Numerical Analysis of the Synchronization of Coupled Chemical Oscillators

A reaction–diffusion model describing a system of coupled oscillators is constructed and investigated. The oscillators in this study are chemical oscillators that represent an oscillatory heterogeneous catalytic reaction in a granular catalyst layer. The oscillators are arranged serially in the reagent stream and are coupled through the gaseous phase. The dynamic behavior of the system is investigated as a function of the main external parameter — the partial pressure of one of the reagents in the gaseous phase. Existence regions of regular and chaotic oscillations are identified. Synchronization conditions are established for the oscillations in such a chain of coupled chemical oscillators.     

Unfolding the torus: Oscillator geometry from time delays

Summary We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.     

Phase-locking for dynamical systems on the torus and perturbation theory for mathieu-type problems

Summary When several oscillators are coupled together and the parameters of their coupling are varied, the oscillators pass through so-called phase-locked regimes. In physical terms this means that the oscillators tend to synchronize their motion. To describe this phenomenon, we frame the concepts ofpartial phase andphase-locking. A partial phase of a toral flow puts emphasis on how orbits of the flow drift around the torus in some fixed direction. The partial phase is locked if it grows in time along some orbit slower than any linear function. When a toral flow is given by a trigonometric polynomial, its phase-locked regions are quite narrow. With the coupling amplitude increasing, each region grows in width as some power of the amplitude. That power can be calculated in terms of both the partial phase and degree of the trigonometric polynomial.     

Heteroclinic Ratchets in Networks of Coupled Oscillators

We analyze an example system of four coupled phase oscillators and discover a novel phenomenon that we call a “heteroclinic ratchet”; a particular type of robust heteroclinic network on a torus where connections wind in only one direction. The coupling structure has only one symmetry, but there are a number of invariant subspaces and degenerate bifurcations forced by the coupling structure, and we investigate these. We show that the system can have a robust attracting heteroclinic network that responds to a specific detuning Δ between certain pairs of oscillators by a breaking of phase locking for arbitrary Δ>0 but not for Δ≤0. Similarly, arbitrary small noise results in asymmetric desynchronization of certain pairs of oscillators, where particular oscillators have always larger frequency after the loss of synchronization. We call this heteroclinic network a heteroclinic ratchet because of its resemblance to a mechanical ratchet in terms of its dynamical consequences. We show that the existence of heteroclinic ratchets does not depend on symmetry or number of oscillators but depends on the specific connection structure of the coupled system.     

A Study of a Sequence of Classical OrthogonalPolynomials of Dimension 2

We construct a sequence ofd-dimensional classical orthogonalpolynomials (d2) that generalize the Gegenbauer polynomials.The cased=2 is fully studied.     

Frustrated Magnetic Systems and the q-Oscillator

We analyze the spherical model with frustration induced by an external gauge field. The case of the infinite-dimensional model has recently been reduced to a problem of q-deformed oscillators with q parametrizing the amount of frustration. We find a complete analytic solution of the model by using a convenient representation of the q-oscillator algebra, the q-Hermite polynomials. The low-temperature phase does not exhibit a glassy behavior. With respect to the usual unfrustrated spherical model, the effect of frustration is only quantitative. A glassy low-temperature phase is expected for the more complicated XY model whose study is in progress. Bibliography: 15 titles.     

多重相邻交联振荡器系统的停振

本文分析多重相邻交联振荡器链的停振现象，构造子多重交联振荡器链的上解与下解，比较了最邻近交联与多重交联的两种不同情况对停振的影响。     

Geometrical properties of coupled oscillators at synchronization

We study the synchronization of N nearest neighbors coupled oscillators in a ring. We derive an analytic form for the phase difference among neighboring oscillators which shows the dependency on the periodic boundary conditions. At synchronization, we find two distinct quantities which characterize four of the oscillators, two pairs of nearest neighbors, which are at the border of the clusters before total synchronization occurs. These oscillators are responsible for the saddle node bifurcation, of which only two of them have a phase-lock of phase difference equals ± π/2. Using these properties we build a technique based on geometric properties and numerical observations to arrive to an exact analytic expression for the coupling strength at full synchronization and determine the two oscillators that have a phase-lock condition of ± π/2.     

Asymptotic Localization of Energy in Nondisordered Oscillator Chains

We study two popular one‐dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete nonlinear Schrödinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter ? > 0, is small. We rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant ?, and we provide strong evidence that it decays faster than any power law in ? as ? → 0. The weak coupling regime also translates into a high‐temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature. To our knowledge, it is the first time that a clear connection has been established between KAM‐like phenomena and thermal conductivity. © 2015 Wiley Periodicals, Inc.     

Nonuniqueness for a bounded, differentiable complete orthonormal system inL 2[0, 1]

The argument of Müsegian and Ovsepjan is adapted to produce a complete orthonormal system on [0, 1] of uniformly bounded functions, differentiable on [0, 1], andC ^∞ on [0, 1], for which the analogue of Cantor's uniqueness theorem is false. We also construct a complete orthonormal system ofC ^∞ functions which vanish to infinite order at both endpoints.     

Asymptotics of the density matrix for a system of a large number of identical particles

The Wigner equation is considered for a system of a large numberN of identical particles with interaction factor of the order of 1/N. In both the Bose and the Fermi cases, we construct the asymptotics of the solution of the Cauchy problem for this equation with regard to the exchange effect for the case in which the Planck constant is of the order ofN ^−1/d , whered is the space dimension. This asymptotics is interpreted in terms of the theory of the complex germ on a curved phase space equivalent to the space of functions with values on the Riemann sphere in the Fermi case and on the Lobachevskii plane in the Bose case. The classical equations of motion in both cases are reduced to the Vlasov equation; since the phase space is infinite-dimensional, the complex germ is subjected to additional conditions depending on the type of statistics. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 84–106, January, 1999.     

Asymptotic stabilization of a class of bilinear systems by a variable structure feedback

We consider the problem of stabilization of a homogeneous bilinear system at zero. We assume that the system can be reduced to a form that admits feedback linearization at all points of the phase space outside a set N of measure zero. For such systems, we construct a variable structure feedback solving the stabilization problem under the condition that N is not an invariant set of the closed system.     

Unit squares intersecting all secants of a square

LetS be a square of side lengths>0. We construct, for any sufficiently larges, a set of less than 1.994s closed unit squares whose sides are parallel to those ofS such that any straight line intersectingS intersects at least one square ofS. It disproves L. Fejes Tóth's conjecture that, for integrals, there is no such configuration of less than 2s−1 unit squares. Supported by “Deutsche Forschungsgemeinschaft”, Grant We 1265/2-1.     

Optimal construction of a selection of a subpopulation

Summary The problem of selecting a subpopulation from a given populationII is to be, on the basis of measurements of members ofII, achieved by choosing those members ofII who satisfy the standards determined by a given selection cirterion and rejecting those who do not. Since the optimum selection depends on the unknown parameter of the probability distribution ofII, it is here considered how to construct a decision function from the space of subsidiary sample having infor-mation on θ to the space of selections. Thus the existence of Bayes and minimax decision functions under the constraint defined by the selection criterion is proved. A necessary and sufficient condition for a decision function satisfying the constraint to be a Bayes decision function is also obtained. The Institute of Statistical Mathematics     

On the regulation number of a multigraph

The regulation number of a multigraphG having maximum degreed is the minimum number of additional vertices that are required to construct ad-regular supermultigraph ofG. It is shown that the regulation number of any multigraph is at most 3. The regulation number of a multidigraph is defined analogously and is shown never to exceed 2. A multigraphG has strengthm if every two distinct vertices ofG are joined by at mostm parallel edges. For a multigraphG of strengthm and maximum degreed, them-regulation number ofG is the minimum number of additional vertices that are required to construct ad-regular supermultigraph ofG having strengthm. A sharp upper bound on the 2-regulation number of a multigraph is shown to be (d+5)/2, and a conjecture for generalm is presented. Research supported by a Western Michigan University faculty research fellowship. Research Professor of Electrical Engineering and Computer Science, Stevens Institute, Hoboken, NJ and Visiting Scholar, Courant Institute, New York University, Spring 1984. Research supported in part by a Western Michigan University research assistantship from the Graduate College and the College of Arts and Sciences.     

On resonances in systems of two weakly coupled oscillators

Assuming that two weakly coupled oscillators are essentially nonlinear we construct the most suitable form of a shortened 3-dimensional system which describes behavior of solutions inside non-degenerate resonance zones. We analyze a model system of that kind and establish the existence of limit cycles of different types and also the existence of nonregular attractors which are explained by the existence of saddle-focus loops.      

The structure of semiclassical asymptotic expansions of antisymmetric solutions of the stationary Schrödinger equation

We consider the semiclassical asymptotics of eigenfunctions for the Hamiltonian of a quantum-mechanical system ofN identical fermions withd degrees of freedom without spin interaction. In the one-dimensional case (d=1), examples are known in which the ground antisymmetric state in the semiclassical limit is the product ofN(N−1)/2 two-particle wave functions. We construct a nontrivial generalization of this property ford>1. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 257–269, February, 2000.     

A family of neighborly polytopes

Ad-polytopeP is said to be neighborly provided each [d/2] vertices determine a face ofP. We construct a family ofd-polytopes that are dual to neighborly polytopes by means of facet splitting. We use this family to find a lower bound on the number of combinatorial types of neighborly polytopes. We also show that all members of this family satisfy the famous Hirsch conjecture. Research supported by NSF Grant # MCS-07466.     

The conjugate gradient method for computing all the extremal stationary probability vectors of a stochastic matrix

Summary The conjugate gradient method is developed for computing stationary probability vectors of a large sparse stochastic matrixP, which often arises in the analysis of queueing system. When unit vectors are chosen as the initial vectors, the iterative method generates all the extremal probability vectors of the convex set formed by all the stationary probability vectors ofP, which are expressed in terms of the Moore-Penrose inverse of the matrix (P−I). A numerical method is given also for classifying the states of the Markov chain defined byP. One particular advantage of this method is to handle a very large scale problem without resorting to any special form ofP. The Institute of Statistical Mathematics     

Resonances for perturbations of a semiclassical periodic Schrödinger operator

In the semi-classical regime we study the resonances of the operatorP _t =h ²Δ+V+t·δV in some small neighborhood of the first spectral band ofP ₀. HereV is a periodic potential, δV a compactly supported potential andt a small coupling constant. We construct a meromorphic multivalued continuation of the resolvent ofP _t , and define the resonances to be the poles of this continuation. We compute these resonances and study the way they turn into eigenvalues whent crosses a certain threshold.