Extremum Research Articles

Fuel Consumption Prediction for Full Flight Phases Toward Sustainable Aviation: A DMPSO-LSTM Model Using Quick Access Recorder (QAR) Data

Reducing emissions in the aviation industry remains a critical challenge for global low-carbon transition. Accurate fuel consumption prediction is essential to achieving emission reduction targets and advancing sustainable development in aviation. Aircraft fuel consumption is influenced by numerous complex factors during flight, resulting in significant nonlinear relationships between segment-specific variables and fuel usage. Traditional statistical and econometric models struggle to capture these relationships effectively. This article first focuses on the different characteristics of QAR data and uses the Adaptive Noise Ensemble Empirical Mode Decomposition (CEEMDAN) method to obtain more significant potential features of QAR data, solving the problems of mode aliasing and uneven mode gaps that may occur in traditional decomposition methods when processing non-stationary signals. Secondly, a dynamic multidimensional particle swarm optimization algorithm (DMPSO) was constructed using an adaptive adjustment dynamic change method of inertia weight and learning factor, which solved the problem of local extremum and low search accuracy in the solution space that PSO algorithm is prone to during the optimization process. Then, a DMPSO-LSTM aircraft fuel consumption model was established to achieve fuel consumption prediction for three flight segments: climb, cruise, and descent. The final proposed model was validated on real-world datasets, and the results showed that it outperformed other baseline models such as BP, RNN, PSO-LSTM, etc. Among the results, the climbing segment MAE index decreased by more than 40%, the RMSE index decreased by more than 38%, and the R2 index increased by more than 6%, respectively. The MAE index of the cruise segment decreased by more than 40%, the RMSE index decreased by more than 40%, and the R2 index increased by more than 5%, respectively. The MAE index of the descending segment decreased by more than 20%, the RMSE index decreased by more than 30%, and the R2 index increased by more than 5%, respectively. The improved prediction accuracy can be used to implement multi-criteria optimization in flight operations: (1) by quantifying weight–fuel relationships, it supports payload–fuel tradeoff decisions; (2) enhanced phase-specific predictions allow optimized climb/cruise profile selections, balancing time and fuel use; and (3) precise consumption estimates facilitate optimal fuel-loading decisions, minimizing safety margins. The high-precision fuel consumption prediction framework proposed in this study provides actionable insights for airlines to optimize flight operations and design low-carbon route strategies, thereby accelerating the aviation industry’s transition toward net-zero emissions.

Molecular Dynamics of High-Pressure Liquid Water: Going from Ambient to Near-Critical Temperatures.

The properties of high-pressure liquid water were investigated along the isobar of 25MPa and in the temperature range 298.15-623.15 K using classical molecular dynamics simulations. Particular attention has been given to the changes in the local structural and related dynamic properties of liquid water. The results obtained have revealed noticeable changes in the shape of the calculated radial distribution functions, as well as the existence of local extrema or crossovers in several structural descriptors and entropic quantities at temperatures around 423.15 K and 498.15 K, where also significant changes in the hydrogen bond network of liquid water have also been observed. The temperature dependence of translational, reorientational, and hydrogen bond dynamics of liquid water has also been investigated by calculating the translational self-diffusion and the corresponding O-H vector Legendre reorientational correlation times and hydrogen bond lifetimes. The corresponding activation energies for each investigated relaxation process have also been presented and discussed.

Global Output‐Feedback Extremum Seeking Control With Source Seeking Experiments

ABSTRACTThis article discusses the design of an extremum‐seeking controller that relies on a monitoring function for a class of SISO uncertain nonlinear systems characterized by arbitrary and uncertain relative degrees. Our demonstration illustrates the feasibility of achieving an arbitrarily small proximity to the desired optimal point through output feedback. The core concept involves integrating a monitoring function with a norm state observer for the unitary relative degree case and its expansion to arbitrary relative degrees by means of the employment of a time‐scaling technique. Significantly, our proposed scheme attains the extremum of an unknown nonlinear mapping across the entire domain of initial conditions, ensuring global convergence and stability for the real‐time optimization algorithm. Furthermore, we provide tuning rules to ensure convergence to the global maximum in the presence of local extrema. To validate the effectiveness of the proposed approach, we present a numerical example and apply it to a source‐seeking problem involving a cart‐track linear positioning servomechanism. Notably, the cart lacks the ability to sense its velocity or the source's position, but can detect the source of a light signal of an unknown concentration field.

Spectroscopic (FT-Raman, FT-IR, UV-Vis, and NMR) and Theoretical Analysis of 1-Methylindole: Structural Characterization, Non-Covalent Interactions, and Electronic Properties

The structural and spectroscopic characterization of 1-Methylindole (1MI) by favors of FT-Raman (4000–200 cm−1), FT- IR (4000–400 cm−1), UV-Vis, and 1H and 13C NMR techniques is presented. The experimental observations were enlightened by density functional theory (DFT) calculations at the B3LYP/6-311G(d,p) level of theory. Geometrical structure of the molecule was obtained, and bond order analysis was performed based on the optimized structure. Non-Covalent Interactions (NCIs) were analyzed by using the Reduced Density Gradient (RDG) with energy density descriptors to visualize the limiting regions of these interactions. Furthermore, molecular charge distribution and isosurface mappings with local extrema were obtained and the critical regions on the molecular surface were visualized. The vibrational spectra were calculated, and the normal modes were assigned based on total energy distribution (TED) calculations. The electronic properties of 1MI were explored experimentally through UV-Vis spectroscopy and analyzed in detail via Atoms in Molecules (AIMs) methodology. Total and partial density of state (TDOS and PDOS) and overlap population density of state (OPDOS) diagrams were calculated and fractional contributions of nonpolar methyl group and aromatic indole to frontier molecular orbitals were obtained through this methodology. Theoretical NMR chemical shifts were assigned based on DFT calculations that use the gauge-invariant atomic orbital (GIAO) method. Inclusion of solvents effect in NMR calculations produces twice less dispersive data and better fitting results to experimental observations. Non-linear optical properties: polarizability, anisotropy of polarizability, and first hyperpolarizability of the molecule were also computed to explore the potential of 1MI as nonlinear spectroscopy agent.

Application of Genetic Algorithm in Training Automatic Speech Recognition

The application of a genetic algorithm in training automatic speech recognition is considered. In the learning process, hidden Markov models are used to evaluate the statistical properties of each word, including the sequence of cepstral coefficients, as well as the transition probabilities between model states. As the model trains, it seeks to improve the fit between the input cepstral coefficients and the target words in order to improve the accuracy of speech recognition. Once trained, the system is used to recognize new speech inputs and identify the corresponding words. This is done by applying trained Hidden Markov Models to new sequences of cepstral coefficients. Genetic algorithms can be useful for optimizing some aspects of speech recognition systems, such as selecting optimal parameters for processing speech signals and choosing the most appropriate models for specific tasks. The results of a comparative analysis based on typical examples are presented, demonstrating the advantages and high efficiency of the genetic algorithm. The main advantages of the GA are indicated, such as independence from initial conditions and the ability to overcome local extrema, which makes it a promising tool for training speech recognition models. The conclusion emphasizes the significance of the research results and their contribution to the development of SMM training methods for speech recognition.

Optimal Design of a Biconvex Airfoil for a Supersonic Aircraft Using the Basin-Hopping and Exhaustive Search Methods

In this study, based on target design conditions, an airfoil is designed for a supersonic aircraft to achieve the maximum lift-to-wave drag ratio, with constraints on the lift coefficient, pitching moment, and maximum thickness. The coefficients of lift and wave drag are calculated numerically using shock/expansion wave theory. To solve the corresponding optimization problem, the Basin-Hopping algorithm—a method commonly used in computational chemical physics for determining minimum energy structures of molecules—is employed. To enhance the search for local extrema, the Sequential Least Squares Programming (SLSQP) method, known for handling constrained optimization problems, is integrated with the Basin-Hopping algorithm. For comparison and validation, the exhaustive search method, a simple technique that evaluates various combinations of design variables to find the optimal solution, is also applied. The results show that while the exhaustive search identifies the optimal design, the Basin-Hopping algorithm yields a slightly better design and requires only about 1/60 of the computation time. This work outlines the design process and demonstrates how advanced optimization algorithms can efficiently address engineering design challenges.

A Fitness Landscape-Based Method for Extreme Point Analysis of Part Surface Morphology

Advancements in Industry 4.0 and smart manufacturing have increased the demand for precise and intricate part surface geometries, making the analysis of surface morphology essential for ensuring assembly precision and product quality. This study presents an innovative fitness landscape-based methodology for extreme point analysis of part surface morphology, effectively addressing the limitations of existing techniques in accurately identifying and analyzing extremum points. The proposed approach integrates adaptive Fitness-Distance Correlation (FDC) with a roughness index to dynamically determine the number and spatial distribution of initial points within the pattern search algorithm, based on variations in surface roughness. By partitioning the feasible domain into subregions according to FDC values, the algorithm significantly reduces optimization complexity. Regions with high ruggedness are further subdivided, facilitating the parallel implementation of the pattern search algorithm within each subregion. This adaptive strategy ensures that areas with intricate surface features are allocated a greater number of initial points, thereby enhancing the probability of locating both regional and global extremum points. To validate the effectiveness and robustness of the proposed method, extensive testing was conducted using five diverse test functions treated as black-box functions. The results demonstrate the method’s capability to accurately locate extremum points across varying surface profiles. Additionally, the proposed method was applied to flatness error evaluation. The results indicate that, compared to using only the raw measurement data, the flatness error increases by approximately 3% when extremum points are taken into account.

Distributed Backface Strain Sensing of Composite Adhesively Bonded Joints under Mode II Fatigue Loading

Distributed Backface Strain Sensing of Composite Adhesively Bonded Joints under Mode II Fatigue Loading

Equivalent local force conditions minimizing the Frank free energy for topological defect equilibria in nematic liquid crystals

Recently, Miyoshi et al . (Miyoshi et al . 2024 Proc. R. Soc. A 480 , 20240405 (doi: 10.1098/rspa.2024.0405 )) derived analytical formulas for the Frank free energy associated with multiple topological defects in nematic liquid crystals confined to an arbitrary simply connected domain. The energy formulas, derived using an analogy with the so-called Kirchhoff–Routh path function in point vortex dynamics, required the evaluation of contour integrals involving analytical formulas associated with the crystal alignment field. Equilibria for topological defects were then obtained by finding local extrema of this Frank free energy. By contrast, in vortex dynamics, it is rare to find point vortex equilibria by minimizing the Kirchhoff–Routh path function that is a regularized kinetic energy; more commonly, a local ‘non-self-induction rule’ is used that is equivalent to each point vortex being free of net force. It is shown here that an analogous equivalent set of local force conditions can also be used to find equilibria for topological defects in nematic liquid crystals in confined domains, and without any need to evaluate the Frank free energy. This observation is significant theoretically and also because, at a technical level, the new local force conditions are easier to formulate and solve.

SOLUTION OF NONLINEAR OPTIMIZATION PROBLEMS OF HEAT TRANSFER THEORY

The article proposes mathematical models and computational methods for the solution of nonlinear problems of finding local extrema of the objective function. These mathematical models and computational methods create a computational structure for improving the accuracy of applied optimization problems. Computational mathematical models of thermal action on a multilayer material and laser action on an embryo have been developed. It should be noted that the computational mathematical model describing the laser effect on the embryo is a nonlocal boundary value problem with a system of evolutionary, nonstationary heat conduction equations, heat flow boundary conditions, and boundary conditions at the beginning and end of the laser effect on the embryo. Applying the traditional theory of existence and uniqueness of the solution of the boundary value problem, due to the multilayer structure of the embryo and different thermal modes of laser action without averaging the values of the thermophysical characteristics of the object of study, it is impossible to substantiate the correctness of the boundary value problem describing the laser action on the embryo. To substantiate the correctness of this boundary value problem, the article proposes to apply methods from the theory of pseudo-differential operators in the space of slowly increasing distributions that depend smoothly on time variables. This made it possible not only to improve the accuracy of solving boundary value problems, but also to increase the accuracy of solving the general problem of improving the quality of laser action on a multilayer material. Reducing the time required to solve boundary value problems was achieved through the use of functionally oriented blocks that almost instantly implement many computational mathematical models (boundary value problems) on computers. This will increase the efficiency of existing technical means and develop new technical means for automating the design of complex systems containing local sources of thermal action.

Intensified Image Retrivel System :Non-Linear Mutation Based Genetic Algorithm

Globally, increasing amount of websites and data bases leads to utilization of enormous image data used in the system. The users interact with the images which are downloaded, stocked, uploaded and transmitting the data on regularly. Conversely, searching the particular images within the huge files and websites is the daunting problem. Traditionally, manual searching is not only time consuming but also less accurate process. Moreover, it requires the assistance of expertise for the image searching mechanism. To resolve the issue, an efficient IRS (Image Retrieval System) is needed for the users to automate the searching process. To address the issue, several conventional model attempted to achieve efficient IRS. However, existing researches limited in accuracy and speed. To overcome the limitations, proposed model utilized non-linear mutation updation based GA (Genetic Algorithm) for Reddit based IRS on eight classes. The GA is used for the capability of global optimization, parallelism and greater set of solution space. Nevertheless, it is sort by limited drawbacks such as unguided mutation, local extremum and convergence. To resolve the issue, non-linear mutation updation is added in the GA technique to improve the IRS efficiency. Moreover, VGG-16 algorithm is utilized for the feature extraction method to extract significant features from the data. Correspondingly, input data is acquired from the popular website called reddit which allows the users to upload and share images with the own post and comments of the users. The data is extracted through the API mechanism with the specific hashtag content. The performance of the proposed model is calculated using performance metrics in order to evaluate the IRS efficacy. Moreover, internal comparison is processed in the respective research to reveals the better efficiency of the proposed model.

Точная оценка третьего коэффициента для ограниченных не обращающихся в нуль голоморфных функций с действительными коэффициентами

Let Ω_0^r be a class of holomorphic functions ω in the unit disk ∆, with real coefficients, and such that |ω(z)|<1, ω(0)=0, z∈∆. The coefficients problem in the class Ω_0^r is formulated as follows: find the necessary and sufficient conditions to be imposed on the real numbers {ω}_1,{ω}_2,… in order for the series {ω}_1 z+{ω}_2 z^2+⋯ to be the Taylor series of a function in the class Ω_0^r. The class B^r consists of holomorphic functions f in ∆ with real coefficients and such that 0<|f(z)|≤1, z∈∆. The classes B_t^r, t≥0, are defined as the sets of functions f∈B^r such that f(0)=e^(-t). The problem of obtaining a sharp estimation of |{f}_n |, n∈N, on the class B^r or B_t^r is commonly referred to as the Krzyz problem (for the class B^r or B_t^r). It is clear that the union of all classes B_t^r exhausts the class B^r up to rotations in the plane of variable w(w=f(z)). Based on the solution of the coefficients problem for the class Ω_0^r the problem of obtaining a sharp estimation of the functional |{f}_3 | on the classes B_t^r for every t≥0 is solved by transitioning to the functional over the class Ω_0^r, after which the problem is reduced to finding the global constrained extremum of a function of two real variables with inequality-type constraints. The extreme functions are found in two forms: as a convex combination of Schwartz kernels related to the Caratheodory class, and as Blaschke products related to the class Ω_0^r.

БАЙЕСОВСКАЯ ОПТИМИЗАЦИЯ МЕТОДА ИДЕНТИФИКАЦИИ МЕЗОМАСШТАБНЫХ ОКЕАНСКИХ ВИХРЕЙ В МОРЕ ЛАБРАДОР В ДАННЫХ ВИХРЕРАЗРЕШАЮЩЕГО МОДЕЛИРОВАНИЯ

Deep convection in the Labrador Sea has a significant role in the formation of the Northern Hemisphere climate. The eddy activity in the Labrador Sea, represented by Irminger rings, influences the spatial and temporal heterogeneity of the mixed layer depth. Automated eddy identification methods are widely used as a tool to study eddy activity in statistically significant samples. However, the most commonly used method for finding local extrema is highly dependent on a variety of parameters chosen by the author or the user of the method. In this paper, a new algorithm for the identification of mesoscale anticyclonic eddies in numerical simulation data is developed. Using Bayesian optimisation method, the optimal values of hyperparameters of the developed eddy identification algorithm were selected. The identification quality in the F1-score measure is improved to 0.232 compared to 0.352 in the baseline configuration.

CROSSOVER AND MUTATION OPERATORS IN STOCHASTIC ALGORITHMS

The relevance of illuminating contemporary stochastic algorithms is highlighted, with the past two decades witnessing a rapid development in stochastic algorithms, attributed to increased research capabilities and growing data volumes. These algorithms prove effective in solving complex optimization problems, garnering attention from the global scientific community and practitioners worldwide. An exploration of the role and an overview of key crossover and mutation operators in stochastic algorithms represent a pertinent scientific and practical endeavor, fostering deeper understanding and popularization of this field. A genome category analysis is conducted, accompanied by a detailed review of primary gene encoding methods for practical application. Through a specific example of a genome corresponding to the optimal design of a two-stage helical gearbox, typical and adapted crossover and mutation operators are examined for efficient solution search. Each operator is provided with a textual description and a graphically illustrated representation, enhancing clarity, quality, and expediency in understanding the essence and operational sequences of the operator. The role and significance of crossover and mutation operators in stochastic algorithms are discussed, emphasizing that crossover operators facilitate the combination of beneficial genetic traits, enhancing offspring adaptability. Balanced utilization of these operators alongside other algorithmic stages, such as mutation and selection, is crucial for achieving an optimal balance between algorithm exploitation and search intensification. The importance and functionality of mutation operators in optimization stochastic algorithms are highlighted, indicating that mutation enables the avoidance of algorithmic stagnation in local extrema, preserving genetic diversity and stimulating the search for new optimal solutions. The particular significance of mutation usage in conditions of complex problem structures or large search spaces is noted. Thus, crossover and mutation operators are deemed key elements for enhancing the effectiveness of optimal solution search.

Поиск оптимального начального состава реакционной смеси процесса полимеризации с помощью искусственных иммунных систем

To establish regularities of polymerization processes, it is advisable to use methods of mathematical modeling. On the basis of mathematical description of the process it is possible to determine the optimal conditions of its course, providing an increase in its efficiency, as well as the quality of produced polymers. One of the problems of optimization of polymerization processes is the problem of determining the initial composition of the reaction mixture, which ensures the achievement of given quantitative or qualitative indicators of the process. The aim of the work is to develop an algorithm for determining the initial concentrations of polymer synthesis reaction. The statement of the problem of searching for optimal concentrations of components of the polymer synthesis process and a numerical algorithm for its solution are given. Since the kinetic model of the polymerization process is an infinite system of ordinary nonlinear differential equations, its solution using classical optimization methods encounters computational difficulties. Therefore, it is proposed to apply the method of artificial immune systems to calculate the optimal initial composition of the reaction mixture of the polymerization process. This method allows to overcome local extrema in multidimensional search spaces and it is easy to implement for a particular process when the number of optimized parameters is changed. The developed algorithm based on the method of artificial immune systems includes a procedure for converting an infinite system of differential equations describing the kinetics of the polymerization process to a finite form using the method of moments. The algorithm has been tested on an industrially significant polymerization process of butadiene on a neodymium-containing catalytic system. The optimum concentrations of reagents that provide the polymer with the desired properties have been calculated.

DEFORMATION OF A THIN FILM AFTER CONTACT LOSS WITH A CYLINDRICAL BASE, WITH ACCOUNTING FOR THE INFLUENCE OF ITS CURVATURE, COMPLIANCE, AND ACTION OF TRANSVERSE FORCES; DELAMINATION LOCATED IN THE CIRCUMFERENTIAL DIRECTION

Delamination of a coating from a cylindrical substrate under compressive stresses is considered. The solution of the problem was obtained within the theory of cylindrical shells. The Mushtari-Donnell-Vlasov equations were used as the equilibrium equations, and the conditions of the generalized elastic clamped type were used as the boundary conditions. Expressions are obtained for the coating displacement components and the energy release rate during delamination along the rectilinear and curvilinear delamination boundaries. The profiles of the exfoliated section of the coating were obtained. The role of the substrate compliance and its curvature on the values of the energy release rate and the angle of rotation at the clamping point, as well as in the change in the shape of the coating profile, is revealed. It is shown that with an increase in the compliance of the substrate, as well as the value of (positive) curvature, the value of the energy release rate and the value of the normal component of the displacement of the coating increase significantly. It has been found that at sufficiently large values of the substrate curvature, a corrugation effect occurs in the profile of the delaminated section of the coating, which results in the local extrema in the graph of the dependence of the energy release rate on the delamination length. This effect was not observed in the delamination located in the longitudinal direction [1]. It is also shown that for sufficiently compliant substrates there is a certain critical delamination width, for which it becomes energetically more favorable for delamination to develop in the circumferential direction than in the axial one. As the positive curvature and compliance of the substrate increase, the value of the critical width decreases. An alternative criterion for the critical delamination width is also considered, which consists in studying the value of the angle of rotation at the embedding point, which corresponds to the overlap of the coating and substrate surfaces at an angle value of zero.

A Similar Feature Point Matching Method for Aerial Electric Power Tower Images Based on a One-Ring Neighborhood of Vertices

Due to the susceptibility of images to various factors such as scale changes, imaging conditions, and image noise, traditional feature point matching methods are difficult to achieve ideal matching accuracy, leading to many challenges in the automatic analysis and processing of power tower images. To improve the matching accuracy of similar feature points in aerial images of electric power tower, this study proposes a method for matching similar feature points in aerial images of electric power tower based on a vertex ring neighborhood, addressing the problem of large matching errors caused by factors such as scale and imaging conditions. This method first adopts the feature point extraction technique of electric power tower image based on decomposition and filtering, and constructs the Laplacian pyramid of the original image. Subsequently, filter banks are used to decompose the pyramid image in different directions, extract local extremum points as candidate feature point sets, and merge them to obtain the final feature point set. On this basis, taking into account the spatial relationships and geometric characteristics around the feature points, a texture mapping algorithm based on vertex ring neighborhood and patch classification is applied to construct a vertex ring neighborhood structure and extract geometrically representative electric power tower features. Finally, by using a texture feature point matching method based on the principle of similarity, the similarity between the real-time image and the reference image features is calculated for matching, and combined with the Babbitt coefficient to remove mismatched feature points, accurate matching of similar feature points in aerial electric power tower images is achieved. The experimental results show that this method has a mean square error of less than 0.1Pixel in matching similar texture feature points of aerial power tower images under various working conditions, significantly improving the matching accuracy and providing an effective tool for automatic analysis and processing of power tower images.

Mechanochemical topological defects in an active nematic.

We propose a reaction-diffusion system that converts topological information of an active nematic into chemical signals. We show that a curvature-activated reaction dipole is sufficient for creating a system that dynamically senses topology by producing a concentration field possessing local extrema coinciding with ±1/2 defects. The enabling term is analogous to polarization charge density seen in dielectric materials. We demonstrate the ability of this system to identify defects in both passive and active nematics. Our results illustrate that a relatively simple feedback scheme, expressed as a system of partial differential equations, is capable of producing chemical signals in response to inherently nonlocal structures in anisotropic media. We posit that such coarse-grained systems can help generate testable hypotheses for regulated processes in biological systems, such as morphogenesis, and motivate the creation of bio-inspired materials that utilize dynamic coupling between nematic structure and biochemistry.

General Variational Principle, General Noether Theorem, Their Classical and Quantum New Physics and Solution to Crisis Deducing All Basic Physics Laws

This paper discovers that current variational principle and Noether theorem for different physics systems with (in) finite freedom systems have missed the double extremum processes of the general extremum functional that both is deduced by variational principle and is necessarily taken in deducing all the physics laws, but these have not been corrected for over a century since Noether's proposing her famous theorem, which result in the crisis deducing relevant mathematical laws and all physics laws. This paper discovers there is the hidden logic cycle that one assumes Euler-Lagrange equations, and then he finally deduces Euler-Lagrange equations via the equivalent relation in the whole processes in all relevant current references. This paper corrects the current key mistakes that when physics systems choose the variational extreme values, the appearing processes of the physics systems are real physics processes, otherwise, are virtual processes in all current articles, reviews and (text) books. The real physics should be after choosing the variational extreme values of physics systems, the general extremum functional of the physics systems needs to further choose the minimum absolute extremum zero of the general extremum functional, otherwise, the appearing processes of physics systems are still virtual processes. Using the double extremum processes of the general extremum functionals, the crisis and the hidden logic cycle in current variational principle and current Noether theorem are solved. Furthermore, the new mathematical and physical double extremum processes and their new mathematical pictures and physics for (in) finite freedom systems are discovered. This paper gives both general variational principle and general Noether theorem as well as their classical and quantum new physics, which would rewrite all relevant current different branches of science, as key tools of studying and processing them.

From deterministic to stochastic: limits of extracting bifurcation diagrams for noisy bistable oscillators with the control-based continuation method

Noise limits the information that can be experimentally extracted from dynamical systems. In this study, we review the Control-based Continuation (CBC) approach, which is commonly used for experimental characterisation of nonlinear systems with coexisting stable and unstable steady states. The CBC technique, however, uses a deterministic framework, whereas in practice, almost all measurements are subject to some level of random perturbation, and the underlying dynamical system is inherently noisy. In order to discover what the CBC is capable of extracting from inherently noisy experiments, we study the Hopf normal form with quintic terms with additive noise. The bifurcation diagram of the deterministic core of this system is well-known, therefore the discrepancies introduced by noise can be easily assessed. First, we utilise the Step-Matrix Multiplication based Path Integral (SMM-PI) method to approximate the system’s steady state probability density function (PDF) for different intensity noise perturbations. We associate the local extrema of the resulting PDFs with limit cycles, and compare the resulting bifurcation diagram to those captured by CBC. We show that CBC estimates the bifurcation diagram of the noisy system well for noise intensities varying from small to moderate, and in practice, the amplitudes provided by CBC may be accepted as a ’best guess’ proxy for the vibration amplitudes characteristic to the near periodic solutions in a wide range of experiments.