Nonlinear Analysis Tools Research Articles

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.

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.

Accelerated 2D Cartesian MRI with an 8-channel local B0 coil array combined with parallel imaging.

In MRI, the magnetization of nuclear spins is spatially encoded with linear gradients and radiofrequency receiverssensitivity profiles to produce images, which inherently leads to a long scan time. Cartesian MRI, as widely adopted for clinical scans, can be accelerated with parallel imaging and rapid magnetic field modulation during signal readout. Here, by using an 8-channel local coil array, the modulation scheme optimized for sampling efficiency is investigated to speed up 2D Cartesian scans. An 8-channel local coil array is made to carry sinusoidal currents during signal readout to accelerate 2D Cartesian scans. An MRI sampling theory based on reproducing kernel Hilbert space is exploited to visualize the efficiency of nonlinear encoding in arbitrary sampling duration. A field calibration method using current monitors for local coils and the ESPIRiT algorithm is proposed to facilitate image reconstruction. Image acceleration with various modulation field shapes, aliasing control, and distinct modulation frequencies are scrutinized to find an optimized modulation scheme. A safety evaluation is conducted. In vivo 2D Cartesian scans are accelerated by the local coils. For 2D Cartesian MRI, the optimal modulation field by this local array converges to a nearly linear gradient field. With the field calibration technique, it accelerates the in vivo scans (i.e., proved safe) by threefold and eightfold free of visible artifacts, without and with SENSE, respectively. The nonlinear encoding analysis tool, the field calibration method, the safety evaluation procedures, and the in vivo reconstructed scans make significant steps to push MRI speed further with the local coil array.

On an Extension of a Spare Regularization Model

In this paper, we would first like to promote an interesting idea for identifying the local minimizer of a non-convex optimization problem with the global minimizer of a convex optimization one. Secondly, to give an extension of their sparse regularization model for inverting incomplete Fourier transforms introduced. Thirdly, following the same lines, to develop convergence guaranteed efficient iteration algorithm for solving the resulting nonsmooth and nonconvex optimization problem but here using applied nonlinear analysis tools. These both lead to a simplification of the proofs and to make a connection with classical works in this filed through a startling comment.

Linear and Nonlinear Dynamic Analysis of Reinforced Concrete Building with V Shape Steel Bracing

In this study, the RC-MRCBFs were used with V braced frame. The core objective of this examination is to understand the earthquake behavior of the RC-MRCBFs in steel V braced frames. Response spectrum analysis (RSA) is used to understand the seismic performance of the steel braced and un-braced RC frames. Total 12 steel braced RC frames and 12 un-braced frames for all 4 story, 8story, 12 stories and 16 stories are studied and observed the seismic parameter such as fundamental time period (FTP), top story displacements, inter-story drift, base shear and story stiffness of the structures. After studying the parametric study of the 4 to 16 story buildings with a linear and nonlinear analysis tool it was observed that to get the effective braced frame with expected failure mechanism, ductility, the columns should be designed such that, they resist at least 50% base shear contributions. It is observed that using the steel V bracing in the low rise to mid-rise buildings, improves the seismic behaviors of the structures. The steel bracing reduces the maximum top story displacements, drift and time period of the building and increases the seismic base shear demand, stiffness of the structures. The result shows that as increasing the base shear contribution in the columns, the drift and displacement of the story increases and base shear decreases. Key Words: Response Spectrum Analysis, displacement, Maximum storey drift, Storey shear, Storey stiffness, braced frame, RC buildings.

Dynamical behavior analysis of the Van der Pol oscillator with sine nonlinearity subjected to non-sinusoidal periodic excitations by the bifurcation structures

The present work studies the dynamical behavior of the Van der Pol oscillator with sine nonlinearity subjected to the effects of non-sinusoidal excitations through the analysis of bifurcation structures. In this work, the classical Van der Pol oscillator is modified by replacing the linear term with the nonlinear term The idea is to see the impact of the strength of this function on the appearance of chaotic dynamics for small and large values of the nonlinear dissipation term ε. To do this, studies using numerical simulation and analogue simulation are proposed. Firstly, using nonlinear analysis tools, a similarity in the bifurcation sequences is observed despite a difference in the ranks at which the particular behaviours and bifurcation points are obtained. This study is confirmed by their respective maximum Lyapunov exponents. Secondly, a real implementation using microcontroller technology, motivated by the use of an artificial pacemaker, is carried out. For a more practical case, the excitation signal is generated by another microcontroller. The study shows a similarity with the results obtained numerically. Thirdly, in order to show that the model can be derived from mathematical modelling, an electronic simulation using OrCAD-Pspice software is also proposed. The results obtained are in qualitative agreement.

Complex dynamics, released energy, and multistability in a single-layered graphene sheet under periodic loads

Graphene was found in 2004. It’s a carbon allotrope built around a single layer of atoms organized in a two-dimensional (2D) honeycomb lattice nanostructure. In the contribution, the complex behavior of a graphene sheet under periodic loads is investigated using the usual nonlinear analysis tools. From the dynamic states and the material’s symmetry investigations, it is found that the graphene model considered in this work can exhibit the simultaneous existence of up to six different patterns by varying initial conditions for fixed parameters. The Helmholtz theorem is also used to calculate the Hamilton energy of excited graphene. As a result, when the damping coefficient and the nonlocal parameter were close to zero, the system’s energy was null. This supports the fact that when graphene is in its resting state, vibrations are null inside the material. In contrast, when the damping coefficient and the nonlocal parameter were simultaneously increased, the energy of the graphene sheet increased until it reached a peak value, justifying the oscillations observed in the material. The dependency of the material on the initial condition is also used to support the coexistence of the energy found.

Fixed point theorems and applications in p-vector spaces

The goal of this paper is to develop new fixed points for quasi upper semicontinuous set-valued mappings and compact continuous (single-valued) mappings, and related applications for useful tools in nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex and p-vector spaces for p in (0, 1]. In particular, we first develop general fixed point theorems for quasi upper semicontinuous set-valued and single-valued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally p-convex spaces and topological vector spaces for p in (0, 1]; then the best approximation results for quasi upper semicontinuous and 1-set contractive set-valued mappings are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous set-valued mappings with either inward or outward set conditions under various boundary situations. The results established in this paper unify or improve corresponding results in the existing literature for nonlinear analysis, and they would be regarded as the continuation of the related work by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022)–(Fixed Point Theory Algorithms Sci. Eng. 2022:26, 2022) recently.

From coexisting attractors to multi-spiral chaos in a ring of three coupled excitation-free Duffing oscillators

The dynamics of a ring of three unidirectional coupled excitation-free double well Duffing oscillators is considered. The coupling technique followed in this work consists in the amplitude modulation of an oscillator with a proportion of the amplitude of its neighboring counterpart. The rate equation of this model is a sixth-order autonomous nonlinear odd symmetric system endowed with twenty seven equilibrium points eight of which undergo Hopf bifurcation. A detailed analysis of the model based on classic nonlinear analysis tools (e.g. bifurcation diagrams, phase portraits, basins of attraction) discloses attractive and interesting dynamical features such as eight parallel bifurcation threes, Hopf bifurcations, and various types of coexisting modes of oscillations (e.g. eight coexisting limit cycles, four competing double spiral chaotic attractors, two coexisting four-spiral chaotic attractors), and eight-spiral chaotic attractor, just to cite a few. The electronic circuit implementation of the three coupled Duffing oscillators system was executed based on simple available electronic components. The simulation study of the proposed analog circuit in PSpice matches well with the findings the theoretical study. Our results yield the conclusion that coupling excitation-free oscillators of Duffing type in a ring topology can be seen as a useful approach to obtain multi-spiral chaotic attractors.