Nonlinear Analysis Tools Research Articles

Six-scroll chaos within the dynamics of the Thomas chaotic system and application to biomedical data encryption

Abstract In this paper, the influence of non-monotonic nonlinearity within the dynamics of the Thomas model
is studied. The system presents various six-scroll chaos over a wide range of parameter values through indepth analysis using conventional nonlinear analysis tools. The new model also reveals multistability with
an elegant configuration of up to nine periodic states. An electronic analog version of the model is designed
and then simulated using PSPICE software to verify the physical implementation of the model. A novel
biomedical image encryption scheme based on the six-scroll chaos system, Arnold transform and diffusion
operation is proposed and its security is analyzed using statistical testing and key space analysis. The results
demonstrate the effectiveness of the proposed system in providing secure and efficient encryption of digital
images.

2Frame: Web Software for Application in Non-Linear Geometric Analysis of 2-Dimensional Frames

This work presents the development of a web software tool for the linear and nonlinear geometric structural analysis of plane frames. The software leverages matrix methods, specifically the Direct Stiffness Method, and incorporates the Two Circle Iterative Method to account for geometric nonlinearity, alongside principles from solid mechanics. The user-friendly interface allows for easy modeling of frames and input of material parameters, facilitating the determination of stiffness matrices and internal forces. The accuracy and reliability of the software are demonstrated through comparative analysis with data obtained from solved examples in existing literature. Results indicate close agreement, validating the efficacy of the developed tool for nonlinear geometric structural analysis of plane frames. This software stands as a valuable resource for education, research, and professional practice in the field of structural engineering, targeting both undergraduate and graduate students as well as structural professionals in need of nonlinear analysis.

Dynamics of Limit Cycle Oscillations in a Multinozzle Lean Direct Injection Combustor

Thermoacoustic oscillations in a high-pressure multinozzle lean direct injection combustor with fuel staging have been examined. The combustor consists of three independently controlled fuel stages, i.e., the pilot, intermediate, and outer fuel stages. Limit-cycle oscillations have been identified for two cases with different fuel supplies to the three stages of the combustor, operating at different equivalence ratios. The oscillation dynamics of the two cases have been characterized and quantified by employing nonlinear time series analysis tools. Phase portraits, recurrence plots, and recurrence quantification methods were used by phase space reconstruction of the scalar pressure measurements. Further comparisons between the two cases were made by correlating the time-resolved OH* chemiluminescence images to the pressure oscillations. Phase averaging and spectral proper orthogonal decomposition were used to understand the flame dynamics between the three fuel stages.

Discrete-time approximation for backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the discrete-time approximation schemes for a class of backward stochastic differential equations driven by [Formula: see text]-Brownian motion ([Formula: see text]-BSDEs) which corresponds to the hedging pricing of European contingent claims. By introducing an auxiliary extended [Formula: see text]-expectation space, we propose a class of [Formula: see text]-schemes to discrete [Formula: see text]-BSDEs in this space. With the help of nonlinear stochastic analysis techniques and numerical analysis tools, we prove that our schemes admit half-order convergence for approximating [Formula: see text]-BSDE in the general case. In some special cases, our schemes can achieve a first-order convergence rate. Finally, we give an implementable numerical scheme for [Formula: see text]-BSDEs based on Peng’s central limit theorem and illustrate our convergence results with numerical examples.

Design and FPGA Implementation of a Hyper-Chaotic System for Real-time Secure Image Transmission

Recently, chaos theory has been widely used in multimedia and digital communications due to its unique properties that can enhance security, data compression, and signal processing. It plays a significant role in securing digital images and protecting sensitive visual information from unauthorized access, tampering, and interception. In this regard, chaotic signals are used in image encryption to empower the security; that's because chaotic systems are characterized by their sensitivity to initial conditions, and their unpredictable and seemingly random behavior. In particular, hyper-chaotic systems involve multiple chaotic systems interacting with each other. These systems can introduce more randomness and complexity, leading to stronger encryption techniques. In this paper, Hyper-chaotic Lorenz system is considered to design robust image encryption/ decryption system based on master-slave synchronization. Firstly, the rich dynamic characteristics of this system is studied using analytical and numerical nonlinear analysis tools. Next, the image secure system has been implemented through Field-Programmable Gate Arrays (FPGAs) Zedboard Zynq xc7z020-1clg484 to verify the image encryption/decryption directly on programmable hardware Kit. Numerical simulations, hardware implementation, and cryptanalysis tools are conducted to validate the effectiveness and robustness of the proposed system.

Error estimates for the robust α-stable central limit theorem under sublinear expectation by a discrete approximation method

In this work, we develop a numerical method to study the error estimates of the α-stable central limit theorem under sublinear expectation with α∈(0,2), whose limit distribution can be characterized by a fully nonlinear integro-differential equation (PIDE). Based on the sequence of independent random variables, we propose a discrete approximation scheme for the fully nonlinear PIDE. With the help of the nonlinear stochastic analysis techniques and numerical analysis tools, we establish the error bounds for the discrete approximation scheme, which in turn provides a general error bound for the robust α-stable central limit theorem, including the integrable case α∈(1,2) as well as the non-integrable case α∈(0,1]. Finally, we provide some concrete examples to illustrate our main results and derive the precise convergence rates.

Analysis of a class of fractal hybrid fractional differential equation with application to a biological model

Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer’s (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end.

Refined time-shift multiscale slope entropy: a new nonlinear dynamic analysis tool for rotating machinery fault feature extraction

Refined time-shift multiscale slope entropy: a new nonlinear dynamic analysis tool for rotating machinery fault feature extraction

Refined composite multiscale slope entropy and its application in rolling bearing fault diagnosis

Rolling bearing is the key component of rotating machinery, and its vibration signal usually exhibits nonlinear and nonstationary characteristics when failure occurs. Multiscale permutation entropy (MPE) is an effective nonlinear dynamics analysis tool, which has been successfully applied to rolling bearing fault diagnosis in recent years. However, MPE ignores the deep amplitude information when measuring the complexity of the time series and the original multiscale coarse-graining is insufficient, which requires further research and improvement. In order to protect the integrity of information structure, a novel nonlinear dynamic analysis method termed refined composite multiscale slope entropy (RCMSlE) is proposed in this paper, which introduced the concept of refined composite to further boost the performance of MPE in nonlinear dynamical complexity analysis. Furthermore, RCMSlE utilizes a novel symbolic representation that takes full account of mode and amplitude information, which overcomes the weaknesses in describing the complexity and regularity of bearing signals. Based on this, a GWO-SVM multi-classifier is introduced to fulfill mode recognition, and then a new intelligent fault diagnosis method for rolling bearing based on RCMSlE and GWO-SVM is proposed. The experimental results show that the proposed method can not only accurately identify different fault types and degrees of rolling bearing, but also has a short computation time and better performance than other comparative methods.

Plastic hinge length in reinforced HPFRCC beams and columns

The use of ductile concrete materials such as High Performance Fiber Reinforced Cementitious Composites (HPFRCCs) within plastic hinge regions of structural components has garnered research interest in order to improve the seismic resistance of reinforced concrete structures. While experimental and numerical results appear promising in reducing component damage and probability of system level collapse, accurate nonlinear analysis tools capable of capturing the influence of axial load well into a component’s inelastic regime is needed. In this study, a series of 180 high fidelity numerical simulations of HPFRCC beam–column elements are simulated and used to calibrate new plastic hinge length expressions for concentrated and distributed plasticity models for use in system level structural analysis. The numerical models cover a range of HPFRCC material properties, reinforcement ratios, shear span lengths, and axial load levels. The ability of the newly developed expressions to predict component inelastic rotations are subsequently compared to hinge length expressions in the literature and the inelastic rotations of 47 experimental components. The results of this study provide new insights into the effects of axial load on the plastic hinge behavior of HPFRCC components, significantly improves on the accuracy of past plastic hinge length expressions allowing for more accurate modeling of HPFRCC component responses and system level behavior.

Evaluating the residual stresses and fracture behavior of glass-ceramics by numerical modeling

Residual stresses often arise after glass crystallization due to the residual glass and crystal phases' thermal and elastic properties mismatch. Using the Positional Finite Element Method, we developed a computational tool for nonlinear 2D geometric analysis to simulate residual stresses from crystal growth in a glass matrix. We implemented a damage model in which interface elements are introduced into the mesh. These elements' elastic modules were penalized after reaching a tensile stress limit to simulate crack propagation. The crystal mesh was generated separately and included in the glass mesh via kinematic compatibility without introducing degrees of freedom nor requiring mesh conformity. Results were consistent with analytical calculations using Selsing's model. The fracture mode in crystal-glass composite microstructures with different signs of thermal expansion mismatch is closely in accordance with the microstructure pictures in the literature. The adapted damage model showed excellent potential in evaluating and predicting crack patterns due to residual stresses in glass-ceramics.

Extreme Homogeneous and Heterogeneous Multistability in a Novel 5D Memristor-Based Chaotic System with Hidden Attractors

This paper proposes a novel five-dimensional (5D) memristor-based chaotic system by introducing a flux-controlled memristor into a 3D chaotic system with two stable equilibrium points, and increases the dimensionality utilizing the state feedback control method. The newly proposed memristor-based chaotic system has line equilibrium points, so the corresponding attractor belongs to a hidden attractor. By using typical nonlinear analysis tools, the complicated dynamical behaviors of the new system are explored, which reveals many interesting phenomena, including extreme homogeneous and heterogeneous multistabilities, hidden transient state and state transition behavior, and offset-boosting control. Meanwhile, the unstable periodic orbits embedded in the hidden chaotic attractor were calculated by the variational method, and the corresponding pruning rules were summarized. Furthermore, the analog and DSP circuit implementation illustrates the flexibility of the proposed memristic system. Finally, the active synchronization of the memristor-based chaotic system was investigated, demonstrating the important engineering application values of the new system.

Inverse problems for evolutionary quasi-variational hemivariational inequalities with application to mixed boundary value problems

The aim of this paper is to examine an inverse problem of parameter identification in an evolutionary quasi-variational hemivariational inequality in infinite dimensional reflexive Banach spaces. First, the solvability and compactness of the solution set to the inequality are established by employing a fixed point argument and tools of non-linear analysis. Then, general existence and compactness results for the inverse problem have been proved. Finally, we illustrate the applicability of the results in the study of an identification problem for an initial-boundary value problem of parabolic type with mixed multivalued and non-monotone boundary conditions and a state constraint.

Inverse problems for evolutionary quasi-variational hemivariational inequalities with application to mixed boundary value problems

The aim of this paper is to examine an inverse problem of parameter identification in an evolutionary quasi-variational hemivariational inequality in in nite dimensional re exive Banach spaces. First, the solvability and compactness of the solution set to the inequality are established by employing a fixed point argument and tools of nonlinear analysis. Then, general existence and compactness results for the inverse problem have been proved. Finally, we illustrate the applicability of the results in the study of an identification problem for an initial-boundary value problem of parabolic type with mixed multivalued and nonmonotone boundary conditions and a state constraint.

A FRACTAL-FRACTIONAL ORDER MODEL TO STUDY MULTIPLE SCLEROSIS: A CHRONIC DISEASE

A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the fractal-fractional order derivative (FFOD) in the Caputo sense. In addition, the tools of nonlinear functional analysis are applied to prove some qualitative results including the existence theory, stability, and numerical analysis. For the recommended results of the existence theory, Banach and Krassnoselski’s fixed point theorems are used. Additionally, Hyers–Ulam (HU) concept is used to derive some results for stability analysis. Additionally, for numerical illustration of approximate solutions of various compartments of the considered model, the modified Euler method is utilized. The aforementioned results are displayed graphically for various values of fractal-fractional orders.

Damage and Failure of Aeroengine Blades Under Small UAV Impacts

Unmanned aerial vehicles (UAVs) may collide with vulnerable parts of other aircrafts, such as engine blades, causing accidents. The research on small UAV impact is limited. Besides, UAV impact with propeller and rotor blades is poorly understood. In this paper, a finite element (FE) model of a UAV with a weight of 4.78 kg, turbofan blades, propeller blades, and helicopter tail rotor blades was developed. The FE modelling was carried out by commercial software named LS‐DYNA R13.1.0, which is one of the most advanced simulation tools for nonlinear dynamic structural analysis, mainly solving numerical simulations of collision and penetration problems. First, the UAV FE model was validated by comparing the simulation results against the experiments. Next, simulations were conducted to investigate the damage of the three blades under UAV impact with different heading angles. The damage type and effect of UAV’s heading angle were analyzed and identified. The simulation results indicated that the turbofan blades experienced the more severe damage, compared with the propeller blades and tail rotor blades. The head orientation of the UAV has a significant effect on the impact force of the three aircraft engine blades. In addition, certain parts of the UAV (e.g., battery) can produce significant damage to the blades. The research findings are helpful for the safety assessment of the UAV impact damages to different aircraft engines.

Metodologias para avaliação da integridade estrutural de cúpulas e abóbadas históricas em alvenaria

Abstract Modern non-destructive investigation techniques and computational tools for nonlinear analysis allow understanding the structural behavior and damage of existing buildings, aiming at the least possible extent of intervention. Careful and minimal intervention is essential to preserve the authenticity of the built cultural heritage. An investigation with a historical, experimental, and numerical approach was carried out in the Theatro Municipal do Rio de Janeiro, in Brazil, a building with an eclectic architecture from the beginning of the twentieth century. Its masonry domes and vault have paintings by renowned artists on their intrados and were strengthened in the 1970s. The adopted methodology was based on anamnesis, characterization and observation of the structure employing non-destructive tests, and on the assessment of its vulnerability by nonlinear analyses of calibrated numerical models. Several hypotheses of differential settlement under gravitational actions were investigated, seeking to reproduce the cracking pattern and to identify the causes of damage to the masonry domes and vault before the strengthening. The nonlinear analysis of the structure allowed to evaluate the causes of the observed damage, assess the level of safety, identify the most vulnerable parts, and characterize the collapse mechanisms, in addition to demonstrating the efficiency of the intervention measures adopted in the past.

Hamilton energy of a complex chaotic system and offset boosting

The complex differential system can be obtained by introducing complex variable in the real differential system. Complex variables can be decomposed into real component and imaginary component, which makes the complex differential systems have more complex dynamic behaviors. Complex chaotic system is used in secure communications to increase the security of cryptographic systems. In this study, we designed a complex differential system by incorporating a complex variable into a 3D differential system. Dynamics of this complex differential system are investigated by applying typical nonlinear analysis tools. Furthermore, Hamilton energy function for complex differential system is obtained based on Helmholtz’s theorem. The values of Hamilton energy with different oscillations of complex differential system are calculated. In addition, offset boosting control for the complex chaotic signal is realized by adding a constant to variable of complex system. Simulation shows that the position of the chaotic attractor in phase space can be flexibly shifted by applying the offset parameter.

Identifying quasi-periodic variability using multivariate empirical mode decomposition: a case of the tropical Pacific

Abstract. A variety of statistical tools have been used in climate science to gain a better understanding of the climate system's variability on various temporal and spatial scales. However, these tools are mostly linear, stationary, or both. In this study, we use a recently developed nonlinear and nonstationary multivariate time series analysis tool – multivariate empirical mode decomposition (MEMD). MEMD is a powerful tool for objectively identifying (intrinsic) timescales of variability within a given spatio-temporal system without any timescale pre-selection. Additionally, a red noise significance test is developed to robustly extract quasi-periodic modes of variability. We apply these tools to reanalysis and observational data of the tropical Pacific. This reveals a quasi-periodic variability in the tropical Pacific on timescales ∼ 1.5–4.5 years, which is consistent with El Niño–Southern Oscillation (ENSO) – one of the most prominent quasi-periodic modes of variability in the Earth’s climate system. The approach successfully confirms the well-known out-of-phase relationship of the tropical Pacific mean thermocline depth with sea surface temperature in the eastern tropical Pacific (recharge–discharge process). Furthermore, we find a co-variability between zonal wind stress in the western tropical Pacific and the tropical Pacific mean thermocline depth, which only occurs on the quasi-periodic timescale. MEMD coupled with a red noise test can therefore successfully extract (nonstationary) quasi-periodic variability from the spatio-temporal data and could be used in the future for identifying potential (new) relationships between different variables in the climate system.

NLDyn - An open source MATLAB toolbox for the univariate and multivariate nonlinear dynamical analysis of physiological data

Background and ObjectiveWe present NLDyn, an open-source MATLAB toolbox tailored for in-depth analysis of nonlinear dynamics in biomedical signals. Our objective is to offer a user-friendly yet comprehensive platform for researchers to explore the intricacies of time series data. MethodsNLDyn integrates approximately 80 distinct methods, encompassing both univariate and multivariate nonlinear dynamics, setting it apart from existing solutions. This toolbox combines state-of-the-art nonlinear dynamical techniques with advanced multivariate entropy methods, providing users with powerful analytical capabilities. NLDyn enables analyses with or without a sliding window, and users can easily access and customize default parameters. ResultsNLDyn generates results that are both exportable and visually informative, facilitating seamless integration into research and presentations. Its ongoing development ensures it remains at the forefront of nonlinear dynamics analysis. ConclusionsNLDyn is a valuable resource for researchers in the biomedical field, offering an intuitive interface and a wide array of nonlinear analysis tools. Its integration of advanced techniques empowers users to gain deeper insights from their data. As we continually refine and expand NLDyn's capabilities, we envision it becoming an indispensable tool for the exploration of complex dynamics in biomedical signals.