Periodic dynamics of coupled cell networks II: cyclic symmetry

A coupled cell network is a directed graph whose nodes represent dynamical systems and whose directed edges specify how those systems are coupled to each other. The typical dynamic behaviour of a network is strongly constrained by its topology. Especially important constraints arise from global (group) symmetries and local (groupoid) symmetries. The H/K theorem of Buono and Golubitsky characterises the possible spatio-temporal symmetries of time-periodic states of group-equivariant dynamical systems. A version of this theorem for group-symmetric networks has been proved by Josi? and Török. In networks, spatial symmetries correspond to synchrony of cells, and spatio-temporal symmetries correspond to phase relations between cells. Associated with any coupled cell network is a canonical class of admissible ODEs that respect the network topology. A pattern of synchrony or phase relations in a hyperbolic time-periodic state of such an ODE is rigid if the pattern persists under small admissible perturbations. We characterise rigid patterns of synchrony and rigid phase patterns in coupled cell networks, on the assumption that the periodic state is fully oscillatory (no cell is in equilibrium) and the network has a basic property, the rigid phase property. We conjecture that all networks have the rigid phase property, and that in any path-connected network an admissible ODE with a hyperbolic periodic state can always be perturbed to make the perturbed periodic state fully oscillatory. Our main result states that in any path-connected network with the rigid phase property, every rigid pattern of phase relations can be characterised in two stages. First, sets of cells form synchronous clumps according to a balanced equivalence relation. Second, the corresponding quotient network has a cyclic group of automorphisms, and the phase relations are induced by associating a fixed phase shift with a generator of this group. Thus the clumps of synchronous cells form a discrete rotating wave. As a corollary, we prove an analogue of the H/K theorem for any path-connected network. We also discuss the non-path-connected case.     

Bifurcations in dynamical systems with interior symmetry

We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. The notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a ‘perturbation term’ which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.     

Partial classification of heteroclinic behaviour associated with the perturbation of hexagonal planforms

Physical systems often exhibit pattern-forming instabilities. Equivariant bifurcation theory is often used to investigate the existence and stability of spatially doubly periodic solutions with respect to the hexagonal lattice. Previous studies have focused on the six- and twelve-dimensional representation of the hexagonal lattice where the symmetry of the model is perfect. Here, perturbation of group orbits of translation-free axial planforms in the six- and twelve-dimensional representations is considered. This problem is studied via the abstract action of the symmetry group of the perturbation on the group orbit of the planform. A partial classification for the behaviour of the group orbits is obtained, showing the existence of homoclinic and heteroclinic cycles between equilibria.     

Reduced dynamics and symmetric solutions for globally coupled weakly dissipative oscillators

Systems of coupled oscillators may exhibit spontaneous dynamical formation of attracting synchronized clusters with broken symmetry; this can be helpful in modelling various physical processes. Analytical computation of the stability of synchronized cluster states is usually impossible for arbitrary nonlinear oscillators. In this paper we examine a particular class of strongly nonlinear oscillators that are analytically tractable. We examine the effect of isochronicity (a turning point in the dependence of period on energy) of periodic oscillators on clustered states of globally coupled oscillator networks. We extend previous work on networks of weakly dissipative globally coupled nonlinear Hamiltonian oscillators to give conditions for the existence and stability of certain clustered periodic states under the assumption that dissipation and coupling are small and of similar order. This is verified by numerical simulations on an example system of oscillators that are weakly dissipative perturbations of a planar Hamiltonian oscillator with a quartic potential. Finally we use the reduced phase-energy model derived from the weakly dissipative case to motivate a new class of phase-energy models that can be usefully employed for understanding effects such as clustering and torus breakup in more general coupled oscillator systems. We see that the property of isochronicity usefully generalizes to such systems, and we examine some examples of their attracting dynamics.     

Spatial heterogeneity enhances and modulates excitability in a mathematical model of the myometrium

The muscular layer of the uterus (myometrium) undergoes profound changes in global excitability prior to parturition. Here, a mathematical model of the myocyte network is developed to investigate the hypothesis that spatial heterogeneity is essential to the transition from local to global excitation which the myometrium undergoes just prior to birth. Each myometrial smooth muscle cell is represented by an element with FitzHugh–Nagumo dynamics. The cells are coupled through resistors that represent gap junctions. Spatial heterogeneity is introduced by means of stochastic variation in coupling strengths, with parameters derived from physiological data. Numerical simulations indicate that even modest increases in the heterogeneity of the system can amplify the ability of locally applied stimuli to elicit global excitation. Moreover, in networks driven by a pacemaker cell, global oscillations of excitation are impeded in fully connected and strongly coupled networks. The ability of a locally stimulated cell or pacemaker cell to excite the network is shown to be strongly dependent on the local spatial correlation structure of the couplings. In summary, spatial heterogeneity is a key factor in enhancing and modulating global excitability.     

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.     

Group-theoretic approach to enhancing the Fourier modal method for crossed gratings of plane group p3

The Fourier modal method for crossed gratings is reformulated by use of a group-theoretic approach when the grating structures, according to plane crystallography, are of plane group p3. By exploiting the structural symmetries of the grating, i.e. the invariance of grating pattern after rotations about the normal of the mean grating plane through angles n(2π/3), a general diffraction problem with asymmetrical incidence is decomposed into three so-called symmetrical basis problems whose field distributions are the symmetry modes of the grating. Then by solving the symmetrical basis problems instead of the original problem, we reduce the memory occupation and time consumption in numerical computation to 1/3 and 1/9 of those of the original formulation, respectively, if the truncated reciprocal lattice of the diffracted field also has the symmetry of the grating structure. For the special case of normal incidence, the reduction factor can be further reduced to 2/27. Numerical examples are provided to illustrate the effectiveness of the new formulation.     

Chaos in the Takens–Bogdanov bifurcation with O(2) symmetry

The Takens–Bogdanov bifurcation is a codimension-two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation-invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens–Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and travelling and standing waves in the nonlinear regime and predicts the presence of modulated travelling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated travelling waves, the normal form for the Takens–Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil'nikov type. Thus chaos is present arbitrarily close to the codimension-two point.     

Thermal and moisture expansion of material elements and macroscopic bodies with geometrical symmetries—wood logs and sawn goods as practical examples

Thermal and moisture expansion states available to material elements with particular symmetries, as well as macroscopic bodies with particular structural symmetries, are investigated in terms of Group Theory and Curie’s principle. It is found that in the case of one-dimensional material symmetry, Eigen shear strain of the material element must be zero within any plane perpendicular to the reflection plane. The same applies to the most simple two-dimensional rotation symmetry, within any plane including the rotation axis. The corresponding reflection-rotation symmetry requires all the Eigen shear strains to be zero. Greater 2-d rotation symmetries require, in addition, normal eigenstrains perpendicular to the rotation axis to be equal, which means transverse isotropy of the eigenstrain tensor. Such a strain state complies with three-dimensional rotation symmetries, as long as not more than one of the symmetries is of order greater than 2. In such case all the normal eigenstrains must be equal, corresponding to isotropy of the eigenstrain tensor. Unlike two-dimensional symmetries, any three-dimensional reflection–rotation symmetry does not place more restrictions on the eigenstrain state than the corresponding rotation symmetry. There is, however, a three-dimensional symmetry which places no restriction on the eigenstrain state. Possible non-uniform eigenstrain states of macroscopic bodies are discussed. Eigendeformations of wood logs are illustrated as examples. It is found that one abstract symmetry group may correspond to several geometrical symmetry groups, and the geometrical symmetry group of a body may depend on the choice of co-ordinate system. Log symmetries translate into sawn goods provided the sawing pattern complies with the log symmetries.     

Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with square symmetry

The Fourier modal method for crossed gratings with square symmetry is reformulated by use of a group-theoretic approach that we developed recently. In the new formulation, a crossed-grating problem is decomposed into six symmetrical basis problems whose field distributions are the symmetry modes of the grating. Then the symmetrical basis problems are solved with symmetry simplifications, whose solutions are superposed to get the solution of the original problem. Theoretical and numerical results show that when the grating is at some Littrow mountings, the computation efficiency can be improved effectively: The memory occupation is reduced by 3/4 and the computation time is reduced by a factor from 25.6 to 64 in different incident cases. Numerical examples are given to show the effectiveness of the new formulation.     

Lie symmetries and conserved quantities of constrained mechanical systems

Summary The Lie symmetries and conserved quantities of constrained mechanical systems are studied. Using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the systems are established. The structure equation and the form of conserved quantities are obtained. We find the corresponding conserved quantity from a known Lie symmetry, that is a direct problem of the Lie symmetries. And then, the inverse problem of the Lie symmetries-finding the corresponding Lie symmetry from a known conserved quantity-is studied. Finally, the relation between the Lie symmetry and the Noether symmetry is given.     

非线性振荡系统的Hopf分叉幅值控制

研究了非线性连续系统Ｈｏｐｆ分叉的状态反馈控制方法,提出了Ｈｏｐｆ分叉幅值可控性的概念,获得了平面系统Ｈｏｐｆ分叉所产生的极限环幅值可控的充分条件,在此基础上采用非线性状态反馈控制来抑制Ｈｏｐｆ分叉引起的自激振动。数值仿真结果表明：Ｈｏｐｆ分叉控制是非线性振荡系统控制的一种切实有效的方法。     

Exact block diagonalization of large eigenvalue problems for structures with symmetry

We consider large eigenvalue problems for skeletal structures with symmetry. We present an algorithm, based upon a novel combination of group-theoretic ideas and substructuring techniques, that block-diagonalizes such systems exactly and efficiently. The procedure requires only the structural matrices of a repeating substructure, together with the symmetry modes, which are obtained from symmetry considerations alone. We first present a simple paradigmatic example and then follow with several non-trivial applications involving large lattice structures.     

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.     

On the Possibility of s+d Wave Superconductivity Within a Two-Band Scenario for High T c Cuprates

A variety of different experimental results show substantial evidence that the order parameter in high-temperature superconducting copper oxides is not of pure d-wave symmetry, but that an s-wave component exists, which especially shows up in experiments that test the c-axis properties. These findings are modeled theoretically within a two-band model with interband interactions, where the superconducting order parameters in the two bands are allowed to differ in symmetry. It is found that the coupling of order parameters with different symmetries (s+d) leads to substantial enhancements of the superconducting transition temperature T _c as compared to order parameters with only s-wave symmetry. An additional enhancement factor of T _c is obtained from the coupling of the bands to the lattice where moderate couplings favor superconductivity while too strong couplings lead to electron (hole) localization and consequently suppress superconductivity.     

On the Possibility of s+d Wave Superconductivity Within a Two-Band Scenario for High Tc Cuprates

A variety of different experimental results show substantial evidence that the order parameter in high-temperature superconducting copper oxides is not of pure d-wave symmetry, but that an s-wave component exists, which especially shows up in experiments that test the c-axis properties. These findings are modeled theoretically within a two-band model with interband interactions, where the superconducting order parameters in the two bands are allowed to differ in symmetry. It is found that the coupling of order parameters with different symmetries (s+d) leads to substantial enhancements of the superconducting transition temperature T _c as compared to order parameters with only s-wave symmetry. An additional enhancement factor of T _c is obtained from the coupling of the bands to the lattice where moderate couplings favor superconductivity while too strong couplings lead to electron (hole) localization and consequently suppress superconductivity.     

耦合非线性振荡系统运动的一种全局分叉

一、引言非线性振荡系统运动的全局分叉是全局分析中重要和复杂问题,至今尚无一般解决方法.对于耦合非线性振荡系统,由于维数大于4更加困难.1980年开始提出的胞映射法,由于将连续的N维相空间离散为N维胞空间,将非线性振荡系统转化为胞胞映射系统,故可以用一种算法完成包括非线性系统的全局分析.     

Structure of the order parameter in iron pnictide-based superconducting materials

Using data on the crystal symmetry of iron pnictides, we have calculated possible symmetries of singlet two-electron states corresponding to the superconducting order parameter. It has been shown that the octet line node structure observed in photoemission spectra is due to interaction between pairs of the same symmetry and that the presence of an odd component of a superconducting state in NMR spectra is associated with states on the Brillouin zone boundary.     

Spatio-temporal symmetries in SN- and SN × 2-equivariant systems

This paper investigates the spatio-temporal symmetries of periodic trajectories in dynamical systems with S_N and S_N × ₂ symmetry. It turns out that trajectories in S_N-equivariant systems cannot exhibit spatio-temporal symmetries beyond the trivial symmetry of all periodic orbits. More complex symmetries in the trajectories require additional constraints on the dynamics. The possibilities offered by S_N × ₂ symmetric systems are considered and a specific S₃ × ₂-equivariant system is investigated numerically.