Cubic Model Research Articles

A Polynomial Least Squares Multiple-Model Estimator: Simple, Optimal, Adaptive, and Practical

Original polynomial least squares (LS) fits data by constructing coefficients that minimize the sum of squared deviations of deterministic samples from assumed polynomials. Contemporary LS estimates existing polynomial coefficients by filtering out corrupting statistical errors. Kalman Filter (KF) state estimation from noisy data is analogous to polynomial LS estimation. A major problem in both LS and KF target tracking is matching estimator order to target dynamics. 2nd order estimators match constant velocity targets; 3rd order estimators match accelerating targets. Filtered error variances from 3rd order estimators are larger than from 2nd order estimators. 2nd order estimators applied to accelerating targets produce increasing biases as more data are filtered, causing their MSEs (variance plus bias-squared) to rapidly exceed 3rd order variances (MSEs). This becomes troublesome when recurrently maneuvering targets make acceleration jumps. A trade-off between 2nd and 3rd order MSEs is needed. The interacting multiple-model (IMM) addresses this problem adaptively by making adjustments between 2nd and 3rd order KFs with model probabilities as functions of likelihoods from KF residuals and transition probabilities of assumed acceleration jumps. The IMM does not address biases, perform variance/bias-squared trade-offs, or minimize MSEs. In this paper linear interpolation is established between the 2nd and 3rd order polynomial estimators creating the LS multiple-model (LSMM), the 2nd order acceleration bias is defined, and the LSMM MSE is minimized in a variance/bias-squared trade-off. A sequence of optimal LSMMs matched to accelerations covering the spectrum of acceleration between zero and assumed maximum are derived and an adaptive algorithm is designed.

Selecting for changes in average "parity curve" pattern of litter size in Large White pigs.

This study aimed to analyse genetic background of variation in reproductive performance between parities of a sow and to investigate selection strategies to change the "parity curve". Total number born (TNB) recorded in Large White sows was provided by Topigs Norsvin. Analysis with basic (BM) and random regression (RRM) models was done in ASReml 4.1. The BM included only a fixed "parity curve", while RRM included 3rd order polynomials for additive genetic and permanent sow effects. Parameters from RRM were used in simulations in SelAction 2.1. Based on Akaike information criterion, RRM was a better model for TNB data. Genetic variance and heritability estimates of TNB from BM and RRM were increasing with parity from parity 2. Genetically, parity 1 is the most different from parities 7 to 10, whereas most similar to parities 2 and 3. This indicates presence of genetic variation to change the "parity curve". Based on simulations, the selection to increase litter size in parity 1 only increases TNB in all parities, but does not change the observed shape of "parity curve", whereas selection for increased TNB in parity 1 and reduced TNB in parity 5 decreases differences between parities, but also reduces overall TNB in all parities. Changing the "parity curve" will be difficult as the genetic and phenotypic relationships between the parities are hard to overcome even when selecting for one parity.

Estimation and validation of daily global solar radiation by day of the year-based models for different climates in China

Day of the year-based (DYB) models can achieve great accuracy in daily global solar radiation estimation without specific meteorological elements. Many empirical models (EMs) and machine learning (ML) methods have been proposed for DYB models. However, the number of their comparative studies based on diverse climates is limited. In this study, a grand total of 14 DYB models are established to estimate daily global solar radiation based on measured data from 1994 to 2015 at 35 meteorological stations in six climate zones of China. Detailed tasks are as follows: (1) Seven EMs and seven ML models are trained for solar radiation estimation. (2) A new EM and two novel ML models are proposed, i.e. hybrid 3rd order polynomial and sine wave model, adaptive neuro-fuzzy inference system (ANFIS) optimized by chaotic firefly algorithm (CFA) and ANFIS optimized by whale optimization algorithm with simulated annealing and roulette wheel selection (WOASAR). (3) Four statistical indicators are utilized to compare those models, and the best model for each station is decided. (4) We discuss the model parameters and climate variances of six specific stations in different climate zones. The comparison results demonstrate superb estimation precision and climate adaptability of the newly proposed models.

Analytical investigation of the practical range and deflection of megavoltage electron beam in the water phantom with the presence of magnetic field

Introduction: Integrated radiation therapy - MRI systems are capable of delivering high doses to the target tissues near sensitive organs and achieve better therapeutic results; however, the Applied magnetic fields for imaging, can influence the charged path, change the penetration depth and deflect the particles, laterally, leading to dose distribution variations. Therefore, investigating the effects of magnetic field on charged particles is important in treatment planning. Therefore, this study aimed to calculate the effects of magnetic field on the range and lateral deflection electrons to determine analytic relations for treatment planning of electron therapy integrated with MRI-systems. Also, these relations will be used to active condensing of charged particle dose in target volume. Materials and methods: An analytical survey based on electron practical range and energy, was done. The penetration depth and lateral deflection of electrons with therapeutic energy ranges in the presence of uniform magnetic field were calculated, analytically. Calculations were done with Mathematica software version 7.0 and MATLAB 7.0 was applied to plot curves and curve fittings. Results: A cubic polynomial with linear coefficients and a power model with constant power model were applied to illustrate lateral deflection and practical range of electrons as a function of initial energy and magnetic field intensity. Electrons go to spiral path with increasing magnetic field intensity about 1.25 Tesla. Conclusion: The proposed analytical approach can calculate the lateral deflection and penetration depth of electrons in the water phantom in the presence of magnetic field of any intensities fast and accurately. This method can be used to calculate dose variations in integrated radiation therapy- MRI imaging systems for treatment planning and research proposes. Due to fast calculations of the presented analytical method in comparison with Monte Carlo based approaches, the calculations time in treatment systems will be decreased.

TYC 5353-1137-1: an enigmatic Doubly Periodic Variable of semiregular amplitude

We have performed a new search for DPVs of short period in the ASAS catalog (Pojmanski, G., 1997), focusing on those stars with orbital periods between 2 to 3 days which also show variations in their brightness. From a total of 244 objects, we have found another candidate to DPV, one whose mean brightness is gradually decreasing. By fitting a 3rd order polynomial to the mean magnitude and then moving it to zero for a second analysis, a gradual decrease over 2500 days was revealed. During the last 1000 days of this decrease, a 42% increase in the variation between the minimum and maximum values of the magnitude was observed. We determined the orbital period by using the PDM IRAF software (Stellingwerf 1978) and estimated the errors for the orbital period and long cycle by visual inspection of the light curves phased with trial periods near the minimum of the periodogram given by PDM.

An investigation of the distribution of gaze estimation errors in head mounted gaze trackers using polynomial functions.

Second order polynomials are commonly used for estimating the point-of-gaze in headmounted eye trackers. Studies in remote (desktop) eye trackers show that although some non- standard 3rd order polynomial models could provide better accuracy, high-order polynomials do not necessarily provide better results. Different than remote setups though, where gaze is estimated over a relatively narrow field-of-view surface (e.g. less than 30×20 degrees on typical computer displays), head-mounted gaze trackers (HMGT) are often desired to cover a relatively wider field-of-view to make sure that the gaze is detected in the scene image even for extreme eye angles. In this paper we investigate the behavior of the gaze estimation error distribution throughout the image of the scene camera when using polynomial functions. Using simulated scenarios, we describe effects of four different sources of error: interpolation, extrapolation, parallax, and radial distortion. We show that the use of third order polynomials result in more accurate gaze estimates in HMGT, and that the use of wide angle lenses might be beneficial in terms of error reduction.

Effect of vibration frequency, size ratio and large particle volume fraction on packing density of binary spherical mixtures

The packing density of a particulate mixture is an important aspect of materials sciences and engineering. The packing density of a mixture depends on several parameters. This study considers the combined effect of vibration frequency, size ratio and large particle volume proportion on packing density of binary spherical mixtures.When all other parameters are constant, the increase of vibration frequency increases the packing density to a maximum and further increase of vibration frequency decreases the packing density of a mixture. The relationship found to be following a 3rd order polynomial curve. The increase of size ratio decreases the packing density linearly. The increase of large particle volume proportion also increases the packing density to a maximum and further increase of large particle proportion reduces the packing density rapidly. This variation also follows a polynomial pattern. These relationships were analyzed to develop a model incorporating combined effect and design graphs were developed from the derived model. The design graphs can be easily used to determine the packing density for a given vibration frequency and given size ratio.

World estimates of PV optimal tilt angles and ratios of sunlight incident upon tilted and tracked PV panels relative to horizontal panels

This study provides estimates of photovoltaic (PV) panel optimal tilt angles for all countries worldwide. It then estimates the incident solar radiation normal to either tracked or optimally tilted panels relative to horizontal panels globally. Optimal tilts are derived from the National Renewable Energy Laboratory’s PVWatts program. A simple 3rd-order polynomial fit of optimal tilt versus latitude is derived. The fit matches data better above 40° N latitude than do previous linear fits. Optimal tilts are then used in the global 3-D GATOR-GCMOM model to estimate annual ratios of incident radiation normal to optimally tilted, 1-axis vertically tracked (swiveling vertically around a horizontal axis), 1-axis horizontally tracked (at optimal tilt and swiveling horizontally around a vertical axis), and 2-axis tracked panels relative to horizontal panels in 2050. Globally- and annually-averaged, these ratios are ∼1.19, ∼1.22, ∼1.35, and ∼1.39, respectively. 1-axis horizontal tracking differs from 2-axis tracking, annually averaged, by only 1–3% at most all latitudes. 1-axis horizontal tracking provides much more output than 1-axis vertical tracking below 65° N and S, whereas output is similar elsewhere. Tracking provides little benefit over optimal tilting above 75° N and 60° S. Tilting and tracking benefits generally increase with increasing latitude. In fact, annually averaged, more sunlight reach tilted or tracked panels from 80 to 90° S than any other latitude. Tilting and tracking benefit cities of the same latitude with lesser aerosol and cloud cover. In sum, for optimal utility PV output, 1-axis horizontal tracking is recommended, except for the highest latitudes, where optimal tilting is sufficient. However, decisions about panel configuration also require knowing tracking equipment and land costs, which are not evaluated here. Installers should also calculate optimal tilt angles for their location for more accuracy. Models that ignore optimal tilting for rooftop PV and utility PV tracking may underestimate significantly country or world PV potential.

Impact of Aging on Inspiratory and Expiratory Work during Exercise

BACKGROUNDIn addition to the reduction in lung elastance and diaphragm function, aging is associated with a greater ventilatory response for a given metabolic rate. Together these age‐related pulmonary alterations may result in a higher energetic cost (i.e. work) of breathing for the elderly during exercise.PURPOSETo examine the impact of aging on the inspiratory and expiratory components of exercise‐induced respiratory work in physically active participants.METHODSPhysical activity (PA) was monitored in seven healthy young (25 ± 4 yr) and older (72 ± 6 yr) participants for one week. The Tiffeneau‐Pinelli Index (TFPI) of pulmonary function (forced expiratory volume in one second (FEV1)/forced vital capacity (FVC)) was quantified prior to exercise. Following a 6 min warmup at 30 W, all participants performed an incremental cycling exercise test (30 W + 30 W/2 min) to exhaustion. The work of breathing (WOB) was acquired during the final minute of each stage by integrating the averaged esophageal pressure‐volume loop (~10 breaths). Furthermore, the WOB was partitioned into inspiratory (elastic + resistive) and expiratory work by 1) drawing a line connecting end‐expiratory lung volume (EELV) and end‐inspiratory lung volume (EILV) and 2) forming a right angle by adding an isovolume line from EILV and an iso‐pressure line from EELV. As there were no differences were found between inspiratory elastic or resistive work, combined inspiratory work is presented. In order to examine age‐related differences in WOB, the following two analyses were performed. First, for each individual, a 3rd order polynomial fit was calculated for the WOB across all minute ventilations (VE) obtained during the incremental test. T‐tests were then performed for each coefficient of the regression equation to determine if the slope for VE and WOB were different between groups. Second, to compare the impact of aging on total, inspiratory, and expiratory WOB, a mixed model ANOVA was performed including 3 matched flow rates (40, 70, and 100 L·min−1).RESULTSPA (~8700 steps/day) and TFPI (~80%) were similar between groups (P > 0.28). During exercise, the young attained a greater peak workrate compared to the older participants (291 ± 66 W vs 184 ± 37 W, respectively, P < 0.05). Furthermore, at any given workrate, VE was 18 to 29% greater in the old and this was reflected in a higher VE/VCO2 compared to the young (P < 0.05). At each matched VE, the older individuals demonstrated a 17 to 28% greater EELV (P < 0.05). While the total WOB increased with VE in both groups, the slope coefficient was greater in the old (P < 0.05). Specifically, both expiratory and total WOB were greater in the old at 100 L·min−1 (representing 58% and 90% of maximal VE in young and old, respectively). Additionally, 5 of the 7 old and 1 of the 7 young participants demonstrated expiratory flow limitation (> 15%) at 100 L·min−1.CONCLUSIONThe WOB increases with advancing age due to both a greater ventilatory response to a given workrate and a greater WOB at a given ventilatory rate. The greater WOB at a given VE is primarily accounted for by the elevated expiratory work which is likely attributable to the age‐related reduction in elastic recoil of the lung.This abstract is from the Experimental Biology 2018 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

Structured-illumination reflectance imaging coupled with phase analysis techniques for surface profiling of apples

Three-dimensional (3-D) geometry information is valuable for fruit quality evaluation. This study was aimed at exploring an emerging structured-illumination reflectance imaging (SIRI) system, coupled with phase analysis, for reconstructing surface profiles of fruit. Phase-shifted sinusoidal patterns, distorted by the fruit geometry, were acquired and processed through phase demodulation, phase unwrapping and other post-processing procedures to obtain phase difference maps relative to the phase of a reference plane. The phase maps were then transformed into height profiles based on phase-to-height calibrations. A reference plane-based approach, in conjunction with the curve fitting technique using polynomials of order 3 or higher, was utilized for phase-to-height calibrations, which achieved superior accuracies with the root-mean-squared errors (RMSEs) of 0.027–0.033 mm for a height measurement range of 0–91 mm. The 3rd-order polynomial curve fitting technique was further tested on two reference blocks with known heights, resulting in relative errors of 3.75% and 4.16%. Tests of the calibrated system for reconstructing the surface of apple samples showed that surface concavities (i.e., stem/calyx regions) could be readily discriminated from bruising defects from both the phase difference images and reconstructed height profiles. This study has laid a foundation for using SIRI for reconstructing the 3-D geometry, and thus expanded the capability of the technique for quality evaluation of horticultural products. Further research is needed to utilize the phase analysis techniques for detecting surface concavities of apples, and optimize the phase demodulation and unwrapping algorithms for faster and more reliable detection.

The sensitivity of catchment hypsometry and hypsometric properties to DEM resolution and polynomial order

Derivation of the hypsometric curve of a catchment, and properties relating to that curve, requires both use of topographical data (commonly in the form of a Digital Elevation Model – DEM), and the estimation of a functional representation of that curve. An early investigation into catchment hypsometry concluded 3rd order polynomials sufficiently describe the hypsometric curve, without the consideration of higher order polynomials, or the sensitivity of hypsometric properties relating to the curve. Another study concluded the hypsometric integral (HI) is robust against changes in DEM resolution, a conclusion drawn from a very limited sample size. Conclusions from these earlier studies have resulted in the adoption of methods deemed to be “sufficient” in subsequent studies, in addition to assumptions that the robustness of the HI extends to other hypsometric properties. This study investigates and demonstrates the sensitivity of hypsometric properties to DEM resolution, DEM type and polynomial order through assessing differences in hypsometric properties derived from 417 catchments and sub-catchments within South Australia. The sensitivity of hypsometric properties across DEM types and polynomial orders is found to be significant, which suggests careful consideration of the methods chosen to derive catchment hypsometric information is required.

Rational cubic spline interpolation for missing solar data imputation

Missing data imputation is an important task in statistical and sciences dscipline. Solar radation data obtained from the solar tracker does not complete and some data are missing due to human error in handling the instrument or the failure of the imtnunent. Thus, missing data imputation can be used to predict and estimate the unknownvalue of the solar radation at cemin time. This study will estimate the solar radation by using rational cubic Ball spline function with three parameters. The interpolating rational Ball spline is able to give good result based on quadratic regression model. © 2018, Medwell Journals.

Near-infrared refractive index of synthetic single crystal and polycrystalline diamonds at high temperatures

We measured the refractive index n(T) and thermo-optical coefficient β(T) = (1/n)(dn/dT) of high quality synthetic diamonds from room temperature to high temperatures, up to 1520 K, in near-infrared spectral range at wavelength 1.56 μm, using a low-coherence interferometry. A type IIa single crystal diamond produced by high pressure–high temperature technique and a transparent polycrystalline diamond grown by chemical vapor deposition were tested and revealed a very close n(T) behavior, with n = 2.384 ± 0.001 at T = 300 K, monotonically increasing to 2.428 at 1520 K. The n(T) data corrected to thermal expansion of diamond are well fitted with 3rd order polynomials, and alternatively, with the Bose-Einstein model with an effective oscillator frequency of 970 cm−1. Almost linear n(T) dependence is observed above 800 K. The thermo-optical coefficient is found to increase monotonically from (0.6 ± 0.1) × 10−5 K−1 (300 K) to (2.0 ± 0.1) × 10−5 K−1 (1300 K) with a tendency to saturation at >1200 K. These β(T) values are an order of magnitude lower than those known for Si, GaAs, and InP. The obtained results significantly extend the temperature range, where the refractive index of diamond was previously measured.

Reply to A.V. Smagin: IV. Surface diffusion or random noise?

Анализируется эксперимент проф. А.В. Смагина, из которого он получает численное значение коэффициента поверхностной диффузии газов в почве. Нами предпринята попытка оценить погрешность этого коэффициента. Обосновывается пессимистическая точка зрения: поскольку погрешность оказалась во много раз больше самого коэффициента диффузии, то не имеет смысла говорить о поверхностной диффузии – из рассмотренного эксперимента мы ничего не можем заключить о ее интенсивности, ибо статистически значимо она не отличается от нуля.

Polynomial approximation pseudo-two-dimensional battery model for online application in embedded battery management system

Physics-based model has been regarded as a promising alternative to equivalent circuit model due to its ability to describe internal electrochemical states of battery. However, the rigorous physics-based model, namely pseudo-two-dimensional (P2D) model, is too complicated for online application in embedded battery management system. In this paper, to simplify the P2D model, a series of polynomial functions are employed to approximate the electrolyte phase concentration profile, solid phase concentration profile, and non-uniform reaction flux profile, respectively. Especially, the accuracy of 2nd-order and 3rd-order polynomial approximations for reaction flux is compared, and the higher-order is validated with more strength. Benefit from the acquisition of above variables, the electrolyte potential is derived directly according to the conservation of charge at electrolyte phase; the accuracy of activation overpotential is also improved by using the non-uniform reaction flux rather than assuming the uniform current density in single particle (SP) model. Finally, the developed model is simulated by different constant current rates, hybrid pulse and driving cycles, and its outputs are compared with P2D model and original SP model. The results demonstrate that the model proposed in this paper could capture the battery characteristics efficiently, and also significantly reduce the computation complexity.

Chest response assessment of post-mortem swine under blast loadings

To better protect soldiers from blast threat, that principally affect air-filled organs such a lung, it is necessary to develop an adapted injury criterion and, prior to this, to evaluate the response of a biological model against that threat. The objective of this study is to provide some robust data to quantify the chest response of post-mortem swine under blast loadings.7 post-mortem swine (54.5 ± 2.6 kg), placed side-on to the threat and against the ground, were exposed to 5 shock-waves of increasing intensities. Their thorax were instrumented with a piezo-resistive pressure sensor, an accelerometer directly exposed to the shock-wave and a target was mounted on the latter in order to track the chest wall displacement.For incident impulses ranging from 47 kPa ms±2% to 173 kPa ms ±6%, the measured maximum of linear chest wall acceleration (Γmax) goes from 5800 m/s2 ±16% to 41,000 m/s2 ± 8%, with a duration of 0.8 ms. Chest wall displacements ranging from 5 mm ± 20% to 20 mm ± 15%, with a duration of 9 ms, are reached. These reproducible data were used to find simple relations (linear, 2nd and 3rd order polynomials) between the kinematic parameters (plus the viscous criterion) and the incident and reflected impulses.Correlating the new reproducible data with the prediction from the Bowen curves showed a lung injury threshold in terms of Γmax similar to that of Cooper (10,000 m/s2). However, the limits defined for the viscous criterion in the automobile field and for non-lethal weapons seems not adapted for the blast threat.

Regeneração Natural de Anadenanthera colubrina (Vell.) Brenan Fabaceae em Brejo de Altitude em Bananeiras, Paraíba

A destruição de habitats atrelada à fragmentação causa sérias consequências para qualquer tipo de vegetação, principalmente para a Floresta Atlântica na Região Nordeste. O estudo teve como objetivo avaliar a dinâmica populacional da regeneração natural da espécie, Anadenanthera colubrina, Fabaceae, ocorrente em fragmento florestal do Centro de Ciências Humanas, Sociais e Agrárias da Universidade Federal da Paraíba. Foi selecionada uma população, e foram plotadas aleatoriamente 10 parcelas (10x10m), totalizando uma área amostral de 1.000m2, onde todos os exemplares menor ou igual a 0,30 cm de altura foram marcados. O estudo foi realizado entre os meses de dezembro de 2014 e julho de 2015. Foram avaliados, altura, diâmetro do colo, número de folhas, número de indivíduos e taxa de mortalidade. As variáveis se ajustaram ao modelo de regressão quadrática, e cúbica. Com relação a taxa de mortalidade, foi de 85,45%. Algumas parcelas, apresenta localização, ou por menor ação antrópica ou competitiva, apresentaram aumento no número total de indivíduos. O crescimento de plântulas de A. colubrina foi maior onde as parcelas estão localizadas ao centro do fragmento estudado, ou seja, em área com menor influência antrópica, ou próximo a fontes hídricas.

Density measurement for polymers by magneto-Archimedes levitation: Simulation and experiments

In a magneto-Archimedes levitation device, a polymer can be levitated in a paramagnetic medium, and its equilibrium levitated height is mainly related to its density. Here, a novel density measurement method for polymers via magneto-Archimedes levitation is proposed, in which the relationship between the density and the levitated height is determined through numerical simulation. COMSOL software is employed to simulate the distribution of magnetic field in the device, and FLUENT-EDEM multiphase software is adopted to predict the polymer's levitated height. A 3rd-order polynomial equation is fitted by simulation results for density calculation. Several experiments with different devices, different magnets, different mediums, and various polymers were carried out to verify the proposed method. Experimental results showed that the proposed method has high accuracy (0.0060–0.0130 g/cm3) in density measurements for small polymer samples (∼25 mm3). In general, the proposed method has prospects of broad application in polymer density-based testing.

Application of simplex lattice mixture design for optimization of active modified atmosphere for pomegranate arils (cv. Wonderful) based on microbial criteria

Abstract Despite the advantages of low O2 and enriched CO2 for extending the storage life of pomegranate arils, limited studies on systematic selection and optimization of gasses to reduce the microbial growth during storage have been reported. Therefore, this study was undertaken to optimize the gas composition for cold storage (10 °C) of pomegranate arils (cv. Wonderful) based on microbiological criteria. Selecting experimental points and optimizing gas concentration were carried out according to the simplex lattice mixture design with three factors (O2, CO2 and N2). Seven gas combinations including low O2 (2–18 kPa), enriched and high CO2 (2–18 kPa) and 80–96 kPa N2 were used in varying concentrations and the bacterial, yeast and mould growth on arils were analysed. Aerobic mesophilic bacteria growth varies between 3.95–5.89 log CFU g−1 and yeast growth was between 3.84–5.91 log CFU g−1, whereas, mould growth was between 2.15–3.63 log CFU g−1 across the modified atmospheres. Data from these analyses were used to fit linear and cubical polynomial models. Pareto analysis and ternary contour plots showed that the main effects (CO2, N2 and O2) as well as their interaction had significant effects on microbial growth. For all microbial criteria’s, the lowest growth was tend to move to the highest CO2 in the modified atmosphere system. On the contrary, the presence of higher O2 concentration stimulated the growth. This was evident that, the gas mixture containing 12.67–18 kPa CO2, 2–4.67 kPa O2 and 80 − 82.67 kPa N2 significantly reduced microbial count on pomegranate arils. Validation of the model showed that the cubical model predicted the microbial counts effectively with high correlation coefficients of R2 > 94% for mould and >99% for aerobic mesophilic bacteria and yeast, whereas, the linear model overestimated bacteria and yeast counts and underestimated mould count.

Experimental and theoretical bifurcation study of a nonlinear standing-wave thermoacoustic system

In this work, both theoretical and experimental bifurcation studies of nonlinear standing-wave thermoacoustic systems are performed. To characterize the unsteady heat release, two different models are proposed to approximate experimental measurements of the response of a premixed flame to acoustic waves. Model A is formulated with 3rd order polynomial equation, while model B is 5th order one. The expansion method of Galerkin series is applied to decouple the unsteady acoustic field in the spatial and time domains. Multiple scales method is proposed to extract the essential dynamic behaviors from the governing ordinary differential equations. The process of the thermoacoustic system's transition to instability is evaluated by continuation methods. Model A is shown to be associated with a supercritical Hopf bifurcation. However, subcritical Hopf bifurcation occurs in the higher-order modelled nonlinear thermoacoustic system, i.e. model B. Furthermore, hysteresis and bistable zones are observed. In addition, the effects of 1) the heating power K, 2) the axial location of the heat source xf/L and 3) the overall acoustic damping/loss ξ on supercritical and subcritical bifurcations are studied one at a time. To validate our theoretical findings on supercritical and subcritical bifurcation, experimental investigation on a standing-wave Rijke tube with a premixed flame confined is conducted. It is observed that varying the flame axial location xf/L can destabilize the thermoacoustic system via either subcritical or supercritical bifurcation on two different operation conditions. In addition, the measured bifurcation behaviors show a good qualitative agreement with the theoretical findings.