Divergence Exponent Research Articles

Observing and quantifying ocean flow properties using drifters with drogues at different depths

AbstractThis paper presents analyses of drifters with drogues at different depths – 1, 10, 30, 50 m – that were deployed in the Mediterranean Sea to investigate frontal subduction and upwelling. Drifter trajectories were used to estimate divergence, vorticity, vertical velocity, and finite-size Lyapunov exponents (FTLEs), and to investigate the balance of terms in the vorticity equation. The divergence and vorticity are O(f) and change sign along trajectories. Vertical velocity is O(1 mm/s), increases with depth, indicates predominant upwelling with isolated downwelling events, and sometimes changes sign between 1 and 50 m. Vortex stretching is one of, but not the only, significant term in the vorticity balance. 2D FTLEs are 2 × 10−51/s after 1 day, twice larger than in a 400-m-resolution numerical model. 3D FTLEs are 50% larger than 2D FTLEs and are dominated by the vertical shear of horizontal velocity. Bootstrapping suggests uncertainty levels of ~10% of the time-mean absolute values for divergence and vorticity. Analysis of simulated drifters in a model suggests that drifter-based estimates of divergence and vorticity are close to the Eulerian model estimates, except when drifters get aligned into long filaments. Drifter-based vertical velocity is close to the Eulerian model estimates at 1 m but differs at deeper depths. The errors in the vertical velocity are largely due to the lateral separation between drifters at different depths, and partially due to only measuring at 4 depths. Overall, this paper demonstrates how drifters, heretofore restricted to 2D near-surface observations, can be used to learn about 3D flow properties throughout the upper layer of the water column.

Approach to Phonon Relaxation Time and Mean Free Path in Nonlinear LatticesSupported by the National Natural Science Foundation of China (Grant Nos. 12075199 and 11675133).

Based on the self-consistent phonon theory, the spectral energy density is calculated by the canonical transformation and the Fourier transformation. Through fitting the spectral energy density by the Lorentzian profile, the phonon frequency as well as the phonon relaxation time is obtained in one-dimensional nonlinear lattices, which is validated in the Fermi–Pasta–Ulam-β (FPU-β) and ϕ 4 lattices at different temperatures. The phonon mean free path is then evaluated in terms of the phonon relaxation time and phonon group velocity. The results show that, in the FPU-β lattice, the phonon mean free path as well as the phonon relaxation time displays divergent power-law behavior. The divergent exponent coincides well with that derived from the Peierls–Boltzmann theory at weak anharmonic nonlinearity. The value of the divergent exponent expects a power-law divergent heat conductivity with system size, which violates Fourier’s law. For the ϕ 4 lattice, both the phonon relaxation time and mean free path are finite, which ensures normal heat conduction.

Motor Adaptation to Cognitive Challenges and Destabilizing Treadmill Walking in Healthy Older Adults

To develop effective fall prevention intervention, it is necessary to understand how older adults respond to challenges that demand cognitive-motor dual-tasking capability, an important capability in the daily lives. The purpose of this study is to investigate how older adults adjust their motor responses when encountering cognitive and walking perturbations simultaneously. We recorded kinematic data as subjects walked on a treadmill with or without 1) continuous random-amplitude treadmill platform sways (Perturbed vs. No-perturbed walking); and 2) each of the four cognitive tasks: Paced Auditory Serial Addition test (PASAT), clock test, visual color-word incongruent test (V-stroop), and auditory pitch-word incongruent test (A-stroop). We computed dynamic margins of stability (MOS), gait variability, and short-term local divergence exponent (LDE) of the trunk motion (local stability). Data of ten older subjects (age: 72.2±4.9) show that cognitive performance did not differ between standing, Perturbed or No-perturbed walking. Subjects demonstrated significantly greater local instability and variability in step measures, joint angle and MOS during Perturbed than No-perturbed walking (p<0.001). During dual-task conditions, subjects walked with significantly larger medio-lateral MOS (MOSML) compared to walking only, especially during early phase of the trial. During Perturbed walking, subjects had significantly larger MOSML during PASAT and Vstroop than walking only. Our data showed that subjects tried to increase their dynamic MOS during Perturbed walking or a cognitive task more difficult or taxing visual attention. However, the adjustments do not sustain throughout the trial. These findings suggest older adults tend to prioritize cognitive over walking tasks even when encountering walking perturbations.

Between-day reliability of IMU-derived spine control metrics in patients with low back pain

Inertial measurement units (IMUs) are a potentially useful tool for clinicians and researchers in assessing spine movement biomechanics and neuromuscular control patterns. This study assessed the between-day reliability of the HIKOB FOX IMU in measuring local dynamic stability (LDS) and variability of trunk movements in patients with chronic low back pain (LBP). The local divergence exponent (λmax) was used to quantify LDS and the mean standard deviation (MeanSD) between cycles was used to quantify variability during 30 repetitive cycles of flexion/extension, rotation, and complex movement tasks. For λmax the average coefficient of variation (CV) was ~10% in the flexion/extension and rotation tasks, and all CV values were <20% when also including the complex task. ICC values for λmax ranged from 0.28 to 0.81. Reliability of λmax was similar between the pelvis and thorax segments (CV: ~10%, ICC: 0.48–0.78) and worse for the lumbar spine (CV: ~15%, ICC: 0.28–0.59). The CV for MeanSD was typically in the range of 20–30%, with even greater CV in the non-primary axes during each task (30–52%). Similarly, ICC values were lowest about the anterior-posterior axis in the flexion/extension task (ICC: 0.15–0.29) and largest about the longitudinal axis in the rotation task (ICC: 0.76–0.88). The moderate between-day reliability of λmax in the sagittal and transverse planes offers improvement over manual and subjective tests with poor reliability that are currently used in clinics. The minimal detectable differences presented give a threshold for change in research and rehabilitation in patients with LBP.

Head orientation and gait stability in young adults, dancers and older adults

BackgroundControl of body orientation requires head motion detection by the vestibular system and small changes with respect to the gravitational acceleration vector could cause destabilization. Research questionWe aimed to compare the effects of different head orientations on gait stability in young adults, dancers and older adults. MethodsThree groups of 10 subjects were evaluated, the first composed of young adults (aged 18–30 years), the second composed of young healthy dancers under high performance dance training (aged 18–30 years), and the third group composed of community-dwelling older adults (aged 65–80 years). Participants walked on a treadmill at their preferred speed in four distinct head orientation conditions for four minutes each: control (neutral orientation); dynamic yaw (following a target over 45° bilaterally); up (15° neck extension), and down (40° neck flexion). Foot and trunk kinematic data were acquired using a 3D motion capture system and the gait pattern was assessed by basic gait parameters (step length, stride width and corresponding variability) and gait stability (local divergence exponents and margins of stability). Main effects of conditions and groups, as well as their interaction effects, were evaluated by repeated-measures analysis of variance. ResultsInteractions of group and head orientation were found for both step length and stride width variability; main effects of head orientation were found for all evaluated parameters and main effects of group were found for step length and its variability and local divergence exponents in all directions. SignificanceAs expected, the older adults group showed less stable gait (higher local divergence exponent), the shortest step length and greater step length variability. However, contrary to expectation, the dancers were not more stable. The yaw condition was the most challenging for all groups and the down condition seemed to be least challenging.

Self-reproducing chaos and bursting oscillation analysis in a meminductor-based conservative system.

In this paper, a new meminductor model with sine function is presented. Based on this meminductor and a capacitor, a simple conservative chaotic system is designed. The proposed system has rich dynamic characteristics, including zero divergence, self-reproducing chaos, bursting oscillations, and symmetric Lyapunov exponent spectra. The corresponding mechanisms of these dynamic behaviors are analyzed theoretically. Furthermore, Multisim simulations and experimental circuit are performed to verify the numerical results.

Asymptotic behavior of the time-dependent divergence exponent.

The divergence rate method, which is used to determine the maximum Lyapunov exponent out of time series, is based on the evaluation of the time-dependent divergence exponent. For chaotic systems and in the small time regime, this exponent grows linearly in time. The asymptotic regime is instead characterized by a time-independent behavior due to the system eventually losing its memory of the starting conditions. The amplitude of this "plateau"-like divergence exponent depends both on the choice of the embedding dimension and lag and on the maximum distance of nearby starting trajectories in a way that is characteristic of the underlying dynamical system. In this paper, upon introducing the basic mathematical tools, we address the plateau evaluation for two classes of time series, those generated by a white noise source and those generated by a finite-dimensional chaotic system. The different behavior provides a novel tool to distinguish purely stochastic sources from deterministic ones, as well as to provide a precise estimate of the correlation dimension in the latter case. The method is also sensitive to correlated noise sources.

Gait stability at early stages of multiple sclerosis using different data sources

BackgroundPeople at early stages of multiple sclerosis have subtle balance problems that may affect gait stability. However, differences in methods of determining stability such as sensor type and placements, may lead to different results and affect their interpretation when comparing to controls and other studies. QuestionsDo people with multiple sclerosis (PwMS) exhibit lower gait stability? Do location and type of data used to calculate stability metrics affect comparisons? Methods30 PwMS with no walking impairments as clinically measured and 15 healthy controls walked on a treadmill at 1.2 ms−1 while 3D acceleration data was obtained from sacrum, shoulder and cervical markers and from an accelerometer placed at the sacrum. The local divergence exponent was calculated for the four data sources. An ANOVA with group (multiple sclerosis and control) and data source as main factors was used to determine the effect of disease, data source and their interaction on stability metrics. ResultsPwMS walked with significantly less stability according to all sensors (no interaction). A significant effect of data source on stability was also found, indicating that the local divergence exponent derived from sacrum accelerometer was lower than that derived from the other 3 sensor locations. SignificancePwMS with no evident gait impairments are less stable than healthy controls when walking on a treadmill. Although different data sources can be used to determine MS-related stability deterioration, a consensus about location and data source is needed. The local divergence exponent can be a useful measure of progression of gait instability at early stages of MS.

Assessment of Human Dynamic Gait Stability With a Lower Extremity Assistive Device.

This paper focuses on assessing gait stability by metrics derived from dynamical systems theory to understand the influence of unilateral robot assistance on the human walking pattern. A motorized assistive robot is applied to the right knee joint to provide stance support. The metrics related to global stability (the maximum Floquet multiplier, max FM), local stability (short-term and long-term divergence exponents, [Formula: see text] and [Formula: see text]), and variability (median absolute deviation, MAD) are considered. These metrics are derived for bilateral hip, knee, and ankle joint angles. Additionally, a biomechanical metric, the minimum margin of stability is assessed. Experiments are conducted on 11 healthy participants with different robot controllers. The max FM and [Formula: see text] yield statistically significant results, showing that the unassisted (left) leg is more stable in right knee assistance conditions when compared to the normal walking condition due to inter-limb coordination. Moreover, MAD and [Formula: see text] show that the variability and chaotic order of walking pattern during assistance are lower than those of normal walking. The proposed control strategy (automatic impedance tuning, AIT) improves local and orbital gait stability compared to existing controllers such as finite-state machine (FSM). The assessment of dynamic gait stability presented in this paper provides insights for further improving control strategies of assistive robots to help a user reach improved gait stability while maintaining appropriate variability.

Effects of inclined surfaces on gait variability and stability in unilateral lower limb amputees.

This study aims to analyze gait variability and stability of individuals with amputation walking on upward (8%), horizontal (0%), and downward (- 8%) inclines, by using linear and nonlinear descriptors. Trunk linear variability and gait spatiotemporal parameters were evaluated. Nonlinear variability (local dynamic stability-LDS), was estimated by the maximum Lyapunov exponent (λ) computed from a trunk marker velocity. The gait descriptors were compared among three distinct groups (N participants): unilateral transtibial amputees (TTA, N = 12); unilateral transfemoral amputees (TFA, N = 13); control group (CT, N = 15). For step width and support phase, the effect of inclination was greater for TFA group, especially in the DOWN condition. Linear variability was higher for amputees (TFA and TTA) especially in the UP condition in the medial-lateral and anterior-posterior directions. TTA and TFA groups presented greater λ values than CT group in medial-lateral direction indicating decreased LDS, but TFA group presented smaller λ values than TTA and CT groups in theV direction. Our findings showed that inclination introduced significant changes in the estimated parameters for all groups, with greater changes for amputee groups. Furthermore, the level of amputation directly affects the analyzed gait parameters being the TFA group the one which presents greater changes. Graphical abstract The objective of this study was to analyze gait variability of individuals with amputation walking on (A) upward (+ 8%), (B) horizontal (0%), and (C) downward (- 8%) inclines, by using linear and nonlinear biomechanical descriptors. Linear measures of variability, such trunk variability and gait spatiotemporal parameters were evaluated. Nonlinear variability was estimated by the exponent of divergence (maximum Lyapunov exponent) of the velocity of a marker fixed in the subject's trunk while walking on inclined surfaces.

Anomalous quantum Griffiths singularity in ultrathin crystalline lead films

Superconductor-insulator/metal transition (SIT/SMT) represents a prototype of quantum phase transition, where quantum fluctuation plays a dominant role and dramatically changes the physical properties of low-dimensional superconducting systems. Recent observation of quantum Griffiths singularity (QGS) offers an essential perspective to understand the subtleties of quantum phase transition in two-dimensional superconductors. Here we study the magnetic field induced SMT in ultrathin crystalline Pb films down to ultralow temperatures. The divergent critical exponent is observed when approaching zero temperature quantum critical point, indicating QGS. Distinctively, the anomalous phase boundary of SMT that the onset critical field decreases with decreasing temperatures in low temperature regime distinguishes our observation from previous reports of QGS in various two-dimensional superconductors. We demonstrate that the anomalous phase boundary originates from the superconducting fluctuations in ultrathin Pb films with pronounced spin-orbit interaction. Our findings reveal a novel aspect of QGS of SMT in two-dimensional superconductors with anomalous phase boundary.

Quantum critical properties of Bose–Hubbard models

The Mott insulator-to-superfluid transition exhibited by the Bose–Hubbard model on a two-dimensional square lattice occurs for any value of the chemical potential, but becomes critical at the tips of the so-called Mott lobes only. Employing a numerical approach based on a combination of high-order perturbation theory and hypergeometric analytic continuation we investigate how quantum critical properties manifest themselves in computational practice. We consider two-dimensional triangular lattices and three-dimensional cubic lattices for comparison, providing accurate parametrizations of the phase boundaries at the tips of the respective first lobes. In particular, we lend strong support to a recently suggested inequality which bounds the divergence exponent of the one-particle correlation function in terms of that of the two-particle correlation function, and which sharpens to an equality if and only if a system becomes critical.

Effects of Pulmonary Rehabilitation on Gait Characteristics in Patients with COPD.

Pulmonary rehabilitation (PR) improves lower-limb muscle function in patients with chronic obstructive pulmonary disease (COPD). However, it remains unclear whether patients improve gait characteristics, in particular stride-to-stride fluctuations that are associated with fall risks. This study aims to identify whether, and to what extent, PR affects positively gait characteristics in COPD. In this prospective observational study, 44 COPD patients (aged: 62 ± 7 years; Forced expiratory volume in 1 s 56 ± 20% predicted) performed self-paced, treadmill 6-min-walk tests (Gait Real-time Analysis Interactive Lab) before and after PR, while spatiotemporal parameters and center of mass position were recorded (100 Hz, Vicon Nexus). Standard deviation, coefficient of variation, predictability (sample entropy), and consistency in organization (local divergence exponent) were calculated. Sub-analysis was performed to identify gait differences between good and poor responders (<30 m change in a 6-min-walk distance). Patients demonstrated shorter stride times (p = 0.001) and improved lower-limb muscle function (p < 0.001) following PR. The good responders had a greater increase in stride length (p < 0.001) and a greater decrease in stride time (p < 0.001) compared to the poor responders. Current PR improved stride time in patients, while movement patterns within stride-to-stride fluctuations did not change. Training programs specifically targeting balance issues and gait function may be beneficial in improving gait characteristics in COPD.

An autonomous chaotic and hyperchaotic oscillator using OTRA

This research paper reports a novel design for third order chaotic and hyperchaotic oscillator with cubic nonlinearity using single operational trans-resistance amplifier (OTRA) and few passive elements. The key nonlinear dynamical characteristics in terms of sensitivity, divergence, equilibrium point and Lyapunov exponent are recorded in this literature. The operational activity of the proposed oscillator based on OTRA is integrated using 0.25 µm TSMC CMOS parameter. For the generation of hyperchaotic oscillator, an external capacitor is added to the third order chaotic oscillator. To justify the theoretical nonlinear dynamics of proposed chaotic oscillator, PSPICE simulation by using CMOS based OTRA and experimental investigation using IC AD844 based OTRA are well implemented.

Temperature-dependent divergence of thermal conductivity in momentum-conserving one-dimensional lattices with asymmetric potential.

In this study we used a nonequilibrium simulation method to investigate the temperature dependent divergence of thermal conductivity in a one-dimensional momentum conserving system with an asymmetric double well nearest-neighbor interaction potential. We show that across all temperatures thermal conductivity exhibits power-law divergence with the chain length and the value of the divergence exponent (α) depends on the temperature of the system. At low and high temperatures α reaches close to ∼0.5 and ∼0.33, respectively. Whereas in the intermediate temperature the divergence of thermal conductivity with the chain length saturates with α∼0.07. Subsequent analysis showed that the estimated value of α in the intermediate temperature may not have reached its thermodynamic limit. Further calculations of local α revealed that its approach towards the thermodynamic limit is crucially dependent on the temperature of the system. At low and high temperatures local α reaches its thermodynamic limits in shorter chain lengths. On the contrary, in the case of intermediate temperature its progress towards the asymptotic limit is nonmonotonic.

On the manipulability of swing foot and stability of human locomotion

Manipulability is a measure that quantifies the range of possible motions of a robotic end-effector and it is also an important measure in the study of coordination of human upper body and grasping tasks. This measure, which is defined on both the kinematic and dynamic level, could be useful in gait, as it could be used to determine potential foot placement possibilities. Kinematic manipulability is defined based on the Jacobian and dynamic manipulability on both the Jacobian and mass-inertia matrix. The main purpose of this study was to evaluate the manipulability of human walking and to explore a possible relation between the manipulability and dynamic stability of walking at different speeds. A 37-DoF tree-like model of the human body was developed to evaluate the manipulability index of human walking. We measured kinematics of 11 healthy male subjects while walking on a treadmill, and mapped the data to the model using inverse kinematics. Jacobian based kinematic/dynamic manipulability measures of walking were evaluated for the swing phase of walking. Manipulability ellipsoids were drawn for geometric determination of this measure in all directions during early, mid- and late swing phases. As stability metrics, the local divergence exponent and Floquet Multipliers were calculated. The results indicated a high kinematic manipulability of the swing foot during early and late swing phases and a drop in kinematic manipulability during mid-swing. Kinematic manipulability of the swing leg during early and late (but not mid-) swing phases increased with walking speed but the average kinematic manipulability of the center of mass and dynamic manipulability of swing foot decreased with increasing walking speed. Moreover, the results showed a weak relation between the manipulability and local and orbital stability.

Stepping behaviour contributes little to balance control against continuous mediolateral trunk perturbations.

Human bipedal gait is exceptionally stable, but the underlying strategies to maintain stability are unclear, especially in the frontal plane. Our study investigated balance strategies of healthy adults subjected to continuous mediolateral oscillations at the trunk during walking. We used a backpack with a passive inverted pendulum to create perturbations that were fixed, in-phase or out-of-phase with subjects' trunk. We evaluated subjects' corrective strategies and whether they yielded equivalent stability, measured by the margin of stability and the local divergence exponent. The margin of stability measure quantified adjustments in step behaviour relative to the centre of mass, and the local divergence exponent measure characterized the chaotic behaviour of the system throughout the entire trial. Among the conditions, there was no significant difference in the step width. We found a higher margin of stability for the out-of-phase condition and the lowest local divergence exponent for the in-phase condition and the highest for the fixed condition. These results indicate that the in-phase condition was more stable with respect to fluctuations throughout gait cycles, and the out-of-phase condition was more stable in terms of foot placement relative to centre of mass. To maintain equivalent or greater gait stability, subjects elected to reduce the motion of their centre of mass rather than alter step width. The reduction in centre of mass motion without a reduction in step width suggests direct control of the centre of mass to maintain stability was preferred over adjusting stepping behaviour.

Local Dynamic Stability of the Locomotion of Lower Extremity Joints and Trunk During Backward Upslope Walking

Backward slope walking was considered as a practical rehabilitation and training skill. However, its gait stability has been hardly studied, resulting in its limited application as a rehabilitation tool. In this study, the effect of walking direction and slope grade were investigated on the local dynamic stability of the motion of lower extremity joints and trunk segment during backward and forward upslope walking (BUW/FUW). The local divergence exponents (λS) of 16 adults were calculated during their BUW and FUW at grades of 0%, 5%, 10%, and 15%. Mean standard deviation over strides (MeanSD) was analyzed as their gait variability. Backward walking showed larger λS for the abduction-adduction and rotational angles of knee and ankle on inclined surface than forward walking, while λS for hip flexion-extension angle at steeper grades was opposite. No grade effect for any joint existed during BUW, while λS increased with the increasing grade during FUW. As to the trunk, walking direction did little impact on λS. Still, significant larger λS for its medial-lateral and vertical motion were found at the steeper grades during both FUW and BUW. Results indicate that during BUW, the backward direction may influence the stability of joint motions, while the trunk stability was challenged by the increasing grades. Therefore, BUW may be a training tool for the stability of both upper and lower body motion during gait.

Quantum Renormalization of Spin Squeezing in Spin Chains

By employing quantum renormalization group (QRG) method, we investigate quantum phase transitions (QPT) in the Ising transverse field (ITF) model and in the XXZ Heisenberg model, with and without Dzyaloshinskii Moriya (DM) interaction, on a periodic chain of N lattice sites. We adopt a new approach called spin squeezing as an indicator of QPT. Spin squeezing, through analytical expression of a spin squeezing parameter, is calculated after each step of QRG. As the scale of the system becomes larger, (after many QRG steps), the ground state (GS) spin squeezing parameters show an abrupt change at a quantum critical point (QCP). Moreover, in all of the studied models, the first derivative of the spin squeezing parameter with respect to the control parameter is discontinuous, which is a signature of QPT. The spin squeezing parameters develop their saturated values after enough QRG iterations. The divergence exponent of the first derivative of the spin squeezing parameter in the near vicinity of the QCP is associated with the critical exponent of the correlation length.

Signature of quantum Griffiths singularity state in a layered quasi-one-dimensional superconductor

Quantum Griffiths singularity was theoretically proposed to interpret the phenomenon of divergent dynamical exponent in quantum phase transitions. It has been discovered experimentally in three-dimensional (3D) magnetic metal systems and two-dimensional (2D) superconductors. But, whether this state exists in lower dimensional systems remains elusive. Here, we report the signature of quantum Griffiths singularity state in quasi-one-dimensional (1D) Ta2PdS5 nanowires. The superconducting critical field shows a strong anisotropic behavior and a violation of the Pauli limit in a parallel magnetic field configuration. Current-voltage measurements exhibit hysteresis loops and a series of multiple voltage steps in transition to the normal state, indicating a quasi-1D nature of the superconductivity. Surprisingly, the nanowire undergoes a superconductor-metal transition when the magnetic field increases. Upon approaching the zero-temperature quantum critical point, the system uncovers the signature of the quantum Griffiths singularity state arising from enhanced quenched disorders, where the dynamical critical exponent becomes diverging rather than being constant.