Quasi-periodic Patterns Research Articles

A fractional-order improved FitzHugh-Nagumo neuron model

Abstract In this paper, we propose a fractional-order improved FitzHugh-Nagumo (FHN) neuron model in terms of a generalised Caputo fractional derivative. Following the existence of a unique solution for the proposed model, we derive the numerical solution using a recently proposed L1 predictor-corrector method. The given method is based on the L1-type discretization algorithm and the spline interpolation scheme. We perform the error and stability analyses for the given method. We perform graphical simulations demonstrating that the proposed FHN neuron model generates rich electrical activities of periodic spiking patterns, chaotic patterns, and quasi-periodic patterns. The motivation behind proposing a fractional-order improved FHN neuron model is that such a system can provide a more nuanced description of the process with better understanding and simulation of the neuronal responses by incorporating memory effects and non-local dynamics, which are inherent to many biological systems.

Soliton wave profiles and dynamical analysis of fractional Ivancevic option pricing model

This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions.

On some new travelling wave solutions and dynamical properties of the generalized Zakharov system.

This study examines the extended version of the Zakharov system characterizing the dispersive and ion acoustic wave propagation in plasma. The genuine, non-dispersive field depicts a shift in plasma ion density from its equilibrium state, whereas the complex, dispersive field depicts the fluctuating envelope of a highly oscillatory field of electricity. The main focus of the analysis is on employing the expanded Fan sub-equation approach to achieve some novel travelling wave structures including the explicit, periodic, linked wave, and other new exact solutions are developed for different values of this parameter. Three dimensional graphs are utilised to examine the properties of the obtained solutions. Furthermore, ideas from planar dynamical theory are applied in this work to analyse the intricate behaviour of the analysed model. Sensitivity analysis, multistability, quasi-periodic and chaotic patterns, Poincaré map, and the Lyapunov characteristic exponent are used to analyse the dynamical features.

Evolution of aberrant brain-wide spatiotemporal dynamics of resting-state networks in a Huntington's disease mouse model.

Huntington's disease (HD) is marked by irreversible loss of neuronal function for which currently no availability for disease-modifying treatment exists. Advances in the understanding of disease progression can aid biomarker development, which in turn can accelerate therapeutic discovery. We characterised the progression of altered dynamics of whole-brain network states in the zQ175DN mouse model of HD using a dynamic functional connectivity (FC) approach to resting-state fMRI and identified quasi-periodic patterns (QPPs) of brain activity constituting the most prominent resting-state networks. The occurrence of the normative QPPs, as observed in healthy controls, was reduced in the HD model as the phenotype progressed. This uncovered progressive cessation of synchronous brain activity with phenotypic progression, which is not observed with the conventional static FC approaches. To better understand the potential underlying cause of the observed changes in these brain states, we further assessed how mutant huntingtin (mHTT) protein deposition affects astrocytes and pericytes - one of the most important effectors of neurovascular coupling, along phenotypic progression. Increased cell-type dependent mHTT deposition was observed at the age of onset of motor anomalies, in the caudate putamen, somatosensory and motor cortex, regions that are prominently involved in HD pathology as seen in humans. Our findings provide meaningful insights into the development and progression of altered functional brain dynamics in this HD model and open new avenues in assessing the dynamics of whole brain states, through QPPs, in clinical HD research. Hyperactivity in the LCN-linked regions within short QPPs observed before motor impairment onset. DMLN QPP presents a progressive decrease in DMLN activity and occurrence along HD-like phenotype development. Breakdown of the LCN DMLN state flux at motor onset leads to a subsequent absence of the LCN DMLN QPP at an advanced HD-like stage.

Quasiperiodic Oscillations of Flare Loops and Slipping Motion of Ribbon Substructures during a C-class Flare

Quasiperiodic oscillations in solar-flaring emission have been observed over the past few decades. To date, the underpinning processes resulting in the quasiperiodic oscillations remain unknown. In this paper, we report a unique event that exhibits both the long-duration quasiperiodic intensity oscillations of flare loops and the quasiperiodic slipping motion of ribbon substructures during a C9.1-class flare (SOL2015-03-15-T01:15), using the observations from the Solar Dynamics Observatory and Interface Region Imaging Spectrograph. The high-temperature flare loops rooted in the straight part of ribbons display a “bright–dim” intensity oscillation, with a period of about 4.5 minutes. The oscillation starts just after the flare onset and lasts over 3 hr. Meanwhile, the substructures within the ribbon tip display the quasiperiodic slipping motion along the ribbon at 1400 Å images, which has a similar periodicity to the stationary intensity oscillation of the flare loops in the straight part of the flare ribbons. We suggest that the quasiperiodic pattern is probably related to the loop-top dynamics caused by the reconnection outflow impinging on the flare loops.

Altered functional connectivity and spatiotemporal dynamics in individuals with sleep disorders.

Idiopathic hypersomnia (IH) is a sleep disorder characterized by highly disruptive symptoms. Like narcolepsy type 1, a well-characterized sleep disorder, individuals with IH suffer from excessive daytime sleepiness, though there is little overlap in metabolic or neural biomarkers across these two disorders. This lack of common pathophysiology, combined with the clear overlap in symptoms presents an ideal paradigm for better understanding the impact of IH on an individual's functional activity and organization, and potentially, the underlying pathophysiology. This study examines the observed functional connectivity in patients with IH, and patients with narcolepsy type 1 (NT1) against healthy control individuals. Static functional connectivity is compared, as are quasi-periodic patterns, acquired from the BOLD timecourse, for all groups. In addition to baseline data comparison, the study also included a post-nap condition, where the individuals included in this analysis napped for at least 10 minutes prior to the scanning session, to explore why individuals with IH do not feel refreshed after a nap like individuals with NT1 do. Assessing the groups' spatiotemporal patterns revealed key differences across both disorders and conditions: static connectivity revealed at baseline higher subcortical connectivity in the NT1 group. There was also observably less connectivity in the IH group both at baseline and post-nap, though none of these static analyses survived multiple comparisons correction to reach significance. The QPP results however found significant differences in the IH group in key networks, particularly the DAN/FPCN correlation is significantly different at baseline vs. post-nap, a trend not observed in either the control or NT1 groups. The DAN and FPCN are drastically altered both at baseline and post-nap when compared to the other groups, and may likely be a disorder-specific result. This study demonstrates that key networks for arousal are more heavily disrupted in IH patients, who are less affected by a nap, confirmed through both subject reporting and functional evidence through spatiotemporal patterns.

Dynamics of quasi-periodic, bifurcation, sensitivity and three-wave solutions for (n + 1)-dimensional generalized Kadomtsev-Petviashvili equation.

This study endeavors to examine the dynamics of the generalized Kadomtsev-Petviashvili (gKP) equation in (n + 1) dimensions. Based on the comprehensive three-wave methodology and the Hirota's bilinear technique, the gKP equation is meticulously examined. By means of symbolic computation, a number of three-wave solutions are derived. Applying the Lie symmetry approach to the governing equation enables the determination of symmetry reduction, which aids in the reduction of the dimensionality of the said equation. Using symmetry reduction, we obtain the second order differential equation. By means of applying symmetry reduction, the second order differential equation is derived. The second order differential equation undergoes Galilean transformation to obtain a system of first order differential equations. The present study presents an analysis of bifurcation and sensitivity for a given dynamical system. Additionally, when an external force impacts the underlying dynamic system, its behavior resembles quasi-periodic phenomena. The presence of quasi-periodic patterns are identified using chaos detecting tools. These findings represent a novel contribution to the studied equation and significantly advance our understanding of dynamics in nonlinear wave models.

Wave transmission in quasi-periodic lattices.

Structural lattices with quasi-periodic patterns possess interesting dynamic features that can be exploited to control, localize and redirect propagating waves. In this work, we show that the properties of a large class of quasi-periodic locally resonant systems (approximated as periodic, with arbitrarily large period) can be performed by defining an equivalent discrete system. Several properties of wave propagation can a priori be demonstrated with reference to this system. Results in terms of bulk spectrum, showing the Hofstadter butterfly pattern, and of topological modes are then discussed in detail with reference to a simple example of quasi-periodic lattice. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability.

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|^{α} in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)10.1103/PhysRevE.104.054102] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)-S(t) can be fit to a power law of the form t^{-(d-α)/2}, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ[over ̃]_{_{V}}(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of α for an infinite self-similar quasicrystal from a modestly sized finite sample. The methods described here offer a simple and general procedure to characterize accurately the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in χ[over ̃]_{_{V}}(k), can be used to estimate their physical properties, such as their effective dynamic dielectric constants, effective dynamic elastic constants, and fluid permeabilities.

Unveiling multi-wave patterns: dynamic characterization and sensitivity analysis of the Yu-Toda-Sasa-Fukuyama model in lattice and liquid

In this study, an examination of the Yu-Toda-Sasa-Fukuyama equation is undertaken, a model that characterizes elastic waves in a lattice or interfacial waves in a two layer liquid. Our emphasis lies in conducting a comprehensive analysis of this equation through various viewpoints, including the examination of soliton dynamics, exploration of bifurcation patterns, investigation of chaotic phenomena, and a thorough evaluation of the model’s sensitivity. Utilizing a simplified version of Hirota’s approach, multi-soliton pattens, including 1-wave, 2-wave, and 3-wave solitons, are successfully derived. The identified solutions are depicted visually via 3D, 2D, and contour plots using Mathematica software. The dynamic behavior of the discussed equation is explored through the theory of bifurcation and chaos, with phase diagrams of bifurcation observed at the fixed points of a planar system. Introducing a perturbed force to the dynamical system, periodic, quasi-periodic and chaotic patterns are identified using the RK4 method. The chaotic nature of perturbed system is discussed through Lyapunov exponent analysis. Sensitivity and multistability analysis are conducted, considering various initial conditions. The results acquired emphasize the efficacy of the methodologies used in evaluating solitons and phase plots across a broader spectrum of nonlinear models.

Dynamic analysis of multi-DOF acoustic resonant system involving highly viscous fluids based on fluid-structure interaction

This paper investigates the dynamic characteristics of a multi-degree-of-freedom (multi-DOF) acoustic resonant system involving highly viscous fluids. First, a fluid-structure interaction (FSI) model is established to describe the interaction between the multi-DOF acoustic resonant system and the two-phase fluids, and then the effects of various excitation parameters on the dynamic response of the coupled system are studied. Under the influence of vertical acoustic vibration, the fluid in the container undergoes violent and irregular motions, which leads to a decrease and fluctuation in the equivalent mass of the fluid. The reduction in fluid equivalent mass causes the excitation frequency to deviate from the natural frequency of the multi-DOF acoustic resonant system, thus significantly reducing the dynamic response of the coupled system. Furthermore, the fluctuation of the fluid equivalent mass induces a quasi-periodic motion pattern of the acoustic resonant system. Although increasing the excitation amplitude can effectively increase the dynamic response of the coupled system, it can also increase the fluctuation level of the dynamic response to a certain extent. By appropriately increasing the excitation frequency, the coupled system can operate at a new resonant frequency, thereby reducing the influence of fluid motion on the dynamic response of the system.

Features of director reorientation in a thin nematic film under the influence of crossed electric and magnetic fields.

The theory based on numerical study of the system of hydrodynamic equations, which includes the director motion, shows that under the influence of crossed electric E and magnetic B fields, the director reorients in such a way that the transient quasiperiodic patterns may arise in microsized nematic volumes if the corresponding distortion mode has the fastest response and thus suppresses all other modes, including uniform ones. It has been shown that there is a threshold value of the amplitude of the thermal fluctuations of the director over the microsized nematic film which provides the nonuniform rotation mode rather than the uniform one, whereas the lower values of the amplitude dominate the uniform mode.

Room-temperature stabilizing strongly competing ferrielectric and antiferroelectric phases in PbZrO3 by strain-mediated phase separation

PbZrO3 has been broadly considered as a prototypical antiferroelectric material for high-power energy storage. A recent theoretical study suggests that the ground state of PbZrO3 is threefold-modulated ferrielectric, which challenges the generally accepted antiferroelectric configuration. However, such a novel ferrielectric phase was predicted only to be accessible at low temperatures. Here, we successfully achieve the room-temperature construction of the strongly competing ferrielectric and antiferroelectric state by strain-mediated phase separation in PbZrO3/SrTiO3 thin film. We demonstrate that the phase separation occurs spontaneously in quasi-periodic stripe-like patterns under a compressive misfit strain and can be tailored by varying the film thickness. The ferrielectric phase strikingly exhibitsa threefold modulation period with a nearly up-up-down configuration, which could be stabilized and manipulated by the formation and evolution of interfacial defects under applied strain. The present results construct a fertile ground for further exploring the physical properties and applications based on the novel ferrielectric phase.

Formation of self-assembled nanodomain structure as a result of motion of oriented planar phase boundary in single crystals of tetragonal PMN-PT

We studied the formation of the domain structure as a result of the motion of a planar interphase boundary in the {001}-cut tetragonal PMN-PT crystals during cooling with oriented temperature gradient. The specialized heating/cooling stage was assembled for this purpose. Piezoresponse force microscopy was used to image the domain structure at the (001) and (010) surfaces of the samples with high spatial resolution, revealing a complex quasi-periodic domain pattern with a periodicity from 300 to 400 nm.

Hyperuniformity in two-dimensional periodic and quasiperiodic point patterns.

We study hyperuniform properties in various two-dimensional periodic and quasiperiodic point patterns. Using the histogram of the two-point distances, we develop an efficient method to calculate the hyperuniformity order metric, which quantifies the regularity of the hyperuniform point patterns. The results are compared with those calculated with the conventional running average method. To discuss how the lattice symmetry affects the order metric, we treat the trellis and Shastry-Sutherland lattices with the same point density as examples of periodic lattices, and Stampfli hexagonal and dodecagonal quasiperiodic tilings with the same point density as examples of quasiperiodic tilings. It is found that the order metric for the Shastry-Sutherland lattice (Stampfli dodecagonal tilings) is smaller than the other in the periodic (quasiperiodic) tiling, meaning that the order metric is deeply related to the lattice symmetry. Namely, the point pattern with higher symmetry is characterized by the smaller order metric when their point densities are identical. Order metrics for several other quasiperiodic tilings are also calculated.

A numerical study of a single isolated human ventricular cell response to a periodic pulse stimulus current using the Rogers–McCulloch model

Among the primary causes of cardiac arrhythmias are abnormalities in the generation of action potentials (APs) by the cardiac cells, which may be induced by alterations in the electrophysiological properties of the cells as well as the stimuli delivered to them. Through numerical simulations using the phenomenological model of Rogers–McCulloch, this study examined how external stimuli modulate the electrical activity of a single isolated human ventricular cell functioning under normal conditions. First, we investigated how the initial conditions of the two model state variables affect how the cell responds to a single rectangular current pulse. This allows for the determination of a critical curve, which separates the initial conditions that allow for AP activation from those that inhibit it. This critical curve can be used to predict AP activation not only when the cell is stimulated by a single current pulse but also by a periodic stimulus. Then, we examined the cell response to a periodic train of rectangular current pulses, with a focus on the effects of the stimulus pacing length and amplitude. Our findings demonstrated that, despite not considering any alteration in the dynamics of ionic currents through the cell membrane, the Rogers–McCulloch model was capable of reproducing some of the most common activation patterns that have been widely identified in several clinical, experimental, and numerical studies. Depending on the stimulus properties, the following activation patterns were identified: (1) AP activation in synchrony with the stimulus, with one AP activation per pulse; (2) quasi-periodic and periodic Wenckebach-like patterns, in which successive AP activations are interrupted by a skipped beat; (3) completely irregular patterns; (4) abnormal potential oscillations that interrupt the repolarization of a previously activated AP, followed or not by a skipped beat; and (5) abnormal potential oscillations that completely inhibit cell repolarization.

Dynamic behavior of explicit elliptic and quasi periodic-wave solutions to the generalized [formula omitted]-dimensional Kundu–Mukherjee–Naskar equation

This paper introduces novel explicit traveling solutions to the generalized (2+1)-dimensional Kundu–Mukherjee–Naskar equation, a model relevant in optical fiber pulses, fluid dynamics, medical imaging, and ocean rogue waves. We propose rational form functions that incorporate both sine and cosine variables or sinh and cosh variables as potential solutions to the model. Subsequently, we utilize 3D diagrams to visually illustrate the characteristics of the identified propagating solutions, which exhibit elliptic-periodic, convex-periodic, cusp-soliton, and quasi-periodic patterns. Furthermore, a graphical analysis is conducted to explore the impact of the model’s parameters; nonlinearity and dispersion. Additionally, we introduce a comparative analysis to assess the superiority of the proposed approach over other existing methods such as Kudryashov-expansion, sine-function, and tanh(coth)-expansion methods. We anticipate that the insights gained from this analysis will prove valuable for understanding and studying other nonlinear models.

Flow over a single dimple recessed in a flat plate

Direct numerical simulations have been conducted to investigate a zero-pressure-gradient boundary layer flow over a single shallow dimple. Here, the dimple depth to dimple diameter ratio (d/D) as well as the Reynolds number (based on D and free-stream velocity) are fixed at 0.05 and 20 000, respectively. The effect of inlet boundary layer thickness δ on a given dimple is investigated by considering δ/D∈[0.023,0.1]. The flow within the dimple exhibits either a horseshoe vortex (a continuous core line through the two spirals within the dimple) or a tornado-like vortex pair (discontinuous core line). For the given parameter range, four different flow patterns have been identified within the single dimple: (i) a steady symmetric horseshoe vortex pattern for δ/D∈[0.053,0.1], (ii) a steady asymmetric horseshoe vortex pattern for δ/D=0.04, (iii) a quasi-periodic asymmetric horseshoe vortex pattern for δ/D=0.033, and (iv) a mixed horseshoe and tornado-like vortex pattern for δ/D=0.023. The growth of the streamwise vorticity, mainly caused by the tilting of the vertical vorticity, plays a key role in the transition between the different flow patterns. Dimple-induced velocity streaks above the single dimple have been investigated in detail for the first time, showing four different streaks: (i) a high-speed streak above the dimple, (ii) two side-low-speed streaks located outside the dimple span, (iii) two side-high-speed streaks, and (iv) a mid-low-speed streak in between them. These are mainly caused by a flow acceleration effect and a flow diffuser effect over the dimple, as well as a “lift-up” mechanism within the downstream part of the dimple, tilting the boundary layer upward.

Is the Atmosphere of the Ultra-hot Jupiter WASP-121 b Variable?

We present a comprehensive analysis of the Hubble Space Telescope observations of the atmosphere of WASP-121 b, an ultra-hot Jupiter. After reducing the transit, eclipse, and phase-curve observations with a uniform methodology and addressing the biases from instrument systematics, sophisticated atmospheric retrievals are used to extract robust constraints on the thermal structure, chemistry, and cloud properties of the atmosphere. Our analysis shows that the observations are consistent with a strong thermal inversion beginning at ∼104 Pa on the dayside, solar to subsolar metallicity Z (i.e., −0.77<log(Z)<0.05 ), and supersolar C/O ratio (i.e., 0.59 < C/O < 0.87). More importantly, utilizing the high signal-to-noise ratio and repeated observations of the planet, we identify the following unambiguous time-varying signals in the data: (i) a shift of the putative hotspot offset between the two phase curves and (ii) varying spectral signatures in the transits and eclipses. By simulating the global dynamics of WASP-121 b's atmosphere at high resolution, we show that the identified signals are consistent with quasiperiodic weather patterns, hence atmospheric variability, with signatures at the level probed by the observations (∼5% to ∼10%) that change on a timescale of ∼5 planet days; in the simulations, the weather patterns arise from the formation and movement of storms and fronts, causing hot (as well as cold) patches of atmosphere to deform, separate, and mix in time.

Time-varying functional connectivity predicts fluctuations in sustained attention in a serial tapping task.

The mechanisms for how large-scale brain networks contribute to sustained attention are unknown. Attention fluctuates from moment to moment, and this continuous change is consistent with dynamic changes in functional connectivity between brain networks involved in the internal and external allocation of attention. In this study, we investigated how brain network activity varied across different levels of attentional focus (i.e., "zones"). Participants performed a finger-tapping task, and guided by previous research, in-the-zone performance or state was identified by low reaction time variability and out-of-the-zone as the inverse. In-the-zone sessions tended to occur earlier in the session than out-of-the-zone blocks. This is unsurprising given the way attention fluctuates over time. Employing a novel method of time-varying functional connectivity, called the quasi-periodic pattern analysis (i.e., reliable, network-level low-frequency fluctuations), we found that the activity between the default mode network (DMN) and task positive network (TPN) is significantly more anti-correlated during in-the-zone states versus out-of-the-zone states. Furthermore, it is the frontoparietal control network (FPCN) switch that differentiates the two zone states. Activity in the dorsal attention network (DAN) and DMN were desynchronized across both zone states. During out-of-the-zone periods, FPCN synchronized with DMN, while during in-the-zone periods, FPCN switched to synchronized with DAN. In contrast, the ventral attention network (VAN) synchronized more closely with DMN during in-the-zone periods compared with out-of-the-zone periods. These findings demonstrate that time-varying functional connectivity of low frequency fluctuations across different brain networks varies with fluctuations in sustained attention or other processes that change over time.