possess an arbitrary operator which affects convergence but may be changed without altering the roots of the original mapping. This, therefore, provides the basis for defining new recursive sequences which may converge in cases where the standard iterative schemes are divergent. To illustrate the approach, two enhanced schemes are developed by making particular selections for the arbitrary operator, and a 1-D boundary layer type field is reconstructed from numerically simulated interferometric data.     

Analyses of vocal tract cross-distance to area mapping: an investigation of a set of vowel images

Traditional models of mappings from midsagittal cross-distances to cross-sectional areas use only local cross-distance information. These are not the optimal models on which to base the construction of a mapping between the two domains. This can be understood because phonemic identity can affect the relation between local cross-distance and cross-sectional area. However, phonemic identity is not an appropriate independent variable for the control of an articulatory synthesizer. Two alternative approaches for constructing cross-distance to area mappings that can be used for articulatory synthesis are presented. One is a vowel height-sensitive model and the other is a non-parametric model called loess. These depend on global cross-distance information and generally perform better than the traditional models.     

Zero-Delay Joint Source Channel Coding for a Bivariate Gaussian Source over the Broadcast Channel with One-Bit ADC Front Ends

In this work, we consider the zero-delay transmission of bivariate Gaussian sources over a Gaussian broadcast channel with one-bit analog-to-digital converter (ADC) front ends. An outer bound on the conditional distortion region is derived. Focusing on the minimization of the average distortion, two types of methods are proposed to design nonparametric mappings. The first one is based on the joint optimization between the encoder and decoder with the use of an iterative algorithm. In the second method, we derive the necessary conditions to develop the optimal encoder numerically. Using these necessary conditions, an algorithm based on gradient descent search is designed. Subsequently, the characteristics of the optimized encoding mapping structure are discussed, and inspired by which, several parametric mappings are proposed. Numerical results show that the proposed parametric mappings outperform the uncoded scheme and previous parametric mappings for broadcast channels with infinite resolution ADC front ends. The nonparametric mappings succeed in outperforming the parametric mappings. The causes for the differences between the performances of two nonparametric mappings are analyzed. The average distortions of the parametric and nonparametric mappings proposed here are close to the bound for the cases with one-bit ADC front ends in low channel signal-to-noise ratio regions.     

Robust transport barriers resulting from strong Kolmogorov-Arnold-Moser stability

Hamiltonian systems that locally violate the twist condition arise in many applications. Numerical simulations reveal that, when systems of this type are perturbed, the degenerate or nontwist tori are remarkably stable. This phenomenon, which we refer to as strong Kolmogorov-Arnold-Moser (KAM) stability, is shown to be linked to very small resonance widths near degenerate tori. Quantitative estimates of degenerate resonance widths are derived and bifurcations of degenerate resonances are described. Strong KAM stability leads to robust transport barriers, which are important in all of the many applications in which Hamilitonians with the nontwist property arise.     

Invariant noncanonical mappings of singular theories. II

In a paper of the same title published in Physical Revview Dit was shown that in singular theories (i.e. theories incorporating constraints) non-canonical infinitesimal mappings that map equivalence classes intact on each other lead to the divergence of a vector field associated with the mapping constant throughout the equivalence class. The infinitesimal mappings form the germ of the group of finite mappings of equivalence classes on each other that change the form of the symplectic tensor field. Any non-canonical coordinate system on phase space thus obtained defines a scalar density field that is constant over an equivalence class. The constant of the motion obtained earlier represents the infinitesimal deviation of this new field from unity.     

A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, which could be applied to a large class of Hamiltonian PDEs containing the derivative ? _x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed Benjamin-Ono equation with periodic boundary conditions, so KAM tori and thus quasi-periodic solutions are obtained for them.     

Streaminglike diffusion in the low-dimensional stochastic pump model

In this paper we analyze a diffusion phenomenon in a few-dimensional Hamiltonian system of coupled mappings in which the principal component of diffusion occurs along resonances. The result is that the diffusion can have power-law dependence in coupling parameter mu and be independent of the stochastic parameter K. For the same range of parameters, the usual analytical Arnold diffusion across resonances is dependent on K and can be much smaller than resonance streaming diffusion. The results are used to qualitatively explain recent results in multidimensional coupled standard maps.     

The classical mechanics of one-dimensional systems of infinitely many particles

We apply the existence theorem for solutions of the equations of motion for infinite systems to study the time evolution of measures on the set of locally finite configurations of particles. The set of allowed initial configurations and the time evolution mappings are shown to be measurable. It is shown that infinite volume limit states of thermodynamic ensembles at low activity or for positive potentials are concentrated on the set of allowed initial configurations and are invariant under the time evolution. The total entropy per unit volume is shown to be constant in time for a large class of states, if the potential satisfies a stability condition.On leave from: Department of Mathematics, University of California, Berkeley, California.     

Electro-optic mapping systems of electric-field using CW laser diodes

Two novel systems are proposed to perform the electro-optic (EO) mapping of the electric-field strength close to electrode structure using CW laser diodes. By these systems, mappings of both static (DC) and high-frequency electric-field can be obtained. For the static field, the field is chopped to increase the sensitivity performance. For the high-frequency field, an intensity modulation laser, instead of a pulse laser, is used to perform the mapping measurement in frequency domain, instead of in time domain. This approach greatly simplifies the system. An experiment set-up using a vertical cavity surface emitted laser diode (VCSEL) of a wavelength 850 nm and its mapping results for a coplanar waveguide (CPW) with the signal frequency around 1 GHz are reported.     

Classical and quantum mechanics of non-abelian gauge fields

Classical and quantum mechanics of non-abelian gauge fields are investigated both with and without spontaneous symmetry breaking. The fundamental subsystem (FS) of Yang-Mills classical mechanics (YMCM) is considered. It is shown to be a Kolmogorov K-system, and hence to have strong statistical properties. Integrable systems are also found, to which in terms of KAM theory Yang-Mills-Higgs classical mechanics (YMHCM) is close. Quantum-mechanical properties of the YM system and their relation to the problem of confinement are discussed.     

Hamiltonian maps in the complex plane

The standard map is a nonintegrable discrete time analog of the vertical pendulum. Detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency. The functional dependence of the coordinate q on the canonical angle variable θ is analtyically continued into the complex θ-plane, where natural boundaries are found at constant absolute values of Im θ. The boundaries represent the appearance of chaotic motion in the complex plane. When the domain of analyticity shrinks to zero, the KAM invariant curve is destroyed. Two independent numerical methods with Fourier analysis in the angle variable were used, one based on a variation-annihilation method and the other on a double expansion. The results were further checked by direct solution of the complex equations of motion. The numerically simpler, but intrinsically complex, semipendulum and semistandard map are also studied. We conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or KAM surfaces of general nonintegrable systems with analytic Hamiltonians.     

One-dimensional Bose-Hubbard model with pure three-body interactions

The extended Bose-Hubbard model with pure three-body local interactions is studied using the Density Matrix Renormalization Group approach. The shapes of the first two insulating lobes are discussed, and the values of the critical tunneling for which the system undergoes the quantum phase transition from insulating to superfluid phase are predicted. It is shown that stability of insulating phases, in contrast to the standard Bose-Hubbard model, is enhanced for larger fillings. It is also shown that, on the tip of the boundary of the insulating phase, the model under consideration belongs to the Berenzinskii-Kosterlitz-Thouless universality class.     

Lie algebra bases for the pseudo-orthogonal groups

The isomorphic mappings between the canonical bases appearing in the general structure theory of semi-simple real Lie algebras and the bases obtained directly from the pseudoorthogonal groups are investigated in detail, and it is shown that these mappings can be cast in a remarkable simple form which is valid for all cases.