Bounds on the unstable eigenvalue for period doubling

Bounds are given for the unstable eigenvalue of the period-doubling operator for unimodal maps of the interval. These bounds hold for all types of behaviour |x| ^r of the interval map near its critical point. They are obtained by finding cones in function space which are invariant under the tangent map to the doubling operator at its fixed point.     

Clustering of exponentially separating trajectories

It might be expected that trajectories of a dynamical system which has no negative Lyapunov exponent (implying exponential growth of small separations) will not cluster together. However, clustering can occur such that the density ρ(Δx) of trajectories within distance |Δx| of a reference trajectory has a power-law divergence, so that ρ(Δx) ∼ |Δx| ^−β when |Δx| is sufficiently small, for some 0 < β < 1. We demonstrate this effect using a random map in one dimension. We find no evidence for this effect in the chaotic logistic map, and argue that the effect is harder to observe in deterministic maps.     

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

The XY model and Kadanoff's variational real space renormalization group

The XY model in d dimensions is studied by means of a variational real space renormalization group transformation. Contrary to an earlier computation in the same framework for the d = 2 case, we find that a low order operator basis truncation is highly unstable. For certain values of the variational parameter p the renormalization group flow can display period doubling sequences towards a chaotic regime. The behavior in the d = 3 case is very similar.     

The Critical Renormalization Fixed Point for Commuting Pairs of Area-Preserving Maps

We prove the existence of the critical fixed point (F, G) for MacKay’s renormalization operator for pairs of maps of the plane. The maps F and G commute, are area-preserving, reversible, real analytic, and they satisfy a twist condition.     

Bulk singularities at critical end points: a field-theory analysis

A class of continuum models with a critical end point is considered whose Hamiltonian [φ,ψ] involves two densities: a primary order-parameter field, φ, and a secondary (noncritical) one, ψ. Field-theoretic methods (renormalization group results in conjunction with functional methods) are used to give a systematic derivation of singularities occurring at critical end points. Specifically, the thermal singularity ∼ | t|^{2 - α} of the first-order line on which the disordered or ordered phase coexists with the noncritical spectator phase, and the coexistence singularity ∼ | t|^{1 - α} or ∼ | t|^β of the secondary density <ψ> are derived. It is clarified how the renormalization group (RG) scenario found in position-space RG calculations, in which the critical end point and the critical line are mapped onto two separate fixed points _CEP ^* and _λ ^*, translates into field theory. The critical RG eigenexponents of _CEP ^* and _λ ^* are shown to match. _CEP ^* is demonstrated to have a discontinuity eigenperturbation (with eigenvalue y = d), tangent to the unstable trajectory that emanates from _CEP ^* and leads to _λ ^*. The nature and origin of this eigenperturbation as well as the role redundant operators play are elucidated. The results validate that the critical behavior at the end point is the same as on the critical line. Received 18 January 2001     

Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in ℝ ^k , called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in ℝ ^k . We also discuss a simple signaling pathway related to cancer research, called p53 module.     

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.     

Bounds on the Unstable Eigenvalue for the Asymmetric Renormalization Operator for Period Doubling

We establish rigorous bounds for the unstable eigenvalue of the period-doubling renormalization operator for asymmetric unimodal maps. Herglotz-function techniques and cone invariance ideas are used. Our result generalizes an established result for conventional period doubling.This research is supported by The Leverhulme Trust (grant number F/00144/W).     

A purely algebraic construction of a gauge and renormalization group invariant scalar glueball operator

This paper presents a complete algebraic proof of the renormalizability of the gauge invariant d=4 operator F _{μ ν} ²(x) to all orders of perturbation theory in pure Yang–Mills gauge theory, whereby working in the Landau gauge. This renormalization is far from being trivial as mixing occurs with other d=4 gauge variant operators, which we identify explicitly. We determine the mixing matrix Z to all orders in perturbation theory by using only algebraic arguments and consequently we can uncover a renormalization group invariant by using the anomalous dimension matrix Γ derived from Z. We also present a future plan for calculating the mass of the lightest scalar glueball with the help of the framework we have set up.     

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.     

Period doubling bifurcations for families of maps on ℝ n

Infinite sequences of period doubling bifurcations in one-parameter families (1-pf) of maps enjoy very strong universality properties: This is known numerically in a multitude of cases and has been shown rigorously for certain 1-pf of maps on the interval. These bifurcations occur in 1-pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 /4.6692 ....In this paper we indicate the main steps of a proof that the same is true for 1-pf of analytic maps from ⁿ to ⁿ , whose restriction to ⁿ is real.Work supported by the Fonds National Suisse, and by the National Science Foundation under Grant PHY-79-16812.     

Self-interacting one-dimensional oscillators

Energy eigenvalues and 〈x ²〉 _n for the oscillators having potential energyV(x)=(ω ² x ²/2)+λ<x ^2r >x ^2s have been determined for various values ofλ, r, s andn using renormalized hypervirial-Padé scheme. In general, the results show an improvement over the findings of earlier workers. Variation of the evaluated quantities and of the renormalization parameter withλ, r, s andn has been discussed. In addition, this potential has been employed as an illustrative example of the applicability of alternative formalism of perturbation theory developed by Kim and Sukhatme (J. Phys. A25 647 (1992)).     

Baryon magnetic moments in a quark model

The magnetic moments of uncharmed and charmed baryons are considered to arise through single-quark and two-quark transitions in a quark model. The magnetic moment operator is taken to transform as:T _β ^α ˜aT ₁ ¹ , +bT ₂ ² +cT ₃ ³ +dT ₄ ⁴ , whereT _β ^α are members of SU(4)20′-plet. The assumption, that the magnetic moment operator obtains contribution from the single and two-quark transitions, yields good results for the magnetic moment values of uncharmed baryons. Magnetic moments of charmed baryons can be expressed in terms of one parameter.     

Generic Instability of Spatial Unfoldings¶of Almost Homoclinic Periodic Orbits

We consider spatially homogeneous time periodic solutions of general partial differential equations. We prove that, when such a solution is close enough to a homoclinic orbit or a homoclinic bifurcation for the differential equation governing the spatially homogeneous solutions of the PDE, then it is generically unstable with respect to large wavelength perturbations. Moreover, the instability is of one of the two following types: either the well-known Kuramoto phase instability, corresponding to a Floquet multiplier becoming larger than 1, or a fundamentally different kind of instability, occurring with a period doubling at an intrinsic finite wavelength, and corresponding to a Floquet multiplier becoming smaller than −1. Received: 5 December 1999 / Accepted: 31 July 2000     

Renormalization group functions of the φ<Superscript>4</Superscript> theory from high-temperature expansions

It has been previously shown that calculation of the renormalization group (RG) functions of scalar ϕ⁴ theory reduces to analysis of thermodynamic properties of the Ising model. Using high-temperature expansions for the latter, RG functions of the four-dimensional theory can be calculated for arbitrary coupling constant g with an accuracy of 10⁻⁴ for the Gell-Mann-Low function β(g) and with an accuracy of 10⁻³–10⁻² for anomalous dimensions. The expansions of the renormalization group functions up to the 13th order in g ^−1/2 have been obtained.     

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.     

Criticality of the mean-field spin-boson model: boson state truncation and its scaling analysis

The spin-boson model has nontrivial quantum phase transitions at zero temperature induced by the spin-boson coupling. The bosonic numerical renormalization group (BNRG) study of the critical exponents β and δ of this model is hampered by the effects of boson Hilbert space truncation. Here we analyze the mean-field spin boson model to figure out the scaling behavior of magnetization under the cutoff of boson states N _b . We find that the truncation is a strong relevant operator with respect to the Gaussian fixed point in 0 < s < 1/2 and incurs the deviation of the exponents from the classical values. The magnetization at zero bias near the critical point is described by a generalized homogeneous function (GHF) of two variables τ = α – α _c and x = 1/N _b . The universal function has a double-power form and the powers are obtained analytically as well as numerically. Similarly, m(α = α _c ) is found to be a GHF of ϵ and x. In the regime s > 1/2, the truncation produces no effect. Implications of these findings to the BNRG study are discussed.     

An exactly solvable linear model giving indication of stochastic resonance

The linear stochastic equation dx _β /dt+[1+f _β (t)]x _β (t)=A sin (Ωt) is discussed. The functionƒ _β (t) is defined as a Poissonian noise dependent on a parameterβ>0,ƒ _β (t)=β Σ _j [δ(t − t _j ⁺ ) −δ (t − t _j ⁻ )]. The mean frequency of the delta-pulses is chosen asβ-dependent in the formλ(β)=2γ(β ⁻² + 1) exp(−β) whereγ is a constant from the interval (0, 0.974). With the stochastic functionƒ _β (t) defined in this way, attention is paid on the oscillational term of the averaged function 〈x(t)〉, 〈x(t)〉_osc=Āsin(Ωt − α). It is found that the dependenceĀ=Ā(β) exhibits one maximum and one minimum. The occurrence of these extrema seems to affirm the presence of stochastic resonance. This work has been supported by the Slovak Grant Agency VEGA under contract No. 1/4319/97.     

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.