New universal bifurcation scenario in one-dimensional trimodal maps

By means of star products and high precision numerical calculation, an abnormal phenomenon is found in period-p-tupling bifurcation processes in one-dimensional trimodal maps. A route of transition to chaos, presented by a right-associative non-normal star product, breaks the Feigenbaum's metric universality, namely, the conventional Feigenbaum's successive rates exhibit a strong divergence. To overcome the divergence, an approximate scheme of accelerating convergence is proposed; and the Feigenbaum scenario is included as a special case in the new bifurcation scenario. It will provide access to understanding non-normal star products and their corresponding renormalization.     

Hierarchy of Chaotic Maps with an Invariant Measure

We give hierarchy of one-parameter family (, x) of maps at the interval [0, 1] with an invariant measure. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent of these maps analytically, where the results thus obtained have been approved with the numerical simulation. In contrary to the usual one-parameter family of maps such as logistic and tent maps, these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor for certain values of the parameter, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at those values of the parameter whose Lyapunov characteristic exponent begins to be positive.     

Critical Slowing Down in One-Dimensional Maps and Beyond

This is a brief review on critical slowing down near the Feigenbaum period-doubling bifurcation points and its consequences. The slowing down of numerical convergence leads to an “operational” fractal dimension D=2/3 at a finite order bifurcation point. There is a cross-over to D ₀=0.538... when the order goes to infinity, i.e., to the Feigenbaum accumulation point. The problem of whether there exists a “super-scaling” for the dimension spectrum D _q ^W that does not depend on the primitive word W underlying the period-n-tupling sequence seems to remain open     

Dynamical behaviors of a plate activated by an induction motor

Dynamics and chaotification of a system consisting of an induction motor activating a mobile plate (with variable contents) fixed to a spring are studied. The dynamical model of the device is presented and the electromechanical equations are formulated. The oscillations of the plate are analyzed through variations of the following reliable control parameters: phase voltage supply of the motor, frequency of the external source and mass of the plate. The dynamics of the system near the fundamental resonance region presents jump phenomenon. Mapping of the control parameters planes in terms of types of motion reveals period-n motion, quasi-periodicity and chaos. Anti-control of chaos of the induction motor is also obtained using the field-oriented control associated to the time delay feedback control.     

Arithmetic properties of mirror map and quantum coupling

We study some arithmetic properties of the mirror maps and the quantum Yukawa couplings for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of theJ-function. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function fieldQ(J). This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman's Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced modp. Under the mirror hypothesis and an integrality assumption, we derive modp congurences for the Fourier coefficients. For the quintics, we deduce, (at least for 5×d) that the degreed instanton numbersn _d are divisible by 5³ — a fact first conjectured by Clemens.Research supported by grant DE-FG02-88-ER-25065     

A superconvergent universality induced by non-associativity

The star products in symbolic dynamics, as effective algebraic operations for describing self-similar bifurcation structure in classical dynamical systems, are found to have either associativity or non-associativity. In this Letter, non-associative star products in trimodal iterative dynamical systems are considered. As the left and right operations have different effects, right-associative star products break the conventional Feigenbaum's metric universality. Through high precision parallel computation, it is found that period-p-tupling bifurcation processes described by right-associative star products exhibit a superconvergent universality of double exponential form.     

Existence and Properties of p-tupling Fixed Points

We prove the existence of fixed points of p-tupling renormalization operators for interval and circle mappings having a critical point of arbitrary real degree r > 1. Some properties of the resulting maps are studied: analyticity, univalence, behavior as r tends to infinity. Received: 10 April 2000 / Accepted: 7 July 2000     

Symmetry and chaos in the motion of the damped driven pendulum

The damped, driven pendulum equation is studied numerically. A relation is pointed out between the symmetry of the initial period-m dynamical state of am×2ⁿ period-doubling sequence and the form of the chaotic attractor for the final chaotic dynamical state reached after completion of the inverse-doubling sequence. Effects of extrinsic noise are also mentioned.     

Two interacting particles in a disordered chain II: Critical statistics and maximum mixing of the one body states

For two particles in a disordered chain of length L with on-site interaction U, a duality transformation maps the behavior at weak interaction onto the behavior at strong interaction. Around the fixed point of this transformation, the interaction yields a maximum mixing of the one body states. When (the one particle localization length), this mixing results in weak chaos accompanied by multifractal wave functions and critical spectral statistics, as in the one particle problem at the mobility edge or in certain pseudo-integrable billiards. In one dimension, a local interaction can only yield this weak chaos but can never drive the two particle system to full chaos with Wigner-Dyson statistics. Received: 22 May 1998 / Received in final form: 24 August 1998 / Accepted: 4 September 1998     

Subharmonic resonances of a mechanical oscillator with periodically time-varying, piecewise-nonlinear stiffness

The subharmonic (period-η, η>1) motions of a piecewise-nonlinear (PN) mechanical oscillator having parametric and external excitations are investigated. The system is formed by a viscously damped, single-degree-of-freedom oscillator subjected to a periodically time-varying, PN stiffness defined by a clearance surrounded by continuous forms of nonlinearity. A multiterm harmonic balance formulation in conjunction with discrete Fourier transforms is used to determine steady-state period-η motions of the system near the parametric instability regions. The accuracy of analytical solutions is verified through a comparison with direct numerical integration results. A parametric study is also presented to demonstrate the combined influence of a clearance and of cubic nonlinearities on period-η motions within typical ranges of system and excitation parameters.     

The effect of non-linear capacitances in the localization properties of aperiodic dual electric transmission lines

This study investigates the localization properties of dual electric transmission lines with non-linear capacitances. The V_C,n voltage across each capacitor is selected as a non-linear function of the electric charge q_n, i.e., V_C,n = q_n(1/C_n -ε_n|q_n|²)where C_n is the linear part of the capacitance and ε_n the amplitude of the non-linear term. We follow a binary distribution of values of ε_n, according to the Thue-Morse m-tupling sequence. The localization behavior of this non-linear case indicates that the case m = 2 does not belong to the m ≥ 3, family because when m changes from m = 2 to m = 3, the number of extended states diminishes dramatically. This proves the topological difference of the m = 2 and m = 3 families. However, by increasing m values, localization behavior of the m-tupling family resembles that of the m = 2, case because the system begins to regain its extended states. The exact same result was obtained recently in the study of linear direct transmission lines with m-tupling distribution of inductances. Consequently, we state that the localization behavior of the m-tupling family as a function of the m value is independent of both the linear and the non-linear system under study, but independent of the kind of transmission line (dual or direct). This is curious behavior of the m-tupling family and thus deserves more scholarly attention.     

Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps

We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the (z₁, z₂)-logarithmic map, corresponds to a generalization of the z-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the z-logistic map is numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy S_q. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to q-Gaussian attractor distributions. We also study the generalized q-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The q-sensitivity indices are obtained as well. Our results are, like those for the z-logistic maps, numerically compatible with the q-generalization of a Pesin-like identity for ensemble averages.     

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.     

Universality for period n-tuplings in complex mappings

The theory of period doublings for real iterative mappings is generalized to period n-tupling for complex iterative mappings. We find an infinity of universal functions associated with different sequences of period n-tuplings.     

振动筛系统的两类余维三分岔与非常规混沌演化

建立了振动筛系统的动力学模型,推导出了其周期运动的Poincaré 映射,基于Poincaré 映射方法着重研究了系统Flip-Hopf-Hopf余维三分岔、三次强共振条件下的Hopf-Hopf余维三分岔以及三种非常规的混沌演化过程.研究结果表明,此两类余维三分岔点附近的动力学行为变得更加复杂和新颖,在分岔点附近出现了三角形吸引子、3T²环面分岔以及“五角星型”、“轮胎型”概周期吸引子,揭示了环面爆破、环面倍化以及T²环面分岔向混沌演化的过程,这些结果对于振动筛系统的动力学优化设计提供了理论参考. 关键词： 余维三分岔 非常规混沌演化 T²环面分岔')" href="#">T²环面分岔     

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1-q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law , where and are the extreme values appearing in the multifractal function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d_f equals unity for all z in contrast with q which does depend on z, it becomes clear that d_f plays no major role in the sensitivity to the initial conditions. Received 5 February 1999     

Proton and Carbon‐13 NMR Studies of the Phencyclone Adducts of Medium Alkyl Chain‐Length N‐n‐Alkylmaleimides. Hindered Rotations and Magnetic Anisotropic Effects. Ab initio Calculations for Optimized Structures

Abstract

The Diels–Alder adducts, 3a–e, of phencyclone, 1, have been prepared from a series of N‐n‐alkylmaleimides, 2, with medium chain‐length n‐alkyl groups. The maleimides were obtained by cyclodehydration of the N‐n‐alkylmaleamic acids, 4, formed from reaction of maleic anhydride with the corresponding n‐alkylamines. The five adducts prepared included derivatives from n‐heptyl, 3a; n‐octyl, 3b; n‐nonyl, 3c; n‐decyl, 3d; and n‐dodecyl, 3e. The NMR spectra of the adducts were studied in CDCl₃ at ambient temperatures at 300 MHz for proton and 75 MHz for carbon‐13, with full proton assignments achieved by high‐resolution COSY45 spectra for the aryl and the alkyl regions. Slow exchange limit (SEL) spectra were observed for both ¹H and ¹³C spectra showing slow rotation on the NMR timescales of the unsubstituted bridgehead phenyl groups. Endo Diels–Alder adduct stereochemistry was supported by striking magnetic anisotropic shielding effects in the ¹H spectra of the alkyl groups, with the NCH₂ CH ₂ signal of each adduct appearing upfield of tetramethylsilane (TMS) at ca. ?0.32 ppm. Proton NMR spectra for precursor maleamic acids and maleimides are reported, with some solvent effects found (CDCl₃ vs. d ₆‐acetone) for the carbon‐bound HC?CH protons of 4. Ab initio molecular modeling calculations at the Hartree‐Fock level using the 6‐31G* basis set have been performed for two key conformers of the phencyclone adduct of N‐n‐octylmaleimide, as a representative structure for these hindered adducts, to estimate geometric parameters for the adduct. A syn conformer, with the alkyl chain directed into the adduct cavity, was found to be ca. 0.23 kcal/mol lower energy than an anti conformer (in which the alkyl chain was directed away from the phenanthrenoid moiety).     

Multifractality of the Feigenbaum Attractor and Fractional Derivatives

It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities (f(α). This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions [Frisch and Matsumoto, J. Stat. Phys. 108:1181, 2002]. The relation between the thermodynamic approach [Vul, Sinai and Khanin, Russian Math. Surveys 39:1, 1984] and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations.     

Review of quasi-elastic charge-exchange data in the nucleon-deuteron breakup reaction

The available data on the forward charge exchange of nucleons on the deuteron up to 2 GeV/nucleon are reviewed. The value of the inclusive nd → pnn/np → pn cross section ratio is sensitive to the fraction of spin-independent neutron-proton backward scattering. The measurements of the polarisation transfer in d(n, p){nn} or the deuteron analysing power in p{d, {pp})n in high-resolution experiments, where the final nn or pp pair emerge at low excitation energy, depend upon the longitudinal and transverse spin-spin up amplitudes. The relation between these types of experiments is discussed and the results compared with predictions of the impulse approximation model in order to see what new constraints they can bring to the neutron-proton database.