Aperiodic States Research Articles

Chaotic advection in a recirculating flow: Effect of a fluid multiple-flexible-solid interaction.

This paper deals with chaotic advection due to a two-way interaction between flexible elliptical-solids and a laminar lid-driven cavity flow in two dimensions. The present Fluid multiple-flexible-Solid Interaction study involves various number N(= 1-120) of equal-sized neutrally buoyant elliptical-solids (aspect ratio β = 0.5) such that they result in the total volume fraction Φ = 10 % as in our recent study on single solid, done for non-dimensional shear modulus G ∗ = 0.2 and Reynolds number R e = 100. Results are presented first for flow-induced motion and deformation of the solids and later for chaotic advection of the fluid. After the initial transients, the fluid as well as solid motion (and deformation) attain periodicity for smaller N ≤ 10 while they attain aperiodic states for larger N > 10. Adaptive material tracking (AMT) and Finite-Time Lyapunov Exponent (FTLE)-based Lagrangian dynamical analysis revealed that the chaotic advection increases up to N = 6 and decreases at larger N(= 6-10) for the periodic state. Similar analysis for the transient state revealed an asymptotic increase in the chaotic advection with increasing N ≤ 120. These findings are demonstrated with the help of two types of chaos signatures: exponential growth of material blob's interface and Lagrangian coherent structures, revealed by the AMT and FTLE, respectively. Our work, which is relevant to several applications, presents a novel technique based on the motion of multiple deformable-solids for enhancement of chaotic advection.

Rectangle-triangle soft-matter quasicrystals with hexagonal symmetry.

Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built up from, e.g., kites and darts, squares and equilateral triangles, rhombi- or shield-shaped tiles, and can have a variety of different symmetries. However, almost all quasicrystals occurring in soft matter are of the dodecagonal type. Here we investigate a class of aperiodic tilings with hexagonal symmetry that are based on rectangles and two types of equilateral triangles. We show how to design soft-matter systems of particles interacting via pair potentials containing two length scales that form aperiodic stable states with two different examples of rectangle-triangle tilings. One of these is the bronze-mean tiling, while the other is a generalization. Our work points to how more general (beyond dodecagonal) quasicrystals can be designed in soft matter.

Appointed-time guaranteed adaptive fault-tolerant attitude tracking for quadrotors with aperiodic data updating

In this paper, an appointed-time guaranteed adaptive fault-tolerant attitude tracking for quadrotors with aperiodic data updating is concerned, where inertial moment variation, extraneous disturbance and unknown actuator faults are simultaneously involved. An adaptive parameter estimation law is updated to identify the partial effectiveness loss of actuation and alleviate the estimation burden of suggested observer. Then through filtering manipulations upon aperiodic system states, a novel event-triggered unknown system dynamics estimator (ETUSDE) is developed to learn the remaining uncertainties with exponential error decaying and decreased updating cost, where an event-triggering criterion is incorporated to schedule the transmission instants for control behavior and angular rate, and the explicit formulation between estimation error upper boundary and design coefficients is derived. An appointed-time prescribed performance control (APPC) is introduced to allow for pregiven attitude behaviors. By fusion of adaptive law, ETUSDE and APPC, an adaptive event-triggered attitude control policy is synthesized that delivers appointed-time convergence, accurate steady response and decreased sampling burden. Meanwhile, Lyapunov analysis shows that all the error variables contained in the closed-loop system are ultimately uniformly bounded. Eventually, experiments on a quadrotor attitude test rig are conducted to confirm the superiorities of suggested control strategy.

Stability and dynamics of the flow past a bullet-shaped blunt body moving in a pipe

The flow past a bullet-shaped blunt body moving in a pipe is investigated through global linear stability analysis (LSA) and direct numerical simulation. A cartography of the bifurcation curves is provided thanks to LSA, covering the range of parameters corresponding to Reynolds number $Re = [50\unicode{x2013}110]$ , confinement ratio $a/A = [ 0.01\unicode{x2013}0.92]$ and length-to-diameter ratio $L/d = [2\unicode{x2013}10]$ . Results show that the first bifurcation is always a steady bifurcation associated to a non-oscillating eigenmode with azimuthal wavenumber $m=\pm 1$ leading to a steady state with planar symmetry. For weakly confined cases ( $a/A<0.6$ ), the second bifurcation is associated to an oscillating mode with azimuthal wavenumber $m=\pm 1$ , as in the unconfined case. On the other hand, for the strongly confined case ( $a/A>0.8$ ), a destabilization of non-oscillating modes with $|m| =2,3$ and a restabilization of the $m=\pm 1$ eigenmodes are observed. The aspect ratio $L/d$ is shown to have a minor influence for weakly confined cases and almost no influence for strongly confined cases. Direct numerical simulation is subsequently used to characterize the nonlinear dynamics. The results confirm the steady bifurcation predicted by LSA with excellent agreement for the threshold Reynolds. For weakly confined cases, the second bifurcation is a Hopf bifurcation leading to a periodic, planar-symmetric state in qualitative accordance with LSA predictions. For more confined cases, more complex dynamics is obtained, including a steady state with $|m|=3$ geometry and aperiodic states.

A hybrid lattice Boltzmann model for simulating viscoelastic instabilities

Low-Reynolds-number viscoelastic fluids exhibit a chaotic behaviour linked to the growth of elastic instabilities when the polymer molecules are stretched excessively. Despite offering various exciting benefits, the numerical difficulties associated with simulating this complex regime have left a lot still unexplored regarding the transition and onset of these low-Reynolds number viscoelastic instabilities. Recently, methods based on the lattice Boltzmann method (LBM) have emerged as a viable numerical tool for studying the behaviour of viscoelastic fluids largely due to the simplicity, adaptability, and intrinsic parallel features of LBM. However, extensive memory requirements and the inability to preserve numerical stability have restricted previous attempts from simulating viscoelastic instabilities. In this work, a hybrid lattice Boltzmann model is applied to simulate low-Reynolds number viscoelastic instabilities. In this approach, the hydrodynamic field is solved using a lattice Boltzmann model, whereas the polymer field is solved separately using a high-resolution finite difference scheme with the logarithmic Cholesky decomposition. The model is first validated for steady viscoelastic flows using the four-roll mill case, where the results are compared against the analytical solution and previous numerical studies. The model is then applied to simulate a chaotic low-Reynolds number viscoelastic instability by initialising a small perturbation to the isotropic polymer stress. The results at different Weissenberg numbers experience complex flow dynamics at sufficiently long time durations, which transition from quasi-periodic to periodic to aperiodic states with increasing elastic effects. The quasi-periodic regime was found to experience the fastest transition into the transient regime, with the transitioning time being delayed as the Weissenberg number increased. This work proves the capability of applying a coupled lattice Boltzmann method for exploring the complex flow behaviour of elastic instabilities, allowing the potential to explore more challenging and practical viscoelastic instability cases.

Aperiodic sleep networks promote memory consolidation.

Hierarchical synchronization of sleep oscillations establishes communication pathways to support memory reactivation, transfer, and consolidation. From an information-theoretical perspective, oscillations constitute highly structured network states that provide limited information-coding capacity. Recent findings indicate that sleep oscillations occur in transient bursts that are interleaved with aperiodic network states, which were previously considered to be random noise. We argue that aperiodic activity exhibits unique and variable spatiotemporal patterns, providing an ideal information-rich neurophysiological substrate for imprinting new mnemonic patterns onto existing circuits. We discuss novel avenues in conceptualizing and quantifying aperiodic network states during sleep to further understand their relevance and interplay with sleep oscillations in support of memory consolidation.

Configurational electronic states in layered transition metal dichalcogenides

Mesoscopic irregularly ordered and even amorphous self-assembled electronic structures were recently reported in two-dimensional metallic dichalcogenides (TMDs), created and manipulated with short light pulses or by charge injection. Apart from promising new all-electronic memory devices, such states are of great fundamental importance, since such aperiodic states cannot be described in terms of conventional charge-density-wave (CDW) physics. In this paper, we address the problem of metastable mesoscopic configurational charge ordering in TMDs with a sparsely filled charged lattice gas model in which electrons are subject only to screened Coulomb repulsion. The model correctly predicts commensurate CDW states corresponding to different TMDs at magic filling fractions Doping away from results either in multiple near-degenerate configurational states, or an amorphous state at the correct density observed by scanning tunnelling microscopy. Quantum fluctuations between degenerate states predict a quantum charge liquid at low temperatures, revealing a new generalized viewpoint on both regular, irregular and amorphous charge ordering in transition metal dichalcogenides.

Separated periodic/aperiodic state feedback control using periodic/aperiodic separation filter based on lifting

Automatic machines have recently advanced into diverse environments. A periodic state and an aperiodic state of the system usually have different physical meanings and require different control objectives in the diverse environments. To enable a new control approach based on the periodic and aperiodic states for the different control objectives, a periodic/aperiodic separation filter (PASF) that separates a state into a periodic state and an aperiodic state is proposed in this paper. Furthermore, using the PASF, the periodic/aperiodic state feedback control and the separation principle are established.

Mixed-mode oscillations and chaos in return maps of an oscillatory chemical reaction

The return maps, as an element of mathematical phenomenology appropriate for general examinations of complex dynamic states of the oscillatory systems were used to detect and explain the evolution of mixed-mode oscillations and chaos in a six-dimensional nonlinear reaction system of the Bray–Liebhafsky (BL) reaction, a well-studied nonlinear chemical reaction system that exhibits complex dynamic behavior. For this purpose principally different Poincare sections were applied and different transition scenarios between periodic and aperiodic states were examined by numerical simulations. It is shown that emergence of new periodic patterns can be detected by return maps already within chaotic windows. Besides, we also show that the higher dimensionality of manifold gives the impression of having several layers of manifolds.

Microscopically Based Calculations of the Free Energy Barrier and Dynamic Length Scale in Supercooled Liquids: The Comparative Role of Configurational Entropy and Elasticity

We compute the temperature-dependent barrier for α-relaxations in several liquids, without adjustable parameters, using experimentally determined elastic, structural, and calorimetric data. We employ the random first order transition (RFOT) theory, in which relaxation occurs via activated reconfigurations between distinct, aperiodic minima of the free energy. Two different approximations for the mismatch penalty between the distinct aperiodic states are compared, one due to Xia and Wolynes (Proc. Natl. Acad. Sci. U. S. A. 2000, 97, 2990), which scales universally with temperature as for hard spheres, and one due to Rabochiy and Lubchenko (J. Chem. Phys. 2013, 138, 12A534), which employs measured elastic and structural data for individual substances. The agreement between the predictions and experiment is satisfactory, given the uncertainty in the measured experimental inputs. The explicitly computed barriers are used to calculate the glass transition temperature for each substance--a kinetic quantity--from the static input data alone. The temperature dependence of both the elastic and structural constants enters the temperature dependence of the barrier over an extended range to a degree that varies from substance to substance. The lowering of the configurational entropy, however, seems to be the dominant contributor to the barrier increase near the laboratory glass transition, consistent with previous experimental tests of the RFOT theory using the XW approximation. In addition, we compute the temperature dependence of the dynamical correlation length, also without using adjustable parameters. These agree well with experimental estimates obtained using the Berthier et al. (Science 2005, 310, 1797) procedure. Finally, we find the temperature dependence of the complexity of a rearranging region is consistent with the picture based on the RFOT theory but is in conflict with the assumptions of the Adam-Gibbs and "shoving" scenarios for the viscous slowing down in supercooled liquids.

Identifying homoclinic orbits in the dynamics of intermittent signals through recurrence quantification

In this paper, we show how the phenomenon of intermittency observed in systems with turbulent flow-sound interaction is related to the formation of homoclinic orbits in the phase space. Such orbits that emerge via the intersection of the stable and unstable manifold of an equilibrium configuration result from interactions that happen at multiple spatial/temporal scales associated with turbulent convection and wave propagation. Through a quantification of the time spent by the dynamics in the aperiodic states using recurrence plots, we show how the presence of homoclinic orbits in the dynamics may be convincingly demonstrated, which is often not possible through a visual inspection of the phase space of the attractor.

Collective spin-wave excitations in a two-dimensional array of coupled magnetic nanodots

A general theory of collective spin-wave excitations in a two-dimensional array of magnetic nanodots coupled by magnetodipolar interaction is developed. The theory allows one to analytically calculate spectra, damping rates, excitation efficiencies, and other characteristics of spin waves in both periodic and aperiodic ground states of an array. It is demonstrated that all the properties of collective spin waves in an array existing in any spatially periodic ground state (e.g., ferromagnetic or chessboard antiferromagnetic) are determined by the same state-independent array's demagnetization tensor ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbit{F}}}_{\mathbit{k}}$, which is determined by the spin-wave wave vector $\mathbit{k}$, the size and shape of the array's elements (nanodots), and the geometry of the array's lattice. The applications of the developed general theory are illustrated on particular examples: (i) spin waves in ferromagnetic and chessboard antiferromagnetic states of a square array, and (ii) localized spin-wave excitations associated with an isolated ``defect'' in a uniform ferromagnetic ground state of a square array.

An intrinsic formation mechanism for midgap electronic states in semiconductor glasses

We argue that semiconducting quenched liquids and frozen glasses may exhibit a set of peculiar electronic states of topological origin. These states reside at strained regions arising during structural reconfigurations between distinct aperiodic states intrinsic to quenched melts. The strained regions are domain walls separating the distinct aperiodic states; their number is about 10(20) cm(-3) in all glassformers owing to the universal dynamics of deeply supercooled melts. Even though located near the middle of the forbidden gap, the topological states are rather extended in one direction while being centered at under- and overcoordinated atoms. The states exhibit the reverse charge-spin relation, the majority of states being diamagnetic and charged. The topological states may be sufficient to account for a number of irradiation-induced phenomena in amorphous semiconductors, including electron spin resonance signal, midgap absorption, photoluminescence, and the fatigue of photoluminescence. We propose experiments to test the present microscopic picture.

Flow in a channel with accelerating or decelerating wall velocity: A comparison between self-similar solutions and Navier–Stokes computations in finite domains

We investigate computationally whether a class of self-similar solutions of the Navier–Stokes equations in infinite channels driven by accelerating or decelerating walls, arises anywhere in a channel restricted to finite length. The self-similar solutions satisfy a nonlinear partial differential equation involving time and the vertical coordinate and have been studied previously. Admissible self-similar solutions include stable and unstable steady branches, time-periodic branches emerging from Hopf bifurcations, as well as chaotic solutions following a period-doubling Feigenbaum cascade. The objective here is to explore whether such solutions emerge when restricted to finite, although long, channels. The problem is addressed numerically using fast algorithms to solve the Navier–Stokes equations. It is established that all branches of the self-similar solutions (including time-periodic ones) can occur in finite channels provided the Reynolds numbers are not too high (approximately 500 for accelerating and 33 for decelerating wall flows, respectively). As the Reynolds number increases it becomes more difficult to recover the self-similar branches with general inflow conditions, and this has been overcome by utilizing the self-similar solutions as inflow Dirichlet conditions. At sufficiently high Reynolds numbers, the self-similar inflow conditions are incapable of producing the exact solution in the interior, but instead new steady or time-periodic states emerge which depend on the Reynolds number and the channel length. It is established numerically by solving the linearized Navier–Stokes equations that such phenomena are due to a spatial instability of the self-similar states to perturbations which are not of self-similar form. The results indicate that caution must be exercised in drawing conclusions from a stability theory restricted to self-similar perturbations and examples are given where such analysis is erroneous. In the case of decelerating wall flows the self-similar solutions are recovered on finite domains but for a much smaller range of Reynolds numbers as compared to accelerating wall flows. The results also show that time-periodic self-similar states are possible in finite channels (the Navier– Stokes equations are solved subject to the spatiotemporal self-similar solution at inflow) for accelerating wall flows but not for decelerating ones. Of special interest are computed aperiodic states of the Navier–Stokes equations in the case of accelerating walls and relatively high Reynolds numbers. These solutions behave intermittently and bounce between long-lived states that are congruent to the self-similar solutions to non-self-similar aperiodic ones. This is a complicated phenomenon due to the dimensionality and nonlinear nature of the system but the results indicate that the self-similar states can be strong attractors in some regions of the phase space.

Lyapunov exponents for small aspect ratio Rayleigh-Bénard convection

Leading order Lyapunov exponents and their corresponding eigenvectors have been computed numerically for small aspect ratio, three-dimensional Rayleigh-Benard convection cells with no-slip boundary conditions. The parameters are the same as those used by Ahlers and Behringer [Phys. Rev. Lett. 40, 712 (1978)] and Gollub and Benson [J. Fluid Mech. 100, 449 (1980)] in their work on a periodic time dependence in Rayleigh-Benard convection cells. Our work confirms that the dynamics in these cells truly are chaotic as defined by a positive Lyapunov exponent. The time evolution of the leading order Lyapunov eigenvector in the chaotic regime will also be discussed. In addition we study the contributions to the leading order Lyapunov exponent for both time periodic and aperiodic states and find that while repeated dynamical events such as dislocation creation/annihilation and roll compression do contribute to the short time Lyapunov exponent dynamics, they do not contribute to the long time Lyapunov exponent. We find instead that nonrepeated events provide the most significant contribution to the long time leading order Lyapunov exponent.

Some limit results for controlled branching processes with random control function*

where the empty sum is considered to be 0, N is a positive integer, and {Xnj :n = 0, 1, . . . ; j = 1, 2, . . . } and {φn(k):n, k = 0, 1, . . . } are independent sets of nonnegative-integer-valued random variables defined on the same probability space. The variables Xnj are independent and identically distributed with common probability law {pk}k≥0, pk := P(X01 = k), k = 0, 1, . . . , called the offspring probability distribution, and for n = 0, 1, . . . , {φn(k)}k≥0 are independent stochastic processes such that φn(k), n = 0, 1, . . . , are identically distributed. Let us denote by m := E[X01], σ := Var[X01], e(k) := E[φ0(k)], and ν(k) := Var[φ0(k)], k = 0, 1, . . . , the mean and variance (assumed finite) of the offspring probability distribution and the control random variables, respectively. From (1), it is easily verified that {Zn}n≥0 is a homogeneous Markov chain with state space on the nonnegative integers. For simplicity, we shall consider that the positive integers form a class of communicating and aperiodic states, and we will assume that P(φ0(0) = 0) = 1 and at least one of the following conditions holds: (i) p0 > 0; (ii) P(φ0(k) = 0) > 0, k = 1, 2, . . . . Under these assumptions, it was proved in [14] that the positive states are transient and, as a consequence, it is verified that P(Zn → 0) +P(Zn → ∞) = 1. (2)

Complex and Chaotic Oscillations in a Model for the Catalytic Hydrogen Peroxide Decomposition under Open Reactor Conditions

Numerous periodic and aperiodic dynamic states obtained in a model for hydrogen peroxide decomposition in the presence of iodate and hydrogen ions (the Bray-Liebhafsky reaction) realized in an open reactor (CSTR), where the flow rate was the control parameter, have been investigated numerically. Between two Hopf bifurcation points, different simple and complex oscillations and different routes to chaos were observed. In the region of the mixed-mode evolution of the system, the transitions between two successive mixed-mode simple states are realized by period-doubling of the initial state leading to a chaotic window in which the next dynamic state emerges mixed with the initial one. It appears in increasing proportions in concatenated patterns until total domination. Thus, with increasing flow rate the period-doubling route to chaos was obtained, whereas with decreasing flow rate the peak-adding route to chaos was obtained. Moreover, in very narrow regions of flow rates, chaotic mixtures of mixed-mode patterns were observed. This evolution of patterns repeats until the end of the mixed-mode region at high flow rates that corresponds to chaotic mixtures of one large and many small amplitude oscillations. Starting from the reverse Hopf bifurcation point and decreasing the flow rate, simple small amplitude sinusoidal oscillations were encountered and then the period-doubling route to chaos. With a further decreasing flow rate, the mixed-mode oscillations emerge inside the chaotic window.

Breaking synchrony by heterogeneity in complex networks.

For networks of pulse-coupled oscillators with complex connectivity, we demonstrate that in the presence of coupling heterogeneity precisely timed periodic firing patterns replace the state of global synchrony that exists in homogenous networks only. With increasing disorder, these firing patterns persist until a critical temporal extent is reached that is of the order of the interaction delay. For stronger disorder the periodic firing patterns cease to exist and only asynchronous, aperiodic states are observed. We derive self-consistency equations to predict the precise temporal structure of a pattern from a network of given connectivity and heterogeneity. Moreover, we show how to design heterogeneous coupling architectures to create an arbitrary prescribed pattern.

A route to Eulerian chaos in a two-layer wind-driven ocean

A numerical study is made of a two-layer quasigeostrophic model for a wind-driven subtropical gyre in a square β-plane basin bounded by no-slip walls. We analyze the time series of basin-averaged potential and kinetic energies to examine changes in dynamical regimes when the strength of the steady winds is gradually increased while the eddy viscosity coefficient is fixed. We have confirmed the following transition from steady to aperiodic states: (i) period-doubling and inverse period-doubling bifurcations associated with mesoscale eddies with a period of several months; (ii) the occurrence of an interannual frequency which gives rise to a quasiperiodic state and eventually is phase-locked with the short-term frequencies; and (iii) successive period-doubling bifurcations leading to the spectral broadening. Therefore, the decadal variations appearing in the present model may be related with the short-term variations through this sequence including a Feigenbaum scenario.

The aperiodic states of zircon: an ab initio molecular dynamics study

We theoretically investigated the local structure of the aperiodic states of zircon (ZrSiO 4 ) using ab initio quantum mechanical calculations. The low and high density liquid and solid glassy phases were obtained by constant volume Car-Parrinello molecular dynamics simulations, using the molar volume of metamict and crystalline zircon, respectively. As in naturally metamict zircons, the polymerization of Si units, the segregation of Zr atoms, and an overall decrease of the Zr coordination were observed. However, the local ordering of our theoretical glasses differs from that of the natural amorphous samples. In the theoretical glasses, the Zr-O distances in the first coordination polyhedra are significantly more distributed and five- and six-coordinated Si species were observed. The relaxation of the glass structure on a time scale exceeding the possibility of molecular modeling is a possible explanation for these discrepancies. The dielectric and 2 9 Si NMR responses of the glassy phases were computed, providing new constraints for the analysis of experimental data recorded from metamict zircons. The calculated NMR spectra are in good agreement with the experimental NMR spectra in the four-coordinated Si region. Our results show that the usual regular systematic decrease of the 2 9 Si chemical shift as a function of the polymerization of Si units (Q n species) cannot be used to interpret the 2 9 Si NMR spectrum of amorphous zircon. In particular, whereas the empirical scale would indicate an average polymerization of Q 3 , the average polymerization of Q 1 . 5 observed in the low density zircon glass can account for the experimental data. In a non-uniform model of the structure of metamict zircon, this lower average polymerization is consistent with a clustering of cations limited to few coordination polyhedra.