HRV Analysis: Unpredictability of Approximate Entropy in Chronic Obstructive Pulmonary Disease

Introduction: Approximate entropy (ApEn) is a widely imposed metric to evaluate a chaotic response and irregularities of RR-intervals from an electrocardiogram. Yet, the technique is problematic due to the accurate choice of the tolerance (r) and embedding dimension (M). We prescribed the metric to evaluate these responses in subjects exhibiting symptoms of chronic obstructive pulmonary disease (COPD) and we strived to overcome this disadvantage by applying different groupings to detect the optimal. Methods: We examined 38 subjects split equally: COPD and control. To evaluate autonomic modulation the heart rate was measured beat-by-beat for 30 min in a supine position without any physical, sensory, or pharmacological stimuli. In the time-series obtained the ApEn was then applied with set values for tolerance, r and embedding dimension, M. Then, the differences between the two groups and their effect size by two measures (Cohen’s ds and Hedges’s gs) were computed. Results: The highest value of statistical significance accomplished for any effect size statistical combinations undertaken was -1.13 for Cohen’s ds, and -1.10 for Hedges’s gs with embedding dimension, M = 2 and tolerance, r = 0.1. Conclusion: ApEn was capable of optimally identifying the decrease in chaotic response in COPD. The optimal combination of r and M for this were 0.1 and 2, respectively. Despite this, ApEn is a relatively unpredictable mathematical marker and the use of other techniques to evaluate a healthy or pathological condition is encouraged.

Avoiding chaos in granular dampers

Granular dampers are passive devices used to attenuate mechanical vibrations. The most common configuration consists in an enclosure, partiallyfilled with particles, attached to the vibrating structure that needs to be damped. The energy is dissipated due to inelastic collisions and friction between the grains and between the grains and the inner walls of the container as the structure vibrates. As a result of the collisions, the mechanical response of the system often results in chaotic motion even if the driving is harmonic. Despite the vibration attenuation achieved, this chaotic response may render the granular damper unsuitable for a range of applications. In this work, we showcase two simple modifications of the enclosure design that are able to mitigate the chaotic response of the granular damper. To this end we use Discrete Element Method simulations of: (a) a granular damper with a conical base, and (b) a granular damper with obstaclesfixed inside the enclosure. We compare results against a standardflat-base enclosure damper. The basic mechanical response of the dampers is characterized by measuring the apparent mass and the loss factor. The suppression of the chaotic response is assessed qualitatively via the phase space diagram.

Unreliability of Approximate Entropy to Locate Optimal Complexity in Diabetes Mellitus via Heart Rate Variability

Introduction: Approximate Entropy (ApEn) is a widely enforced metric to evaluate the chaotic response and irregularities of RR intervals from an electrocardiogram. We applied the metric to estimate these responses in subjects with type 1 diabetes mellitus (DM1). So far, as a technique it has one key problem – the accurate choices of the tolerance (r) and embedding dimension (M). So, we attempted to overcome this drawback by applying different groupings to detect the optimum. Methods: We studied 46 subjects split into two equal groups: DM1 and control. To evaluate autonomic modulation the heart rate was measured for 30 min in a supine position without any physical, sensory, or pharmacological stimuli. For the time-series, the ApEn was applied with set values for r (0.1→0.5 in intervals of 0.1) and M (1→5 in intervals of 1) and the differences between the two groups and their effect size by two measures (Cohen’s ds and Hedges’s gs) were computed. Results: The highest value of statistical significance accomplished for the effect sizes (ES) for any of the combinations performed was -0.7137 for Cohen’s ds and -0.7015 for Hedges’s gs with M = 2 and r = 0.08. Conclusion: ApEn was able to identify the reduction in chaotic response in DM1 subjects. Still, ApEn is relatively unreliable as a mathematical marker to determine this.

A New Chaotic Jerk System with Double-Hump Nonlinearity

In this paper, we report a new third-order chaotic jerk system with double-hump (bimodal) nonlinearity. The bimodal nonlinearity is of basic interest in biology, physics, etc. The proposed jerk system is able to exhibit chaotic response with proper choice of parameters. Importantly, the chaotic response is also obtained from the system by tuning the nonlinearity preserving its bimodal form. We analytically obtain the symmetry, dissipativity and stability of the system and find the Hopf bifurcation condition for the emergence of oscillation. Numerical investigations are carried out and different dynamics emerging from the system are identified through the calculation of eigenvalue spectrum, two-parameter and single parameter bifurcation diagrams, Lyapunov exponent spectrum and Kaplan–Yorke dimension. We identify that the form of the nonlinearity may bring the system to chaotic regime. Effect of variation of parameters that controls the form of the nonlinearity is studied. Finally, we design the proposed system in an electronic hardware level experiment and study its behavior in the presence of noise, fluctuations, parameter mismatch, etc. The experimental results are in good analogy with that of the analytical and numerical ones.

Research on Chaos Response of the Nonlinear Vibration System of Giant Magnetostrictive Actuator

In the present study, the chaotic response of the nonlinear magnetostrictive actuator (GMA) vibration system is investigated. The mathematical model of the nonlinear GMA vibration system is established according to J-A hysteresis nonlinear model, quadratic domain rotation model, Newton’s third law, and principle of GMA structural dynamics by analyzing the working principle of GMA. Then, the Melnikov function method is applied to the threshold condition of the chaotic response of the system to obtain the sense of Smale horseshoe transformation. Furthermore, the mathematical model is solved to investigate the system response to the excitation force and frequency. Accordingly, the corresponding displacement waveform, phase plane trajectory, Poincaré map, and amplitude spectrum are obtained. The experimental simulation is verified using Adams software. The obtained results show that the vibration equation of the nonlinear GMA vibration system has nonlinear and complex motion characteristics with different motion patterns. It is found that the vibration characteristics of the system can be controlled through adjusting the excitation force and frequency.