Synthetically non-Hermitian nonlinear wave-like behavior in a topological mechanical metamaterial.

Topological mechanical metamaterials have enabled new ways to control stress and deformation propagation. Exemplified by Maxwell lattices, they have been studied extensively using a linearized formalism. Herein, we study a two-dimensional topological Maxwell lattice by exploring its large deformation quasi-static response using geometric numerical simulations and experiments. We observe spatial nonlinear wave-like phenomena such as harmonic generation, localized domain switching, amplification-enhanced frequency conversion, and solitary waves. We further map our linearized, homogenized system to a non-Hermitian, nonreciprocal, one-dimensional wave equation, revealing an equivalence between the deformation fields of two-dimensional topological Maxwell lattices and nonlinear dynamical phenomena in one-dimensional active systems. Our study opens a regime for topological mechanical metamaterials and expands their application potential in areas including adaptive and smart materials and mechanical logic, wherein concepts from nonlinear dynamics may be used to create intricate, tailored spatial deformation and stress fields greatly transcending conventional elasticity.

Coupled atmospheric chemistry, radiation, and dynamics of an exoplanet generate self-sustained oscillations.

Nonlinearity in photochemical systems is known to allow self-sustained oscillations, but they have received little attention in studies of planetary atmospheres. Here, we present a unique, self-oscillatory solution for ozone chemistry of an exoplanet from a numerical simulation using a fully coupled, three-dimensional (3D) atmospheric chemistry-radiation-dynamics model. Forced with nonvarying stellar insolation and emission flux of nitric oxide (NO), atmospheric ozone abundance oscillates by a factor of thirty over a multidecadal timescale. As such self-oscillations can only occur with biological nitrogen fixation contributing to NO emission, we propose that they are a unique class of biosignature. The resulting temporal variability in the atmospheric spectrum is potentially observable. Our results underscore the importance of revisiting the spectra of exoplanets over multidecadal timescales to characterizing the atmospheric chemistry of exoplanets and searching for exoplanet biosignatures. There are also profound implications for comparative planetology and the evolution of the atmospheres of terrestrial planets in the solar system and beyond. Fully coupled, 3D atmospheric chemistry-radiation-dynamics models can reveal new phenomena that may not exist in one-dimensional models, and hence, they are powerful tools for future planetary atmospheric research.

Electrostatic Control of Nonlinear Photonic-Crystal Polaritons in a Monolayer Semiconductor.

Integration of 2D semiconductors with photonic crystal slabs provides an attractive approach to achieving strong light-matter coupling and exciton-polariton formation in a chip-compatible geometry. However, for the development of practical devices, it is crucial that polariton excitations are easily tunable and exhibit a strong nonlinear response. Here we study neutral and charged exciton-polaritons in an electrostatically gated photonic crystal slab with an embedded monolayer semiconductor MoSe2 and experimentally demonstrate a novel approach to optical control based on polariton nonlinearity. We show that spatial modulation of the dielectric environment within the photonic crystal unit cell results in the formation of two distinct excitonic species with significantly different nonlinear responses of the corresponding charged exciton-polaritons under optical pumping. This behavior enables optical switching with ultrashort laser pulses and can be sensitively controlled via an electrostatic gate voltage. Our results open new avenues toward the development of active polaritonic devices in a compact chip-compatible implementation.

Adaptive Signal Modulation Evolved by the Inherent Nonlinearity of Phase-Change Quantum-Dot String.

To simulate a topological neural network handling weak signals via stochastic resonance (SR), it is necessary to introduce an inherent nonlinearity into nanoscale devices. We use the self-assembly method to successfully fabricate a phase-change quantum-dot string (PCQDS) crossing Pd/Nb:AlNO/AlNO/Nb:AlNO/Pd multilayer. The inherent nonlinearity of phase change couples with electron tunneling so that PCQDS responds to a long signal sequence in a modulated output style, in which the pulse pattern evolves to that enveloped by two sets of periodic wave characterized by neural action potential. We establish an SR mode consisting of several two-state systems in which dissipative tunneling is coupled to environment. Size oscillations owing to NbO QDs adaptively adjust barriers and wells, such that tunneling can be periodically modulated by either asymmetric energy or local temperature. When the external periodic signals are applied, the system first follows the forcing frequency. Subsequently, certain PCQDs oscillate independently and consecutively to produce complicated frequency and amplitude modulations.

Probing Temperature-Induced Plasmonic Nonlinearity: Unveiling Opto-Thermal Effects on Light Absorption and Near-Field Enhancement.

Precise measurement and control of local heating in plasmonic nanostructures are vital for diverse nanophotonic devices. Despite significant efforts, challenges in understanding temperature-induced plasmonic nonlinearity persist, particularly in light absorption and near-field enhancement due to the absence of suitable measurement techniques. This study presents an approach allowing simultaneous measurements of light absorption and near-field enhancement through angle-resolved near-field scanning optical microscopy with iterative opto-thermal analysis. We revealed gold thin films exhibit sublinear nonlinearity in near-field enhancement due to nonlinear opto-thermal effects, while light absorption shows both sublinear and superlinear behaviors at varying thicknesses. These observations align with predictions from a simple harmonic oscillation model, in which changes in damping parameters affect light absorption and field enhancement differently. The sensitivity of our method was experimentally examined by measuring the opto-thermal responses of three-dimensional nanostructure arrays. Our findings have direct implications for advancing plasmonic applications, including photocatalysis, photovoltaics, photothermal effects, and surface-enhanced Raman spectroscopy.

Optimization of a deep mutational scanning workflow to improve quantification of mutation effects on protein-protein interactions.

Deep Mutational Scanning (DMS) assays are powerful tools to study sequence-function relationships by measuring the effects of thousands of sequence variants on protein function. During a DMS experiment, several technical artefacts might distort non-linearly the functional score obtained, potentially biasing the interpretation of the results. We therefore tested several technical parameters in the deepPCA workflow, a DMS assay for protein-protein interactions, in order to identify technical sources of non-linearities. We found that parameters common to many DMS assays such as amount of transformed DNA, timepoint of harvest and library composition can cause non-linearities in the data. Designing experiments in a way to minimize these non-linear effects will improve the quantification and interpretation of mutation effects.

Nonlinear life table response experiment analysis: Decomposing nonlinear and nonadditive population growth responses to changes in environmental drivers.

Life table response experiments (LTREs) decompose differences in population growth rate between environments into separate contributions from each underlying demographic rate. However, most LTRE analyses make the unrealistic assumption that the relationships between demographic rates and environmental drivers are linear and independent, which may result in diminished accuracy when these assumptions are violated. We extend regression LTREs to incorporate nonlinear (second-order) terms and compare the accuracy of both approaches for three previously published demographic datasets. We show that the second-order approach equals or outperforms the linear approach for all three case studies, even when all of the underlying vital rate functions are linear. Nonlinear vital rate responses to driver changes contributed most to population growth rate responses, but life history changes also made substantial contributions. Our results suggest that moving from linear to second-order LTRE analyses could improve our understanding of population responses to changing environments.

Abnormally low sensorimotor α band nonlinearity serves as an effective EEG biomarker of Parkinson's disease.

Biomarkers obtained from the neurophysiological signals of patients with Parkinson's disease (PD) have objective value in assessing their motor condition for effective diagnosis, monitoring, and clinical intervention. Prominent cortical biomarkers of PD have typically been derived from various ß band wave features. This study approached the topic from an alternative perspective and attempted to estimate a recently suggested measure representing α band nonlinear autocorrelative memory from a publicly available EEG dataset that involves 15 patients with earlier-stage PD (dopaminergic medication OFF and ON states) and 16 age-matched healthy controls. The cortical nonlinearity was elevated for the PD ON state compared with the OFF state for bilateral sensorimotor channels C3 and C4 (n = 26; P = 0.003). A similar statistical difference was also identified between PD OFF state and healthy subjects (n = 26; P = 0.049). Analysis over all channels revealed that the α band nonlinearity induced upon medication was constrained to sensorimotor regions. The α nonlinearity measure was compared with a well-accepted cortical biomarker of ß-γ phase-amplitude coupling (PAC). They were in moderate negative correlation (r = -0.412; P = 0.036) for only healthy subjects, but not for the patients. The nonlinearity measure was found to be insusceptible to the nonstationary variations within the particular data. Our study provides further evidence that the α band nonlinearity measure can serve as a promising cortical biomarker of PD. The suggested measure can be estimated from a noninvasive low-resolution single scalp EEG channel of patients with relatively early-stage PD, who did not yet need to undergo deep brain stimulation operation.NEW & NOTEWORTHY This study suggests a nonlinearity measure that differentiates Parkinson's disease (PD) dopamine OFF-state scalp EEG data from those of dopamine ON-state patients and healthy subjects. Unlike typical PD cortical biomarkers based on ß band activity, this metric shows elevation upon dopaminergic medication in the α band. We provide evidence supporting its potential as an early-stage promising PD biomarker that can be estimated from noninvasive EEG recordings with low resolution and SNR.

2D Memory Selectors with Giant Nonlinearity Enabled by Van der Waals Heterostructures.

The integration of one-selector-one-resistor crossbar arrays requires the selectors featured with high nonlinearity and bipolarity to prevent leakage currents and any crosstalk among distinct cells. However, a selector with sufficient nonlinearity especially in the frame of device miniaturization remains scarce, restricting the advance of high-density storage devices. Herein, a high-performance memory selector is reported by constructing a graphene/hBN/WSe2 heterostructure. Within the temperature range of 300-80 K, the nonlinearity of this selector varies from ≈103 - ≈104 under forward bias, and increases from ≈300 - ≈105 under reverse bias, the highest reported nonlinearity among 2D selectors. This improvement is ascribed to direct tunneling at low bias and Fowler-Nordheim tunneling at high bias. The tunneling current versus voltage curves exhibit excellent bipolarity behavior because of the comparable hole and electron tunneling barriers, and the charge transport polarity can be effectively tuned from N-type or P-type to bipolar by simply changing source-drain bias. In addition, the conceptual memory selector exhibits no sign of deterioration after 70 000 switching cycles, paving the way for assembling 2D selectors into modern memory devices.

Flexible Anionic Groups-Activated Structure Dissymmetry for Strong Nonlinearity in Ln2 Ae3 MIV 3 S12 Family.

Structural dissymmetry and strong second-harmonic generation (SHG) responses are key conditions for nonlinear optical (NLO) crystals, and targeted combinatorial screening of suitable anionic groups has become extremely effective. Herein, optimal combination of flexible SnSn (n = 5, 6) groups and highly electropositive cations (lanthanides (Ln3+ ) and alkaline earth (Ae2+ : Sr, Ca) metals) affords the successful synthesis of 12 NLO thiostannates including Ln2 Sr3 Sn3 S12 (Pmc21 ) and Ln2 Ca3 Sn3 S12 (P-62m); whereas 17 rigid GeS4 or SiS4 tetrahedra-constructed Ln2 Ae3 Ge3 S12 and Ln2 Ae3 Si3 S12 crystallize in the centrosymmetric (CS) Pnma. This unprecedented CS to noncentrosymmetric (NCS) structural transformation (Pnma to P-62m to Pmc21 ) in the Ln2 Ae3 MIV 3 S12 family indicates that chemical substitution of the tetrahedral GeS4 /SiS4 units with SnSn breaks the original symmetry to form the requisite NCS structures. Remarkably, strong polarization anisotropy and hyperpolarizability of the Sn(4+) S5 unit afford huge performance improvement from the nonphase-matching (NPM) SHG response (1.4 × AgGaS2 and Δn = 0.008) of La2 Ca3 Sn3 S12 to the strong phase-matching (PM) SHG effect (3.0 × AgGaS2 and Δn = 0.086) of La2 Sr3 Sn3 S12 . Therefore, Sn(4+) S5 is proven to be a promising "NLO-active unit." This study verifies that the coupling of flexible SnSn building blocks into structures opens a feasible path for designing targeted NCS crystals with strong nonlinearity and optical anisotropy.

Input-size dependence of the baroreflex neural arc transfer characteristics during Gaussian white noise inputs.

Although Gaussian white noise (GWN) inputs offer a theoretical framework for identifying higher-order nonlinearity, an actual application to the data of the neural arc of the carotid sinus baroreflex did not succeed in fully predicting the well-known sigmoidal nonlinearity. In the present study, we assumed that the neural arc can be approximated by a cascade of a linear dynamic (LD) component and a nonlinear static (NS) component. We analyzed the data obtained using GWN inputs with a mean of 120 mmHg and standard deviations (SDs) of 10, 20, and 30 mmHg for 15 min each in anesthetized rats (n = 7). We first estimated the linear transfer function from carotid sinus pressure to sympathetic nerve activity (SNA) and then plotted the measured SNA against the linearly predicted SNA. The predicted and measured data pairs exhibited an inverse sigmoidal distribution when grouped into 10 bins based on the size of the linearly predicted SNA. The sigmoidal nonlinearity estimated via the LD-NS model showed a midpoint pressure (104.1 ± 4.4 mmHg for SD of 30 mmHg) lower than that estimated by a conventional stepwise input (135.8 ± 3.9 mmHg, P < 0.001). This suggests that the NS component is more likely to reflect the nonlinearity observed during pulsatile inputs that are physiological to baroreceptors. Furthermore, the LD-NS model yielded higher R2 values compared with the linear model and the previously suggested second-order Uryson model in the testing dataset.NEW & NOTEWORTHY We examined the input-size dependence of the baroreflex neural arc transfer characteristics during Gaussian white noise inputs. A linear dynamic-static nonlinear model yielded higher R2 values compared with a linear model and captured the well-known sigmoidal nonlinearity of the neural arc, indicating that the nonlinear dynamics contributed to determining sympathetic nerve activity. Ignoring such nonlinear dynamics might reduce our ability to explain underlying physiology and significantly limit the interpretation of experimental data.

Does natural resource dependence restrict green development? An investigation from the "Belt and road" countries.

Addressing natural resource dependence is integral to achieving the Sustainable Development Goals by promoting economic diversification, environmental sustainability, and climate resilience. This study explores the effect of natural resource dependence on green development by adopting the balanced panel dataset from the "Belt and Road" countries from 2005 to 2019. Notably, the novelty of our analysis lies in the empirical analysis using instrument-based techniques that consolidate the "green development curse hypothesis" in the Belt and Road countries. The mechanism analysis reveals that natural resource dependence curbs green development by weakening innovative capability, disturbing institutional quality, reducing population density, and crowding out human capital. Further, the dynamic panel threshold model handling endogeneity verifies the nonlinear relationship between natural resource dependence and green development. Interestingly, digital trade offers greater "resilience" than traditional trade, correcting the resource curse dilemma. Finally, heterogeneity analyses indicate that the green development curse hypothesis only exists in countries with high-level environmental regulations and resource-based countries.

Long-term noise exposures and cardiovascular diseases mortality: A study in 5 U.S. states.

BACKGROUND: Previous studies have linked noise exposure with adverse cardiovascular events. However, evidence remains inconsistent, and most previous studies only focused on traffic noise, excluding other anthropogenic sources like constructions, industrial process and commercial activities. Additionally, few studies have been conducted in the U.S. or evaluated the non-linear exposure-response relationships. METHODS: We conducted a relative incidence analysis study using all cardiovascular diseases mortality as cases (n = 936,019) and external causes mortality (n = 232,491) as contrast outcomes. Mortality records geocoded at residential addresses were obtained from five U.S. states (Indiana, 2007; Kansas, 2007-2009, Missouri, 2010-2019, Ohio, 2007-2013, Texas, 2007-2016). Time-invariant long-term noise exposure was obtained from a validated model developed based on acoustical measurements across 2000-2014. Noises from both natural sources (natural activities, including animals, insects, winds, water flows, thunder, etc.) and anthropogenic sources (human activities, including transportation, industrial activities, community facilities & infrastructures, commercial activities, entertainments, etc.) were included. We used daytime and nighttime total anthropogenic noise & day-night average sound pressure level combining natural and anthropogenic sources as exposures. Logistic regression models were fit controlling for Census tract-level & individual-level characteristics. We examined potential modification by sex by interaction terms and potential non-linear associations by thin plate spline terms. RESULTS: We observed positive associations for daytime anthropogenic L50 (sound level exceeded 50% of time) noise (10-dBA OR = 1.047, 95%CI 1.025-1.069), nighttime anthropogenic L50 noise (10-dBA OR = 1.061, 95%CI 1.033-1.091) in a two-exposure-term model, and overall Ldn (day-night average) sound pressure level (10-dBA OR = 1.064, 95%CI 1.040-1.089) in single-exposure-term model. Females were more susceptible to all three exposures. All exposures showed monotonic positive associations with cardiovascular mortality up to certain thresholds around 45-55 dBA, with a generally flattened or decreasing trend beyond those thresholds. CONCLUSIONS: Both daytime anthropogenic and nighttime anthropogenic noises were associated with cardiovascular disease mortality, and associations were stronger in females.

A nonlinear correlation between the serum uric acid to creatinine ratio and the prevalence of hypertension: a large cross-sectional population-based study.

OBJECTIVE: To explore the relationship between the serum uric acid to creatinine (UA/Cr) ratio and the prevalence of hypertension. METHODS: In this cross-sectional study, we included 8571 individuals from the China Health and Nutrition Survey. Logistic regression analysis and restricted cubic spline (RCS) were used to analyze the relationship between the UA/Cr ratio and hypertension. RESULTS: Compared with individuals without hypertension, individuals with hypertension had higher UA/Cr ratios. Multivariate logistic regression analysis showed that a higher UA/Cr ratio was closely related to a higher risk of hypertension (as a continuous variable, OR: 1.054, 95% CI: 1.014-1.095, p = 0.007; as a categorical variable, Q3 vs. Q1, OR: 1.183, 95% CI: 1.011-1.384, p = 0.035; Q4 vs. Q1, OR: 1.347, 95% CI: 1.146-1.582, p < 0.001). Subgroup analysis revealed that the correlation between the UA/Cr ratio and hypertension risk was stable in all subgroups except for the subgroup with diabetes and the subgroup with a BMI ≥ 28 kg/m2 (p < 0.05). Sensitivity analysis confirmed the robustness of the relationship between a higher UA/Cr ratio and a higher risk of hypertension (p < 0.05). The RCS showed that the UA/Cr ratio was nonlinearly related to hypertension risk. Further threshold effect showed that only a UA/Cr ratio less than 5.0 was related to hypertension risk (OR: 1.178, 95% CI: 1.086-1.278, p < 0.001), and the 2-piecewise linear regression model was superior to the 1-line linear regression model (p < 0.05). CONCLUSION: The UA/Cr ratio was associated with the prevalence of hypertension.

Correcting the Non-Linear Response of Silicon Photomultipliers.

The finite number of pixels in a silicon photomultiplier (SiPM) limits its dynamic range to light pulses up to typically 80% of the total number of pixels in a device. Correcting the non-linear response is essential to extend the SiPM's dynamic range. One challenge in determining the non-linear response correction is providing a reference linear light source. Instead, the single-step method used to calibrate PMTs is applied, based on the difference in responses to two light sources. With this method, the response of an HPK SiPM (S14160-1315PS) is corrected to linearity within 5% while extending the linear dynamic range by a factor larger than ten. The study shows that the response function does not vary by more than 5% for a variation in the operating voltage between 2 and 5 V overvoltage in the gate length between 20 and 100 ns and for a time delay between the primary and secondary light of up to 40 ns.

Understanding the Nonlinear Response of SiPMs.

A systematic study of the nonlinear response of Silicon Photomultipliers (SiPMs) was conducted through Monte Carlo (MC) simulations. The MC code was validated against experimental data for two different SiPMs. Nonlinearity mainly depends on the balance between the photon rate and the pixel recovery time. Additionally, nonlinearity has been found to depend on the light pulse shape, the correlated noise, the overvoltage dependence of the photon detection efficiency, and the impedance of the readout circuit. Correlated noise has been shown to have a minor impact on nonlinearity, but it can significantly affect the shape of the SiPM output current. Considering these dependencies and a previous statistical analysis of the nonlinear response of SiPMs, two phenomenological fitting models were proposed for exponential-like and finite light pulses, explaining the roles of their various terms and parameters. These models provide an accurate description of the nonlinear responses of SiPMs at the level of a few percentages for a wide range of situations.

The Influence of Nonlinear High-Intensity Dynamic Processes on the Standing Wave Precession of a Non-Ideal Hemispherical Resonator.

The properties of small size, low noise, high performance and no wear-out have made the hemispherical resonator gyroscope a good choice for high-value space missions. To enhance the precision of the hemispherical resonator gyroscope for use in tasks with large angular velocities and angular accelerations, this paper investigates the standing wave precession of a non-ideal hemispherical resonator under nonlinear high-intensity dynamic conditions. Based on the thin shell theory of elasticity, a dynamic model of a hemispherical resonator is established by using Lagrange's second kind equation. Then, the dynamic model is equivalently transformed into a simple harmonic vibration model of a point mass in two-dimensional space, which is analyzed using a method of averaging that separates the slow variables from the fast variables. The results reveal that taking the nonlinear terms about the square of the angular velocity and the angular acceleration in the dynamic equation into account can weaken the influence of the 4th harmonic component of a mass defect on standing wave drift, and the extent of this weakening effect varies with the dimensions of the mass defects, which is very important for steering the development of the high-precision hemispherical resonator gyroscope.

An Adaptive Control Scheme Based on Non-Interference Nonlinearity Approximation for a Class of Nonlinear Cascaded Systems and Its Application to Flexible Joint Manipulators.

Control design for the nonlinear cascaded system is challenging due to its complicated system dynamics and system uncertainty, both of which can be considered some kind of system nonlinearity. In this paper, we propose a novel nonlinearity approximation scheme with a simplified structure, where the system nonlinearity is approximated by a steady component and an alternating component using only local tracking errors. The nonlinearity of each subsystem is estimated independently. On this basis, a model-free adaptive control for a class of nonlinear cascaded systems is proposed. A squared-error correction procedure is introduced to regulate the weight coefficients of the approximation components, which makes the whole adaptive system stable even with the unmodeled uncertainties. The effectiveness of the proposed controller is validated on a flexible joint system through numerical simulations and experiments. Simulation and experimental results show that the proposed controller can achieve better control performance than the radial basis function network control. Due to its simplicity and robustness, this method is suitable for engineering applications.

Metrology of Short-Length Measurers-Development of a Comparator for the Calibration of Measurers Based on Image Processing and Interferometric Measurements.

For the calibration of linear scales, comparators are generally used. Comparators are devices that enable the movement of an evaluation apparatus over a calibrated scale along a linear base with high precision. The construction of a comparator includes a movable carriage that carries the device for the evaluation of the position of the given edge of the line scale relative to the beginning of the scale. In principle, it involves a camera capturing the scale of the measurer, where the position of the camera's projection center is measured using an interferometer. This article addresses the development of a comparator assembled from low-cost components, as well as the description of systematic influences related to the movement of individual parts of the system, such as the inclination and rotation of the camera and directional and height deviations during the carriage's movement. This article also includes an evaluation of the edge of the given scale with subpixel accuracy, addressing distortion elimination and excluding the influences of impurities or imperfections on the scale. The proposed solution was applied to linear-scale measurers, such as leveling rods with coded and conventional scales and measuring tapes. The entire process of measurement and evaluation was automated.

IoT-Based Heartbeat Rate-Monitoring Device Powered by Harvested Kinetic Energy.

Remote patient-monitoring systems are helpful since they can provide timely and effective healthcare facilities. Such online telemedicine is usually achieved with the help of sophisticated and advanced wearable sensor technologies. The modern type of wearable connected devices enable the monitoring of vital sign parameters such as: heart rate variability (HRV) also known as electrocardiogram (ECG), blood pressure (BLP), Respiratory rate and body temperature, blood pressure (BLP), respiratory rate, and body temperature. The ubiquitous problem of wearable devices is their power demand for signal transmission; such devices require frequent battery charging, which causes serious limitations to the continuous monitoring of vital data. To overcome this, the current study provides a primary report on collecting kinetic energy from daily human activities for monitoring vital human signs. The harvested energy is used to sustain the battery autonomy of wearable devices, which allows for a longer monitoring time of vital data. This study proposes a novel type of stress- or exercise-monitoring ECG device based on a microcontroller (PIC18F4550) and a Wi-Fi device (ESP8266), which is cost-effective and enables real-time monitoring of heart rate in the cloud during normal daily activities. In order to achieve both portability and maximum power, the harvester has a small structure and low friction. Neodymium magnets were chosen for their high magnetic strength, versatility, and compact size. Due to the non-linear magnetic force interaction of the magnets, the non-linear part of the dynamic equation has an inverse quadratic form. Electromechanical damping is considered in this study, and the quadratic non-linearity is approximated using MacLaurin expansion, which enables us to find the law of motion for general case studies using classical methods for dynamic equations and the suitable parameters for the harvester. The oscillations are enabled by applying an initial force, and there is a loss of energy due to the electromechanical damping. A typical numerical application is computed with Matlab 2015 software, and an ODE45 solver is used to verify the accuracy of the method.