Periodic dynamics of coupled cell networks I: rigid patterns of synchrony and phase relations

It has recently been proved by Golubitsky and coworkers that in any network of coupled dynamical systems, the possible 'rigid' patterns of synchrony of hyperbolic equilibria are determined by purely combinatorial properties of the network, known as 'balanced equivalence relations'. A pattern is 'rigid' if it persists under small 'admissible' perturbations of the differential equation — ones that respect the network structure. We discuss a natural generalisation of these ideas to time-periodic states, and motivate two basic conjectures, the Rigid Synchrony Conjecture and the Rigid Phase Conjecture. These conjectures state that for rigid hyperbolic time-periodic patterns, cells with synchronous dynamics must have synchronous input cells, and cells with phase-related dynamics must have input cells that have the same phase relations. We provide evidence supporting the two conjectures, by proving them for a special class of periodic orbits, which we call 'tame', under strong assumptions on the network architecture and the symmetries of the periodic state. The discussion takes place in the formal setting of coupled cell networks. We prove that rigid patterns of synchrony are balanced, together with the analogous result for rigid patterns of phase relations. The assumption on the network architecture simplifies the geometry of admissible vector fields, while tameness rules out patterns with non-trivial local or multilocal symmetry. The main idea is to perturb an admissible vector field in a way that retains sufficient control over the associated perturbed periodic orbit. We present two techniques for constructing these perturbations, both using a general theorem on groupoid-symmetrisation of vector fields, which has independent interest. In particular we introduce a method of 'patching' that makes local changes to an admissible vector field. Having established these results for all-to-all coupled networks and tame periodic orbits we prove more general versions that require these assumptions only on a suitable quotient network. These conditions are weaker and encompass a larger class of networks and periodic orbits. We give an example to show that rigidity cannot be relaxed to hyperbolicity. We also prove, without any technical assumptions, that rigidly synchronous or phase-related cells must be input-isomorphic, a necessary precondition for the two conjectures to hold.     

Interior symmetry and local bifurcation in coupled cell networks

A coupled cell system is a network of dynamical systems, or 'cells', coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the 'symmetry groupoid', has shown that global group-theoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structure-preserving bijections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: 'interior symmetry'. This concept is closely related to the groupoid structure, but imposes stronger constraints of a group-theoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for 'synchrony-breaking' bifurcations of the Equivariant Branching Lemma for steady-state bifurcation, and the Equivariant Hopf Theorem for bifurcation to time-periodic states.     

Bonding configurational changes associated with α ferrite/γ austenite phase stability in steels due to transition metal alloy elements

Abstract

An explanation of the stability of the α ferrite or γ austenite phase in steels containing transition metal alloying elements that lie either to the left or right hand side of iron in the periodic table (except manganese which has an anomalous behaviour) is usually based on the empirical electron state and thermokinetic considerations evolved by Hume-Rothery and Zener. An alternative interpretation of the phase stability is proposed, based on the moments theorem of Cyrot-Lackman, i.e. an electron theory approach. It is also proposed that if the moments theorem approach can be coupled to thermokinetic aspects of phase transformations, it may be possible to predict some structural properties of steels from first principles.

MST/2066     

Irreversible thermodynamics of nonlocal systems

Global laws of balance of momentum, moment of momentum and energy, together with local conservation of mass are reduced to point statements involving localization residuals. A fundamental functional inequality is then obtained which reduces to the Clausius-Duhem inequality for local theories. For nonlocal theories, the functional inequality states that the total internal production of heat of a material body at any given time is non-negative. This inequality is reformulated in terms of a functional inequality on a Hilbert space of ordered collections of L₂ functions. The general solution of this functional inequality is obtained and this leads to all admissible constitutive relations. The existence of dissipation potentials and symmetry relations are established for material bodies with admissible constitutive relations. Nonlocal analogues of Maxwell's reciprocity relations are also obtained as well as a proof of consistency with the results of thermostatics. Satisfaction of nonlinear forms of Onsager's reciprocity relations are shown to be equivalent to the requirement that the operators generating the admissible constitutive relations be potential operators. It is also shown the certain functionals of curves in a function space are odd under time reversal if and only if the nonlinear form of Onsager's reciprocity relations are satisfied. Thus, Gurtin's results [10] for processes which may be approximated by linear departures from equilibrium are extended to all processes with admissible constitutive relations. Similar results are established for significantly less restrictive sets of histories than those used by Gurtin and for a wide class of generalizations of the time reversal operator on such histories. Indications are given that satisfaction of invariance under superimposed rigid body motions implies satisfaction of the zero mean conditions for all localization residuals.     

Patterns on growing square domains via mode interactions

Numerical simulations of reaction-diffusion systems with Neumann boundary conditions (NBC) on growing square domains by Maini et al. exhibit square and stripe (or roll) patterns that are usually associated with bifurcations from a trivial equilibrium on a square lattice. However, these patterns change as the domain grows. In this article we discuss several of these transitions; namely, transitions between different types of squares and between squares and stripes (or rolls). We show that these transitions can be understood by tracing paths through the unfoldings of certain co-dimension two mode interactions. To understand these transitions, we need to discuss two issues: the speed at which the domain size changes and the relations between NBC and periodic boundary conditions (PBC) on a square. First, in the simulations, the domain growth takes place on a time scale that is longer than the one needed for pattern formation. Therefore, we can assume that the domain growth is identified with quasistatic variation of time; that is, the domain grows slowly enough that the PDE solution of the time-dependent system tracks equilibria of the reaction-diffusion systems posed on a fixed size domain. Second, as is well-known, NBC problems on a square of side length l can be embedded in PBC problems on a square with side length 2l. The PBC problem has translation symmetries that are not present in the NBC problem. These additional symmetries are called hidden symmetries in the literature. Moreover, solutions to PBC that restrict to the smaller square and satisfy NBC are just those solutions that satisfy certain symmetry constraints. We show further that the transitions between different patterns can be understood by considering relevant mode interaction bifurcation problems on the larger square and then restricting to the smaller square. We have found that a generic continuous transition can occur between two types of squares. Also, the transition between squares and stripes can generically occur either via steady states and time-periodic states (standing waves) or via a jump. Interestingly, in mode interactions, the symmetry constraints induced by NBC are important in understanding which solutions exist and which solutions are stable. For example, diagonal stripes cannot occur as a primary branch in the NBC problem but do in the PBC problem. Also, patterns can be stable in the NBC problem that are not stable in the PBC problem. As a consequence, in the NBC problem we see standing wave time-periodic solutions as stable patterns leading to stable stripes, whereas in the PBC problem we see wavy rolls steady states as stable patterns leading to stable stripes. In principle, a classification of all transitions in NBC mode interactions is possible. However, we concentrate only on those transitions that are relevant to the numerically observed transitions.     

A class of universal relations for constrained,isotropic elastic materials

Summary A class of universal relations for all kinematically constrained, isotropic, elastic materials is described by the equationSB=BS relating the symmetric extra stress and the Cauchy-Green deformation tensors. This rule generates easily at most three universal relations for all kinematically admissible deformations of any constrained, isotropic body for which these tensors are nondiagonal. New universal formulae for homogeneous, compressible and incompressible materials reinforced by inextensible fibers in a variety of arrangements are presented for several kinds of homogeneous and nonhomogeneous, controllable universal deformations.     

The Structure of Fraunhofer Diffraction Patterns of Apertures with N-fold Rotational Symmetry

The moment theorem is used to show that the innermost part of the Fraunhofer diffraction pattern of any real aperture with higher than two-fold rotational symmetry is rotationally invariant. Then a formalism is presented in which aperture transmission-functions are represented by series of Zernike circle polynomials and diffracted field-amplitudes by series of Bessel functions, from which it is easily shown that the diffraction patterns of such apertures consist of regions, contained between well-defined values of the radius, whose rotational symmetries are integral multiples of that of the aperture. The central region, extending from = 0 to , N ( measures the diffraction angle, and N is the degree of rotational symmetry of the aperture) is rotationally invariant, and successive circumjacent regions have progressively higher rotational symmetries. The diffraction patterns of sectoral apertures and of rings of pinholes are derived and shown to exemplify these general conclusions. Finally it is shown how the diffraction patterns of some apertures (‘chiral apertures’) with rotational symmetries but no mirror symmetry can be deduced from the diffraction pattern of a related aperture with mirror symmetries, to which a chiral perturbation is applied.     

Multi-cluster dynamics in coupled phase oscillator networks

In this paper, we examine robust clustering behaviour with multiple nontrivial clusters for identically and globally coupled phase oscillators. These systems are such that the dynamics is completely determined by the number of oscillators N and a single scalar function g(?) (the coupling function). Previous work has shown that (a) any clustering can stably appear via choice of a suitable coupling function and (b) open sets of coupling functions can generate heteroclinic network attractors between cluster states of saddle type, though there seem to be no examples where saddles with more than two nontrivial clusters are involved. In this work, we clarify the relationship between the coupling function and the dynamics. We focus on cases where the clusters are inequivalent in the sense of not being related by a temporal symmetry, and demonstrate that there are coupling functions that give robust heteroclinic networks between periodic states involving three or more nontrivial clusters. We consider an example for N = 6 oscillators where the clustering is into three inequivalent clusters. We also discuss some aspects of the bifurcation structure for periodic multi-cluster states and show that the transverse stability of inequivalent clusters can, to a large extent, be varied independently of the tangential stability.     

Analysis of Transient Heat Conduction in a Hollow Sphere Using Duhamel Theorem

In this article, analytical modeling of two-dimensional heat conduction in a hollow sphere is presented. The hollow sphere is subjected to time-dependent periodic boundary conditions at the inner and outer surfaces. The Duhamel theorem is employed to solve the problem where the periodic and time-dependent terms in the boundary conditions are considered. In the analysis, the thermophysical properties of the material are assumed to be isotropic and homogenous. Moreover, the effects of the temperature oscillation frequency, the thickness variation of the hollow sphere, and thermophysical properties of the sphere are studied. The temperature distribution obtained here contains two characteristics, the dimensionless amplitude (A) and the dimensionless phase difference (j{\varphi}). Moreover, the obtained results are shown with respect to Biot and Fourier numbers. Comparison between the present results and the findings from a previous study for a hollow sphere subjected to the reference harmonic state show good agreement.     

Special Cases and Substitutes for Rigid E-Unification

The simultaneous rigid E-unification problem arises naturally in theorem proving with equality. This problem has recently been shown to be undecidable. This raises the question whether simultaneous rigid E-unification can usefully be applied to equality theorem proving. We give some evidence in the affirmative, by presenting a number of common special cases in which a decidable version of this problem suffices for theorem proving with equality. We also present some general decidable methods of a rigid nature that can be used for equality theorem proving and discuss their complexity. Finally, we give a new proof of undecidability of simultaneous rigid E-unification which is based on Post's Correspondence Problem, and has the interesting feature that all the positive equations used are ground equations (that is, contain no variables). Received: March 6, 1997; revised version: August 13, 1999     

Long-Range Ordered Amorphous Atomic Chains as Building Blocks of a Superconducting Quasi-One-Dimensional Crystal

Crystalline and amorphous structures are two of the most common solid-state phases. Crystals having orientational and periodic translation symmetries are usually both short-range and long-range ordered, while amorphous materials have no long-range order. Short-range ordered but long-range disordered materials are generally categorized into amorphous phases. In contrast to the extensively studied crystalline and amorphous phases, the combination of short-range disordered and long-range ordered structures at the atomic level is extremely rare and so far has only been reported for solvated fullerenes under compression. Here, a report on the creation and investigation of a superconducting quasi-1D material with long-range ordered amorphous building blocks is presented. Using a diamond anvil cell, monocrystalline (TaSe₄)₂I is compressed and a system is created where the TaSe₄ atomic chains are in amorphous state without breaking the orientational and periodic translation symmetries of the chain lattice. Strikingly, along with the amorphization of the atomic chains, the insulating (TaSe₄)₂I becomes a superconductor. The data provide critical insight into a new phase of solid-state materials. The findings demonstrate a first ever case where superconductivity is hosted by a lattice with periodic but amorphous constituent atomic chains.     

Algebraic solution of master equations in quantum optics

The techniques in thermo field dynamics are applied to solve a certain class of master equations associated with su(1,1)?×?su(1,1) symmetries exactly by an algebraic approach using disentanglement theorem.     

Pseudospectral method for assessing stability robustness for linear time-periodic delayed dynamical systems

The article presents a pseudospectral approach to assess the stability robustness of linear time-periodic delay systems, where periodic functions potentially present discontinuities and the delays may also periodically vary in time. The considered systems are subject to linear real-valued time-periodic uncertainties affecting the coefficient matrices, and the presented method is able to fully exploit structure and potential interdependencies among the uncertainties. The assessment of robustness relies on the computation of the pseudospectral radius of the monodromy operator, namely, the largest Floquet multiplier that the system can attain within a given range of perturbations. Instrumental to the adopted novel approach, a solver for the computation of Floquet multipliers is introduced, which results into the solution of a generalized eigenvalue problem which is linear w.r.t. (samples of) the original system matrices. We provide numerical simulations for popular applications modeled by time-periodic delay systems, such as the inverted pendulum subject to an act-and-wait controller, a single-degree-of-freedom milling model and a turning operation with spindle speed variation.     

Analytic parametric solutions for the HRR nonlinear elastic field with low hardening exponents

Summary In this paper, we restore the already constructed approximate asymptotic solutions extracted in [10] concerning the HRR [1] strongly nonlinear fourth-order ordinary differential equation (ODE) for plane strain conditions in nonlinear elastic (plastic) fracture. It is proved that the above equation, for low strain hardening exponents (0 < N ? 1), is reduced to a strongly nonlinear ODE of the second order. The method of the total differentials is used so that the last equation is reduced to Abels' equations of the second kind of the normal form, that can be analytically solved in parametric form. In addition, the case of rigid perfect-plasticity (N=0) is extensively investigated and several important results are extracted.     

THz Wave Emission from Intrinsic Josephson Junctions Controlled by Surface Impedance

Recently terahertz wave emission from intrinsic Josephson junctions without external magnetic fields has been intensively studied, and some emission states have been proposed numerically or theoretically. For the surface impedance Z=1, the McCumber-like state with small spatial dependence of the electric field in the junction becomes stable, while for large and complex Z, the π-phase kink state characterized by translational symmetry breaking along the layered structure becomes stable. In the present study, the relations between these two extreme cases are clarified numerically by solving the coupled equations of the Josephson relations and the Maxwell equations for an experimental width of the junction, 86 μm. The dynamical phase diagram in the surface impedance (Z)–current (J) plane and the optimal value of Z for the strongest emission are evaluated.     

A digitally programmable multitone generator for coherent signaldetection measurement systems

A digitally programmable multitone generator is presented. The various output signals can be of any shape but periodic, and show absolutely stable phase relations. The proposed generator is useful in modern measurement systems such as in homodyne vector network analyzers or in other coherent measurement systems. It might be realized as a two-chip integrated circuit, one chip for logic and one read-only memory     

Interacting loop-current model of superconducting networks

We review our recent approximation scheme to calculate the normal-superconducting phase boundary, T _c(H), of a superconducting wire network in a magnetic field in terms of interacting loop currents. The theory is based on the London approximation of the linearized Ginzburg-Landau equation. An approximate general formula is derived for any two-dimensional space-filling lattice comprising tiles of two shapes. We provide many examples illustrating the use of this method with a particular emphasis on the fluxoid distribution. In addition to periodic lattices, we also discuss quasiperiodic lattices and fractal Sierpinski gaskets.     

Uniqueness and reciprocal theorems in linear micropolar electro-magnetic thermoelasticity with two relaxation times

A general model for the linear micropolar electro-magnetic thermoelastic continuum based on the hyperbolic heat equation, which is physically more relevant than the classical thermoelasticity theory in analyzing problems involving very short intervals of time and/or very high heat fluxes, is introduced. An integral identity that involves two admissible processes at different instants is established. Uniqueness theorem is proved, with no definiteness assumption on the elastic constitutive coefficients and no restrictions on the electro-elastic coupling moduli, magneto-elastic coupling moduli, and thermal coupling coefficients other than symmetry conditions. The reciprocity theorem is derived, without the use of Laplace transforms. The integral representation formula is obtained in case instantaneous concentrated, time-continuous or time-harmonic loads are applied. The Maysel’s, Somigliana’s and Green’s formulas are derived. The mixed boundary value problem is considered and a system of five singular Fredholm integral equations is obtained. The results for dynamic classical coupled theory can be easy deduced from the given general model formulated for the temperature-rate dependent thermoelasticity.