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.     

Existence and uniqueness of periodic orbits in the cubic autocatalator

Periodic orbits are sought in a mathematical model of a simple prototype chemical reaction involving essentially only two reacting species. Physically, these periodic orbits correspond to time-periodic oscillations in the concentrations of the two chemicals. Using the results for the existence and uniqueness of periodic orbits for Lienard systems, necessary and sufficient conditions are obtained for the existence of exactly one periodic orbit and of no periodic orbits. The results apply to a closed system where the quadratic autocatalytic reaction and decay step are present but the uncatalysed reaction is not and where there is only one physically relevant equilibrium solution     

Homoclinic intersections and Mel'nikov method for perturbed sine-Gordon equation

We describe and characterize rigorously the homoclinic structure of the perturbed sine-Gordon equation under periodic boundary conditions. The existence of invariant manifolds for a perturbed sine-Gordon equation is established. The Mel'nikov method, together with geometric analysis are used to assess the persistence of the homoclinic orbits under bounded and time-periodic perturbations.     

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.     

Stabilization of periodic orbits near a subcritical Hopf bifurcation in delay-coupled networks

We study networks of delay-coupled oscillators with the aim to extend time-delayed feedback control to networks. We show that unstable periodic orbits of a network can be stabilized by a noninvasive, delayed coupling. We state criteria for stabilizing the orbits by delay-coupling in networks and apply these to the case where the local dynamics is close to a subcritical Hopf bifurcation, which is representative of systems with torsion-free unstable periodic orbits. Using the multiple scale method and the master stability function approach, the network system is reduced to the normal form, and the characteristic equations for Floquet exponents are derived in an analytical form, which reveals the coupling parameters for successful stabilization. Finally, we illustrate the results by numerical simulations of the Lorenz system close to a subcritical Hopf bifurcation. The unstable periodic orbits in this system have no torsion, and hence cannot be stabilized by the conventional time delayed-feedback technique.     

Statistical analysis reveals the onset of synchrony in sparse swarms of Photinus knulli fireflies

Flash synchrony within firefly swarms is an elegant but elusive manifestation of collective animal behaviour. It has been observed, and sometimes demonstrated, in a few populations across the world, but exactly which species are capable of large-scale synchronization remains unclear, especially for low-density swarms. The underlying question which we address here is: how does one qualify a collective flashing display as synchronous, given that the only information available is the time and location of flashes? We propose different statistical approaches and apply them to high-resolution stereoscopic video recordings of the collective flashing of Photinus knulli fireflies, hence establishing the occurrence of synchrony in this species. These results substantiate detailed visual observations published in the early 1980s and made at the same experimental site: Peña Blanca Canyon, Coronado National Forest, AZ, USA. We also remark that P. knulli’s collective flashing patterns mirror those observed in Photinus carolinus fireflies in the Eastern USA, consisting of synchronous flashes in periodic bursts with rapid accretion and quick decay.     

Robust cycles unfolding from conservative bifocal homoclinic orbits

We prove that suspended robust heterodimensional cycles and suspended robust homoclinic tangencies can be found arbitrarily close to any non-degenerate bifocal homoclinic orbit of a Hamiltonian vector field. In order to achieve this result, we show that any diffeomorphism with a saddle-node periodic point, which has both quasi-transversal and tangential strong homoclinic intersections, can be approximated by diffeomorphisms with both, robust homoclinic tangencies and a robust heterodimensional cycles. As explained in the paper, Hamiltonian vector fields with non-degenerate bifocal homoclinic orbits are exhibited in the limit family of any generic unfolding of a four-dimensional nilpotent singularity of codimension four. Supported by the achieved results, we conjecture that suspended robust cycles can be generically unfolded from such singularities.     

Genericity in equivariant dynamical systems and equivariant Fuller index theory

We define a notion of equivariant non-degeneracy of G-maps to introduce the class of equivariantly non-degenerate flows on smooth compact manifolds with compact Lie group action. We prove genericity of this class and use this result to construct an equivariant version of the Fuller index, which detects group orbits of periodic orbits of the flow, distinguished by their isotropy.     

Nonreversible homoclinic snaking

Homoclinic snaking refers to the sinusoidal ‘snaking’ continuation curve of homoclinic orbits near a heteroclinic cycle connecting an equilibrium E and a periodic orbit P. Along this curve the homoclinic orbit performs more windings about the periodic orbit. Typically, this behaviour appears in reversible Hamiltonian systems. Here we discuss this phenomenon in systems without any particular structure. We give a rigorous analytical verification of homoclinic snaking under certain assumptions on the behaviour of the stable and unstable manifolds of E and P. We show how the snaking behaviour depends on the signs of the Floquet multipliers of P. Further we present a nonsnaking scenario. Finally, we show numerically that these assumptions are fulfilled in a model equation.     

基于偶极子模型的双小行星系统平衡点特性研究

针对主星是细长型小行星,而次星是小而规则天体的双小行星系统,采用偶极子—粒子模型,建立了普适性的引力场模型,研究了系统平衡点附近的局部动力学及周期轨道问题。研究了同步状态下系统参数对平衡点位置、稳定性和变化趋势的影响,并给出了非共线平衡点的线性稳定域,计算了在非同步双小行星系统的等效平衡点的轨迹。结合路径搜索修正法和伪弧长延拓方法得到同步双小行星系统共线平衡点附近的1∶1共振轨道族。该研究能为双小行星系统探测中轨道设计问题提供理论基础。     

Homeomorphisms of the Sierpinski curve with periodic properties

In this paper, we study the three following types of homeomorphisms of the Sierpinski curve of the two sphere : pointwise periodic, periodic, and almost periodic, and we prove that they are equivalent. We show that a subgroup of homeomorphisms whose orbits are all finite, is a finite subgroup.     

Symmetric and non-symmetric periodic orbits for the digital filter map

We exhibit instances of non-symmetric periodic orbits for the digital filter map, resolving a question posed in the literature as to whether such orbits can exist. This piecewise irrational rotation, depending on a parameter a = 2cos θ, is an isometry of [?1, 1) × [?1, 1) and reflections in the two diagonals are time-reversing symmetries for the map. Symmetric orbits are plentiful and have been much investigated. Each periodic orbit is paired with a symbolic string, from the alphabet {?, 0, +}, arising under iteration of the map because of the presence of a line of discontinuity. We prove the existence of an infinite family of non-symmetric orbits where the period N starts at 29 and increases in steps of 5; they correspond to the strings (+00)⁵(+?)² 0 ^N?19. We describe several computer algorithms to find non-symmetric periodic orbits and their symbolic strings and list non-symmetric strings both for a = 0.5, and for N ≤ 100 across the parameter range. Our evidence suggests that non-symmetric orbits, though not plentiful, are characteristic of the dynamics of the map for all parameter values.     

Synchronization In Lattices Of Coupled Oscillators With Neumann /Periodic Boundary Conditions

We consider a lattice of coupled Duffing oscillators with external periodic forces and Neumann or periodic boundary conditions. We prove that asymptotic synchronization occurs provided the coupling system is dissipative and coefficients of coupling are sufficiently large. We determine dependence of synchronization coefficients on lattice size.     

Essential instabilities of fronts: bifurcation, and bifurcation failure

Various instability mechanisms of fronts in reaction-diffusion systems are analysed; the emphasis is on instabilities that have the potential to create modulated (i.e. time-periodic) waves near the primary front. Hopf bifurcations caused by point spectrum with associated localized eigenfunctions provide an example of such an instability. A different kind of instability occurs if one of the asymptotic rest states destabilizes: these instabilities are caused by essential spectrum. It is demonstrated that, if the rest state ahead of the front destabilizes, then modulated fronts are created that connect the rest state behind the front with small spatially periodic patterns ahead of the front. These modulated fronts are stable provided the spatially periodic patterns are stable. If, on the other hand, the rest state behind the front destabilizes, then modulated fronts that leave a spatially periodic pattern behind do not exist.     

Torus square tilings

We consider periodic square tilings of the plane. By extending a formalism introduced in 1940 for tiling of rectangles by squares we build a correspondence between periodic plane maps endowed with a periodic harmonic vector and periodic square tilings satisfying a regularity condition. The space of harmonic vectors is isomorphic to the first homology group of a torus. So, periodic plane square tilings are described by two parameters and the set of parameters is split into angular sectors. The correspondence between symmetry of the square tiling and symmetry of the corresponding plane map and harmonic vector is discussed and a method for enumerating the regular periodic plane square tilings having $r$ orbits of squares is outlined.     

非线性Mathieu方程的混沌及其控制

用数值方法揭示了非线性Mathieu方程的一种特殊形式——在纵向简谐激励、非线性阻尼和联接质量惯性力作用下的欧拉弯曲问题的分岔现象和混沌行为。利用耦合反馈控制方法,实现了对这种混沌行为的控制,得到受控后系统的稳定周期(包括低周期、高周期和准周期)振动的结果。     

Chaotic double cycling

We study the dynamics of a generic vector field in the neighbourhood of a heteroclinic cycle of non-trivial periodic solutions whose invariant manifolds meet transversely. The main result is the existence of chaotic double cycling: there are trajectories that follow the cycle making any prescribed number of turns near the periodic solutions, for any given bi-infinite sequence of turns. Using symbolic dynamics, arbitrarily close to the cycle, we find a robust and transitive set of initial conditions whose trajectories follow the cycle for all time and that is conjugate to a Markov shift over a finite alphabet. This conjugacy allows us to prove the existence of infinitely many heteroclinic and homoclinic subsidiary connections, which give rise to a heteroclinic network with infinitely many cycles and chaotic dynamics near them, exhibiting themselves switching and cycling. We construct an example of a vector field with Z ₃ symmetry in a five-dimensional sphere with a heteroclinic cycle having this property.     

Stability properties of divergence-free vector fields

A divergence-free vector field satisfies the star property if any divergence-free vector field in some C ¹-neighbourhood has all singularities and all closed orbits hyperbolic. In this article, we prove that any divergence-free vector field defined on a Riemannian manifold and satisfying the star property is Anosov. It is also shown that a C ¹-structurally stable divergence-free vector field is Anosov. Moreover, we prove that any divergence-free vector field can be C ¹-approximated by an Anosov divergence-free vector field, or else by a divergence-free vector field exhibiting a heterodimensional cycle.