On the construction of stable and unstable manifolds of two-dimensional invertible maps

An equation is proposed for describing stable and unstable manifolds for a wide class of two-dimensional invertible maps. Several branches of the stable and unstable manifolds of the dissipative mapx _n+1 =1–a|x _n |+bz _n ,z _n+1 =x _n are constructed explicitly. The limiting case when the strange attractor disappears is discussed.     

On the construction of invariant curves of period-two points in two-dimensional maps

We present an equation describing invariant curves associated with periodic points of period two in a wide class of two-dimensional invertible maps. Several branches of the unstable manifolds for the map x_n+1 = 1 - a|x_n| + bz_n, z_n+1 = x_n are constructed in a situation when they are related to a two-piece strange attractor.     

Fluctuation Bounds for Chaos Plus Noise in Dynamical Systems

We are interested in time series of the form y _n =x _n +ξ _n where {x _n } is generated by a chaotic dynamical system and where ξ _n models observational noise. Using concentration inequalities, we derive fluctuation bounds for the auto-covariance function, the empirical measure, the kernel density estimator and the correlation dimension evaluated along y ₀,…,y _n , for all n. The chaotic systems we consider include for instance the Hénon attractor for Benedicks-Carleson parameters.     

Some more universal scaling laws for critical mappings

We study such nonlinear mappingsx _n +1=F(x _n ;b _cr) of an intervalI into itself for which the Feigenbaum scaling laws hold (i.e., for which b_cr is an accumulation point of bifurcation points). Letx ₀ be a random variable with some absolutely continuous distribution inI. We show in particular that (i) the geometric average distance ofx _n from the nearest point of the attractor decreases liken ^–1.93387; (ii) the geometric average of ¦x _n /x ₀¦ increases liken ^0.60; (iii) the geometric mean distance ¦x _n –y _n ¦ between the iterates of two close-by pointsx ₀,y ₀ asymptotically tends towards a value ¦x ₀–y ₀¦^0.77. These-and other-properties are also borne out from a simple probabilistic model which depicts the evolution as a random walklike process.     

Dynamic behaviours of 2∞ attractor and q-phase transitions at bifurcations in logistic map

Dynamic behaviours of the 2^∞ attractor at the accumulation of period doubling in the logistic map are studied by the sum of the local expansion rates S_n(x₁) of nearby orbits. The variance 〈[S_n(x)]²〉 and algebraic exponent ß_n(x₁) = S_n(x₁)/ln(n) exhibits self-similar structures. The critical bifurcations such as intermittency, band merging and crisis-sudden widening of the chaotic attractor are studied in terms of a q-weighted average Λ(q), (− ∞ < q < ∞) of the coarse-grained local expansion rates Λ of nearby orbitals.     

The mechanism of the increase of the generalized dimension of a filtered chaotic time series

The determination of the attractor dimension from an experimental time series may be affected by the influence of filters which are incorporated into many measuring processes. While this is expected from the Kaplan-Yorke conjecture, we show that for one-dimensional maps a weak filter can induce a self-similarity which is responsible for the increase of the Hausdorff dimension. We are able to calculate the increase of the generalized dimensionD _q for the filtered time series of the logistic mapx _i +1=rx _i (1–x _i ) atr=4 analytically.     

Statistical dynamics at critical bifurcations in Duffing-van der Pol oscillator

We study the characteristic features of certain statistical quantities near critical bifurcations such as onset of chaos, sudden widening and band-merging of chaotic attractor and intermittency in a periodically driven Duffing-van der Pol oscillator. At the onset of chaos the variance of local expansion rate is found to exhibit a self-similar pattern. For all chaotic attractors the variance Σ_n(q) of fluctuations of coarse-grained local expansion rates of nearby orbits has a single peak. However, multiple peaks are found just before and just after the critical bifurcations. On the other hand, Σ_n (q) associated with the coarse-grained state variable is zero far from the bifurcations. The height of the peak of Σ_n(q) is found to increase as the control parameter approached the bifurcation point. It is maximum at the bifurcation point. Power-law variation of maximal Lyapunov exponent and the mean value of the state variablex is observed near sudden widening and intermittency bifurcations while linear variation is seen near band-merging bifurcation. The standard deviation of local Lyapunov exponent λ(X,L) and the local mean valuex(L) of the coordinatex calculated after everyL time steps are found to approach zero in the limitL → ∞ asL ^-Β. Β is sensitive to the values of control parameters. Further weak and strong chaos are characterized using the probability distribution of ak-step difference quantity δx_k = x_i+k x _i.     

Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,

which are algebraically solvable. Here l is the “discrete-time” independent variable taking integer values (l = 0, 1, 2, . . .), x_n ≡ x_n(l) are 2 dependent variables, and are the corresponding 2 updated variables. In a previous paper the 2 functions F⁽ⁿ⁾(x₁, x₂), n = 1, 2, were defined as follows: F⁽ⁿ⁾(x₁, x₂) = P₂ (x_n, x_n+1), n = 1, 2 mod[2], with P₂(x₁, x₂) a specific second-degree homogeneous polynomial in the 2 (indistinguishable!) dependent variables x₁(l) and x₂(l). In the present paper we further clarify some aspects of that model and we present its extension to the case when a specific homogeneous function of arbitrary (integer) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x₁(l) and x₂(l).     

Integrability Conditions for Complex Homogeneous Kukles Systems

In this paper we study the existence of local analytic first integrals for complex polynomial differential systems of the form ? = x + P_n(x, y), ? = ?y, where P_n(x, y) is a homogeneous polynomial of degree n, called the complex homogeneous Kukles systems of degree n. We characterize all the homogeneous Kukles systems of degree n that belong to the Sibirsky ideal. Finally, we provide necessary and sufficient conditions when n = 2,?. . .?, 7 in order that the complex homogeneous Kukles system has a local analytic first integral computing the saddle constants and using Gröbner bases to find the decomposition of the algebraic variety into its irreducible components.     

Central Limit Theorems for Nonlinear Hierarchical Sequences of Random Variables

Let a random variable x ₀ and a function f:[a, b] ^k [a, b] be given. A hierarchical sequence {x _n :n=0, 1, 2,...} of random variables is defined inductively by the relation x _n =f(x _{n–1, 1}, x _{n–1, 2}..., x _{n–1, k} ), where {x _{n–1, i} :i=1, 2,..., k} is a family of independent random variables with the same distribution as x _n–1. We prove a central limit theorem for this hierarchical sequence of random variables when a function f satisfies a certain averaging condition. As a corollary under a natural assumption we prove a central limit theorem for a suitably normalized sequence of conductivities of a random resistor network on a hierarchical lattice.     

Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-Hand Sides

The evolution equations mentioned in the title of this paper read as follows:

where ? is the “discrete-time” independent variable taking integer values (? = 0, 1, 2,?…?), xn ≡ xn(?) are the 2 dependent variables, , and the 2 functions P⁽ⁿ⁾(x₁, x₂), n = 1, 2, are 2 polynomials in the 2 dependent variables x₁(?) and x₂(?). The results reported in this paper have been obtained by an appropriate modification of a recently introduced technique to obtain analogous results in continuous-time t—in which case x_n ≡ x_n(t) and the above recursion relations are replaced by first-order ODEs. Their potential interest is due to the relevance of this kind of evolution equations in various applicative contexts.     

Numerical study of the oscillatory convergence to the attractor at the edge of chaos

This paper compares three different types of “onset of chaos” in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three window; the transition to chaos in the generalized logistic with inflection 1/2 (x_n+1 = 1-μx_n^1/2), in which the main bifurcation cascade, as well as the bifurcations generated by the periodic windows in the chaotic region, collapse in a single point. The occupation number and the Tsallis entropy are studied. The different regimes of convergence to the attractor, starting from two kinds of far-from-equilibrium initial conditions, are distinguished by the presence or absence of log-log oscillations, by different power-law scalings and by a gap in the saturation levels. We show that the escort distribution implicit in the Tsallis entropy may tune the log-log oscillations or the crossover times.     

N.M.R. spectra of the XnAA′ Xn′ type

The method described previously for the calculation of the X spectrum for spin systems of of the X_nAA′X_n′ type is extended by means of magnetic equivalence factoring to cases in which the long-range coupling constant J_x is non-zero. This coupling is treated as a first-order perturbation to the energy levels obtained when J_x =0, and it is found that J_x may be measured directly from the separations between easily identified lines in the spectrum. A detailed account of the X ₃ AA′X ₃′ system is given, and the N.M.R. spectrum of fluoro-N,N′-dimethyl-1,3,2,4-diazadiphosphetidine is analysed. The spectral parameters obtained are discussed. A computer programme which calculates the spectra of systems involving magnetic equivalence is used to examine the validity of the approximations in the method.     

Relaxation properties of chaotic dynamical systems: Operator approach

A method of analytically determining eigenvalues and the piecewise-continuous eigenfunction systems of the Perron-Frobenius operator for Rényi chaotic map x _n+1 = βx _n mod 1, 1 < β < 2 based on introducing the generating function for the eigenfunctions is described. These characteristics define the relaxation properties and decay of correlations in discrete dynamical systems.     

Phase interactions in TiN<Subscript>x</Subscript>(TiB<Subscript>x</Subscript>)-<Emphasis Type="Italic">n</Emphasis>-Si-<Emphasis Type="Italic">n</Emphasis><Superscript>+</Superscript>-Si contacts and their thermal degradation due to rapid thermal annealing

The temperature stability of TiN_x(TiB_x)-n-Si-n ⁺-Si, Au-TiN_x(TiB_x)-n-Si-n ⁺-Si, and Au-Ti(Mo)-TiN_x(TiB_x)-n-Si-n ⁺-Si Schottky-barrier contacts subjected to rapid thermal annealing in hydrogen at temperatures T=400, 600, and 800°C is studied. It is shown that structural and morphological transformations and the related degradation of electrophysical characteristics in interstitial alloys (titanium nitrides and borides) start at 600°C. Reasons for the degradation of the barrier properties of titanium borides and nitrides are discussed.     

A variational problem for a system of magnetic monopoles joined by Abrikosov vortices

An action functional, related to the Higgs model to field theory, depending on a complex scalar field and aU(1) connection is defined. The complex scalar field is a section of a line bundle associated to a principalU(1)-bundle with base space ³\{x ₁,...,x _n }. The pointsx ₁,...,x _n are the positions ofn magnetic monopoles of magnetic chargesm ₁,...,m _n, with . The existence of minimizers of the action functional is proven using direct methods of the calculus of variation. Regularity and decay properties of the minimizers are obtained. By constructing explicit comparison field configurations, we establish accurate upper and lower bounds for the action of the minimizers in a variety of special situations, e.g.n=2 andm ₁=–m ₂.     

Raman studies and optical properties of some (PbO)x(Bi2O3)0.2(B2O3)0.8−x glasses

Five (PbO)_x(Bi₂O₃)_0.2(B₂O₃)_0.8−x glasses, where x = 0, 0.2, 0.3, 0.4 and 0.6, were prepared. The dilatometric glass transition temperature (T_g) was found in the region 470 (x = 0)≥ T_g ( °C) ≥ 347 (x = 0.6), and the density (ρ) varied within 4.57 (x = 0) ≤ ρ (g/cm³) ≤ 8.31 (x = 0.6). Raman spectra indicated the conversion of BO₃ to BO₄ entities for low x values but for x > 0.3, namely, for x → 0.6, back‐conversion occurred, most probably. From the measurements of the optical transmission on very thin bulk samples, the room temperature optical gap values (E_g) were determined to be in the range 4.03 (x = 0)≥ E_g (eV) ≥ 3.08 (x = 0.6). The temperature (T) dependence of the optical gap (E_g(T)) in the region 300 ≤ T(K) ≤ 600 was examined and approximated by a linear relationship of the form of E_g(T) = E_g(0)− γT, where γ × 10⁻⁴(eV/K) varied from 5.1 to 6.8. The non‐linear refractive index (n₂) was estimated from the optical gap values and it was found to correspond to the n₂ values calculated from the experimental third‐order non‐linear optical susceptibility taken from the literature. Copyright © 2008 John Wiley & Sons, Ltd.     

Asymptotic behavior of orbits of C 0-semigroups and solutions of linear and semilinear abstract differential equations

In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + t ⁿ f(t, u(t)), where A is the generator of a C ₀-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C ₀-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class S ^p-A defined similarly to the case of S ^p-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to A _u:= A ∩ BUC(ℝ, X) if n = 0 and to t ⁿ A _u ⊕ w _n C ₀ (ℝ, X) if n ∈ ℕ, where w _n(t) = (1 + |t|)ⁿ. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0. Dedicated to the memory of B. M. Levitan     

Assessment of the possibility of determination of the electrophysical characteristics of CdxHg1–xTe p+–n-structures using differential hall measurements

The process of reconstuction of the distribution profile of hole concentration in the p ⁺–n structure by the method of differential Hall measurements upon implantation of ions As⁺ (Е = 190 keV, D = 3.10¹⁴ cm^-2, j = 0.025 μA/cm²) into epitaxial films Cd _x Hg_1–x Te for x ~ 0.2, with the initial electron concentration and mobility n = 10¹⁴ cm^-3 and μ = 2∙10⁵ cm²∙V^–1∙s^–1 is numerically simulated. The dependences of degree of reconstruction of the hole-concentration distribution profile on the depth of a shunting n-layer and magnitude of the magnetic field, at which the electrophysical parameters of the p ⁺–n structure are measured, are calculated. The dependence of the limiting magnetic field determining the magnetic-field range for measurements on the n-layer depth is found. It is shown that in calculations one should use the conduction values measured at the same magnetic fields as the Hall coefficients for determination of the holeconcentration distribution profile using the Petritz model.     

Structural and electronic properties of GaAsBi

The structural and electronic properties of the GaAs_1−xBi_x ternary alloy are investigated by means of two first principles and full potential methods, the linear augmented plane waves (FPLAPW) method and a recent version of the full potential linear muffin-tin orbitals method (FPLMTO) which enables an accurate treatment of the interstitial regions. In particular, we have found that the maximal GaBi mole fraction x for which GaBi_xAs_1−x remains a semiconductor is probably around x=0.5. The electronic properties of (GaAs)_m/(GaBi)_n quantum well superlattices (SLs) have also been calculated and it is found that such SLs are semiconductors when m is larger or equal to n.