Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations

Two infinite sequences of orbits leading to turbulence in a five-mode truncation of the Navier-Stokes equations for a 2-dimensional incompressible fluid on a torus are studied in detail. Their compatibility with Feigenbaum's theory of universality in certain infinite sequences of bifurcations is verified and some considerations on their asymptotic behavior are inferred. An analysis of the Poincaré map is performed, showing how the turbulent behavior is approached gradually when, with increasing Reynolds number, no stable fixed point or periodic orbit is present and all the unstable ones become more and more unstable, in close analogy with the Lorenz model.     

On the fundamentals of extended thermodynamics (ET) of a one-dimensional rarefied gas

Extended thermodynamics (ET) of degreer for a one-dimensional rarefied gas based, by definition, on a finite set A^r={a⁰, a²,..., a^r} of the firstr–1(3r) direct internal moments of the one-point distribution functionf is carefully investigated. With the aid of the second axiom of thermodynamics, the new representation forf, depending in a local and nonlinear way onA ^r , is explicitly derived. It is demonstrated that in ET of degreer an infinite sequence {b^r+1, b^r+2,...} ofhigher order Hermite coefficients, which normally drops out of Grad's proposition forf fashioned by mathematical apparatus such as the Hermite polynomials, cannot be considered negligible in the case when nonlinear constitutive functions are established. Using Ma's kinetic equation corresponding to a one-dimensional rarefied gas as well as the generalized representation forf, collision productions in the nonconservative moment equations are then calculated for a special choice of the rate of collisions between particles.     

Bifurcations of circle maps: Arnol'd tongues,bistability and rotation intervals

We study the bifurcations of two parameter families of circle maps that are similar tof _b,w (x)=x+w+(b/2) sin (2x) (mod1). The bifurcation diagram is constructed in terms of setsT _r , whereT _r is the set of parameter values (b, w) for whichf _{b, w} has an orbit with rotation numberr. We show that the known structure whenb<1 (forr rational,T _r is an Arnol'd tongue and forr irrational, it is an arc) extends nicely into the regionb>1, wheref _{b, w} is no longer injective and can have an interval of rotation numbers. Specifically, the tongues overlap in a uniform, monotonic manner and forr irrational,T _r opens into a tongue. Our other theorems give information about the dynamics off _{b, w} (e.g., bistability or aperiodicity) for (b, w) in various subsets of parameter space.     

Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.     

Oscillatory convection and chaos in a Lorenz-type model of a rotating fluid

A four-mode model of convection in a rotating fluid layer is studied. The model is an extension of the Lorenz model of Rayleigh-Bénard convection, the extra mode accounting for the regeneration of vorticity by rotation. Perturbation theory is applied to show that the Hopf bifurcations from conductive and steady convective solutions can be either supercritical or subcritical. Perturbation theory is also used at large Rayleigh numbersr to predict novel behavior. Supercritical oscillatory convection of finite amplitude is found by numerical integration of the governing equations. The general picture is of a series of oscillatory solutions stable over larger intervals, interspersed by short bursts of chaos.     

Universal behaviour in families of area-preserving maps

We have investigated numerically the behaviour, as a perturbation parameter is varied, of periodic orbits of some reversible area-preserving maps of the plane. Typically, an initially stable periodic orbit loses its stability at some parameter value and gives birth to a stable orbit of twice the period. An infinite sequence of such bifurcations is accomplished in a finite parameter range. This period-doubling sequence has a universal limiting behaviour: the intervals in parameter between successive bifurcations tend to a geometric progression with a ratio of $\text{1}\text{δ}\text{=}\text{1}\text{8.721097200}$…, and when examined in the proper coordinates, the pattern of periodic points reproduces itself, asymptotically, from one bifurcation to the next when the scale is expanded by α = ?4.018076704… in one direction, and by β = 16.363896879… in another. Indeed, the whole map, including its dependence on the parameter, reproduces itself on squaring and rescaling by the three factors α, β and δ above. In the limit we obtain a universal one-parameter, area-preserving map of the plane. The period-doubling sequence is found to be connected with the destruction of closed invariant curves, leading to irregular motion almost everywhere in a neighbourhood.     

Symmetry breaking on a model of five-mode truncated Navier-Stokes equations

A symmetryless model of nonlinear first-order differential equations, obtained by perturbing a known model of five-mode truncated Navier-Stokes equations, is studied. Some interesting phenomena, such as the existence of an infinite sequence of bifurcations in a very narrow range of the parameter and the simultaneous presence of a strange attractor either with two fixed attracting points or with a periodic attracting orbit, are shown. Furthermore, two new sequences of period doubling bifurcations are found in the unperturbed model.     

Lyapunov number for a noisy 2 n cycle

A stochastic one-dimensional map which produces a sequence of period doubling bifurcations is theoretically studied. We obtain analytic expressions, to a second-order approximation, of the local distribution function of fluctuating orbital points and the Lyapunov number for a noisy 2 ⁿ cycle. The expressions satisfy scaling laws and well agree with the results of numerical experiments when the external noise is weak. A scaling factor for the noise level is formulated in terms of the derivatives of a deterministic map. From it, the scaling factor is refined to be 6.6190 .... The Lyapunov number shows that, when the external noise is weaker than some extent, the noisy orbit is more stable rather than the deterministic one.     

Frei zerfallende Magnetfelder von Kugelschalen

The decay constants of free decaying magnetic fields of sphere shells are calculated. In the regionr ₁<r<r ₀ the electrical conductivity isσ. For different values of the parameterγ=r ₀/(r ₀-r ₁) the following models are considered: The conductivity is zero forr>r ₀ andr<r ₁ (model 1). The conductivity is zero forr>r ₀, and forr<r ₁ the magnetic field is excluded by infinite conductivity (model 2). The magnetic field is excluded forr>r ₀ andr<r ₁ (model 3).     

一类可变禁区的不连续系统的加周期分岔

研究了一类可变禁区不连续系统的加周期分岔行为, 发现由可变禁区导致不同类型的加周期分岔. 研究表明, 系统的迭代轨道和禁区的上下两个边界均可发生边界碰撞, 从而产生加周期分岔. 基于边界碰撞分岔理论, 定义基本的迭代单元, 解析推导出了相应的分岔曲线, 在全参数空间中给出了不同加周期所出现的范围. 与数值模拟结果比较, 理论分析结果与数值结果高度一致.     

Probabilistic bootstrap percolation

In bootstrap percolation, sites are occupied with probabilityp, but those with less thanm occupied first neighbors are removed. This culling process is repeated until a stable configuration (all occupied sites have at leastm occupied first neighbors or the whole lattice is empty) is achieved. Formm ₁ the transition is first order, while form<m ₁ it is second order, withm-dependent exponents. In probabilistic bootstrap percolation, sites have probabilityr or (1–r) of beingm- orm-sites, respectively (m-sites are those which need at leastm occupied first neighbors to remain occupied). We have studied the model on Bethe lattices, where an exact solution is available. Form=2 andm=3, the transition changes from second to first order atr ₁=1/2, and the exponent is different forr<1/2,r=1/2, andr>1/2. The same qualitative behavior is found form=1 andm=3. On the other hand, form=1 andm=2 the transition is always second order, with the same exponents ofm=1, for any value ofr>0. We found, form=z–1 andm=z, wherez is the coordination number of the lattice, thatp _c=1 for a value ofr which depends onz, but is always above zero. Finally, we argue that, for bootstrap percolation on real lattices, the exponents and form=2 andm=1 are equal, for dimensions below 6.On leave from Universidade Federal de Santa Catarina, Depto. de Fisica, 88049, Florianópolis, SC, Brazil     

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.     

Between Poisson and GUE Statistics: Role of the Breit–Wigner Width

We consider the spectral statistics of the superposition of a random diagonal matrix and a GUE matrix. By means of two alternative superanalytic approaches, the coset method and the graded eigenvalue method, we derive the two-level correlation functionX₂(r) and the number varianceΣ²(r). The graded eigenvalue approach leads to an expression forX₂(r) which is valid for all values of the parameterλgoverning the strength of the GUE admixture on the unfolded scale. A new twofold integration representation is found which can be easily evaluated numerically. Forλ?1 the Breit–Wigner widthΓ₁measured in units of the mean level spacingDis much larger than unity. In this limit, closed analytical expressions forX₂(r) andΣ²(r) can be derived by (i) evaluating the double integral perturbatively or (ii) anab initioperturbative calculation employing the coset method. The instructive comparison between both approaches reveals that random fluctuations ofΓ₁manifest themselves in modifications of the spectral statistics. The energy scale which determines the deviation of the statistical properties from GUE behavior is given by. This is rigorously shown and discussed in great detail. The Breit–WignerΓ₁width itself governs the approach to the Poisson limit forr→∞. Our analytical findings are confirmed by numerical simulations of an ensemble of 500×500 matrices, which demonstrate the universal validity of our results after proper unfolding.     

Exact results for a dilute Potts model

Exact results are obtained for the annealed, dilute,q-component Potts model on the decorated square lattice. The phase diagram is found to consist of a high-temperature region, a low-temperature region, and a two-phase region in between which arises only forq>4: exact expressions for the phase boundary and the critical probability are derived. At the critical point the specific heat is generally finite and has a cusp; the slope of the cusp is finite forq=4 and infinite (vertical) forq=2 and 3.Work supported in part by NSF Grant No. DMR 78-18808.     

对均匀的数学描述及其与混沌的关系

基于独占球的概念定义的瞬时混沌度和k步混沌强度是混沌轨道的稳定特征,应用独占球的概念定义了均匀度,它对均匀性的描述与人对均匀的理解非常符合.被含均匀度是一个过渡概念,它与均匀度非常相似,但有更好的数学性质,对于随机轨道,被含均匀度统计收敛于1/V_n（1）（V_n（1）是n维欧氏空间的单位球体积）,而当轨道上的点充分多时,均匀度与被含均匀度近似相等.只要适当选择包含动力系统吸引盆的多 关键词： 独占球 均匀度 混沌 瞬时混沌强度     

Convexity of internal energy of cubic crystal deformed to orthorhombic structure

A generalized set of strain variablesq _r ^N , has been defined to develop the expression for a generalized set of second order and third-order elastic moduliC _rs ^N andC _rst ^N for a cubic crystal deformed to orthorhombic structure. The HessainC _rs ^N δq_rδq_s andC _rst ^N δq_rδq_sδq_t (r=1, 2……6; summation convention) are calculated in the new variables and compared withG-strength andS-strength, for both positive and negative loading environment. The convexity of the internal energy relative to various choice of strain measure is examined considering up to third degree terms in the internal energy expression. The computational results forbcc iron is presented according to the new moduli. The stable ranges thus obtained for iron under hydrostatic compressive and tensile stresses is found to generate the classical stable range, green-stable range and stretch-stable range as the specific cases. However,bcc iron does not seem to follow any conventional stable ranges under hydrostatic compression, where the present generalized stable range is found satisfactory.     

In search of the Griffiths shield region

The van der Waals equation of state for binary mixtures has been used to determine the location and shape of the Griffiths shield region (where three tricritical lines intersect). If one takes the geometric mean fora ₁₂, the arithmetic mean forb ₁₂, and the configurational free energy as ideal, the center of the Griffiths shield region is found only when the ratio of molecular sizes is infinite. When the Flory equation for the configurational free energy for mixtures of chain molecules is substituted for the ideal form, the results appear to be somewhat different. However, for all the cases studied, with systems which approach geometric mean behavior one finds the shield region only when the ratio of molecular size is very large and when the internal pressure of the small molecule is very much greater than that of the long-chain molecule.This paper is dedicated to our colleague Howard Reiss on the occasion of his 66th birthday.     

Ferromagnetism of the electron gas

The ground-state energy of the ferromagnetic electron gas is calculated for the relative polarizationζ=0−1 and the interelectron separationr _s =5−12. The method consists in describing the electron gas approximately by a quadratic boson Hamiltonian, and contains the random-phase approximation as a special case. Numerical studies show that in both the random-phase and the present approximations the paramagnetic state has the lowest energy: the energy increases withζ for all values ofr _s considered. In the present approximation instabilities are found to occur forr _s above a critical value, due to exchange processes of finite momentum transfers. Forζ=0 this critical value ofr _s is 9.4; it decreases with increasingζ. However, the fully-polarized state (ζ=1), which lies above the rest, is always stable. The conclusions are as follows: (1) Forr _s <9.4 the electron gas is paramagnetic. (2) Atr _s =9.4 it goes over to the fully-polarized ferromagnetic state. (3) This phase transition requires an energy absorption of 0.03 rydberg per electron. (4) The fully-polarized state is not obtainable as the limitζ→1.     

The dynamics of coupled nonlinear model Boltzmann equations

A multispecies gas described by coupled nonlinear Boltzmann equations is studied as a dynamical system. Properties are determined of theN coupled nonlinear ODEs for the number densities obtained from the Boltzmann equations for the spatially uniform system ofN species undergoing binary scattering, removal, and regeneration in the presence of an external force field and a reservoir of background gas. The physically realizable setQ, the nonnegative cone in theN-dimensional phase space of species number densities, is established as invariant under the flow. The fixed-point equations for the ODEs are shown to be equivalent to 2 ^N linear systems, and conditions for the stability and instability of the fixed points are then established. Stable fixed points are demonstrated to exist inQ by showing that they enter via a sequence of transcritical bifurcations as physical parameters are varied. For the two-species case the typical global structure of the solutions is established. Various particular cases are described including one which possesses an infinite family of periodic solutions and one that depends delicately upon initial conditions due to a separatrix that separatesQ into two invariant sets.