Order Dynamical Systems Research Articles

Solving convex optimization problems via a second order dynamical system with implicit Hessian damping and Tikhonov regularization

AbstractThis paper deals with a second order dynamical system with a Tikhonov regularization term in connection to the minimization problem of a convex Fréchet differentiable function. The fact that beside the asymptotically vanishing damping we also consider an implicit Hessian driven damping in the dynamical system under study allows us, via straightforward explicit discretization, to obtain inertial algorithms of gradient type. We show that the value of the objective function in a generated trajectory converges rapidly to the global minimum of the objective function and depending the Tikhonov regularization parameter the generated trajectory converges weakly to a minimizer of the objective function or the generated trajectory converges strongly to the element of minimal norm from the $$\mathop {\text {argmin}}$$ argmin set of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Our analysis reveals that the Tikhonov regularization parameter and the damping parameters are strongly correlated, there is a setting of the parameters that separates the cases when weak convergence of the trajectories to a minimizer and strong convergence of the trajectories to the minimal norm minimizer can be obtained.

Citric acid functionalized chitosan hydrogel beads for enhanced Cr(VI) removal: Experimental assessment and COMSOL modelling of adsorption in a fixed-bed system

Chitosan hydrogel beads (CHB) and cross-linked CHB functionalized with tricarboxylic citric acid (CA-GLA-CHB) were synthesized and used for the removal of Cr(VI) from aqueous solutions in batch and dynamic adsorption systems. Applicability of COMSOL Multiphysics software as a tool for predictive modelling of column dynamics using a minimum set of experimental parameters was tested and compared with experimental results. Batch experiments demonstrated that CA-GLA-CHB showed almost 20% higher adsorption capacity and wider operational pH range compared to non-functionalized CHB. In a dynamic system, CA-GLA-CHB was used as column packing material and the column dynamics was investigated at different conditions. The breakthrough and exhaustion time of the column increased with an increase in bed height and a decrease in feed flow rate and initial Cr(VI) concentration. Computational modeling by COMSOL software with implemented Advection-Dispersion-Reaction (ADR) equation was used to simulate breakthrough curves for the applied dynamic system in order to validate the possibility of its application for process design in the scaled-up fixed-bed column systems for wastewater treatment. The simulated breakthrough curves of Cr(VI) adsorption matched well with the experimental, thus proving that COMSOL Multiphysics software can be used for scaling-up fixed-bed column systems packed with low-cost and effective CA-GLA-CHB bioadsorbent.

A discrete fractional order cournot duopoly game model with relative profit delegation: Stability, bifurcation, chaos, 0-1 testing and control

Due to the memory effect, fractional order dynamical systems provide more realistic results compared with ordinary counterparts. In this study, we consider a Cournot-duopoly game model with relative profit delegation in the sense of Caputo fractional derivative. To describe richer dynamical behavior such as chaos in the model, a discrete dynamical system is needed. As a result of the discretization method based on the use of piecewise constant arguments, we obtain a two dimensional system of difference equations. The stability conditions of all equilibrium points of the discrete dynamical system are given comprehensively. The existence of the flip bifurcation in the system has been demonstrated theoretically. Lyapunov exponents and 0–1 test chaos imply that chaotic structures are formed as a result of this bifurcation. In addition, we present the chaos control technique such as Pyragas method to eliminate chaos in the model. All theoretical results dealing with the stability, bifurcation and chaos in the model are stimulated by numerical simulations.

Study of complex dynamics and novel soliton solutions of the Kraenkel-Manna-Merle model describing saturated ferromagnetic materials

This paper introduces a framework for nonlinear short wave propagation in an external field for saturated ferromagnetic materials with zero conductivity: the Kraenkel-Manna-Merle system. New solutions are produced via the G′/(bG′+G+a)-expansion technique. The suggested method is easier to understand, more precise, and simpler to compute. Specifically, a set of singular-periodic and kink-shaped exact soliton solutions is produced. Contour plots, 3D, and 2D visualizations with suitable parametric values are employed to present the computed solutions, which are derived through constraint conditions. For the development of certain novel soliton structures, the arbitrary functions in the solutions are selected. Moreover, bifurcation and chaos theory are applied to the primary dynamical system in order to provide a qualitative analysis of the system under study. Phase portraits of bifurcation are presented at fixed locations to illustrate different situations about parameter values in the dynamic system. However, adopting techniques for identifying chaos in a dynamical system and applying an external force verifies the occurrence of chaotic behavior in system. These results offer new perspectives that can enhance understanding of the dynamics of the KMM system.

Indirect adaptive observer control (I-AOC) design for truck–trailer model based on T–S fuzzy system with unknown nonlinear function

Tracking is a crucial problem for nonlinear systems as it ensures stability and enables the system to accurately follow a desired reference signal. Using Takagi–Sugeno (T–S) fuzzy models, this paper addresses the problem of fuzzy observer and control design for a class of nonlinear systems. The Takagi–Sugeno (T–S) fuzzy models can represent nonlinear systems because it is a universal approximation. Firstly, the T–S fuzzy modeling is applied to get the dynamics of an observational system in order to estimate the unmeasurable states of an unknown nonlinear system. There are various kinds of nonlinear systems that can be modeled using T–S fuzzy systems by combining the input state variables linearly. Secondly, the T–S fuzzy systems can handle unknown states as well as parameters known to the indirect adaptive fuzzy observer. A simple feedback method is used to implement the proposed controller. As a result, the feedback linearization method allows for solving the singularity problem without using any additional algorithms. A fuzzy model representation of the observation system comprises parameters and a feedback gain. The Lyapunov function and Lipschitz conditions are used in constructing the adaptive law. This method is then illustrated by an illustrative example to prove its effectiveness with different kinds of nonlinear functions. A well-designed controller is effective and its performance index minimizes network utilization—this factor is particularly significant when applied to wireless communication systems.

MODEL ORDER REDUCTION USING EIGEN SPECTRUM ANALYSIS AND ROUTH ARRAY METHOD

The author suggests a hybrid technique to reduce the high order dynamic systems. The suggested approach uses the Routh array method to obtain the coefficients of the numerator and the Eigen Spectrum Analysis of the original system to generate the denominator polynomial of the lower order system, which is preserved in the reduced model. This technique is easy and simple and generates stable reduced models if the original high- order system is stable. The suggested technique is illustrated with the help of the numerical example taken from the literature

On the stability of a Caputo fractional order predator-prey framework including Holling type-II functional response along with nonlinear harvesting in predator

In this study, we look at a fractional-order two-species predator-prey framework that uses Caputo Sense's fractional derivative to try to understand its dynamics, which include a simplified Holling type II functional response and nonlinear harvesting in the predator. Initially, we establish certain mathematical facts, such as the solutions of fractional order dynamical systems being unique, non-negative, bounded, and existing. The dissipativeness of the FDE system solution is addressed. Our investigation also includes a look at the local stability requirements for every possible equilibrium point. As a bifurcation constraint, the order of derivatives has been considered in our discussion of the existence condition of Hopf bifurcation. Our analysis has been strengthened by the incorporation of fractional Hopf bifurcation, and the dynamics of the proposed model are influenced by selective non-linear harvesting. Furthermore, the investigation focuses on the impact of the predator's pace of harvesting about the complexities of the prey-predator relationship. Through the use of certain setting values, empirical evidence indicates as the non-integer order framework displays stable to unstable, when the non-integer order is increased. The analytical findings are shown by various instances presented in the numerical part.

Third Order Dynamical Systems for the Sum of Two Generalized Monotone Operators

In this paper, we propose and analyze a third-order dynamical system for finding zeros of the sum of two generalized operators in a Hilbert space H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document}. We establish the existence and uniqueness of the trajectories generated by the system under appropriate continuity conditions, and prove exponential convergence to the unique zero when the sum of the operators is strongly monotone. Additionally, we derive an explicit discretization of the dynamical system, which results in a forward–backward algorithm with double inertial effects and larger range of stepsize. We establish the linear convergence of the iterates to the unique solution using this algorithm. Furthermore, we provide convergence analysis for the class of strongly pseudo-monotone variational inequalities. We illustrate the effectiveness of our approach by applying it to structured optimization and pseudo-convex optimization problems.

A second order dynamical system method for solving a maximally comonotone inclusion problem

In this paper a second order dynamical system model is proposed for computing a zero of a maximally comonotone operator in a Hilbert space. Under mild conditions, we prove existence and uniqueness of a strong global solution of the proposed dynamical system. A proper tuning of the parameters can allow us to establish fast convergence properties of the trajectories generated by the dynamical system. The weak convergence of the trajectory to a zero of the maximally comonotone operator is also proved. Furthermore, a discrete version of the dynamical system is considered and convergence properties matching to that of the dynamical system are established under a same framework. Finally, the validity of the proposed dynamical system and its discrete version is demonstrated by two numerical examples.

Modeling ship-wind turbine dynamics for optimal energy generation and navigation

This paper deals with modeling the dynamics of the ship-wind turbine system in order to optimize power generation and ensure safe navigation. The focus is on developing mathematical models describing the interaction of wind forces, turbine performance and ship dynamics. Numerical simulations are used to investigate the effects of varying wind conditions, turbine design and navigation parameters on energy efficiency. Control strategies are also developed to maximize the energy output of wind turbines while maintaining stability and safety of ship navigation. Model validation is performed through comparative analysis with experimental data or cases from existing wind-assisted ship propulsion systems. The aim of the study is to provide valuable practical recommendations for the implementation of wind energy technologies in maritime operations, contributing to sustainable development and environmental standards in shipping.

Second order dynamical systems governed by maximally monotone operators: Quantitative stability

Second order dynamical systems governed by maximally monotone operators: Quantitative stability

The Infinite State Representation of Fractional Order Differential Systems: A Survey - Part 2

Since the first fractional integration operator dedicated to simulation 25 years ago, the Infinite State representation has evolved into a general methodology for the analysis of fractional order system dynamics. Based on the frequency-distributed model of the fractional integrator, a fractional system is decomposed into an infinite dimensional set of ODEs, whose state variables are the frequency-distributed state variables of the fractional system. Thus, the Infinite State approach provides a conventional integer-order solution to the initialization problem, a new interpretation of the initial conditions of fractional derivatives, and a definition of fractional energy, preliminary to the analysis of system stability based on the Lyapunov approach. Its finite dimensional approximation enables the study of fractional transients thanks to numerical simulation and the discovery of hidden properties of fractional systems. Moreover, it is a helpful and necessary technique for the analysis of nonlinear and chaotic systems and represents a powerful characterization tool.

The Infinite State Representation of Fractional Order Differential Systems: A Survey - Part 1

Since the first fractional integration operator dedicated to simulation 25 years ago, the Infinite State representation has evolved into a general methodology for the analysis of fractional order system dynamics. Based on the frequency-distributed model of the fractional integrator, a fractional system is decomposed into an infinite dimensional set of ODEs, whose state variables are the frequency-distributed state variables of the fractional system. Thus, the Infinite State approach provides a conventional integer-order solution to the initialization problem, a new interpretation of the initial conditions of fractional derivatives, and a definition of fractional energy, preliminary to the analysis of system stability based on the Lyapunov approach. Its finite dimensional approximation enables the study of fractional transients thanks to numerical simulation and the discovery of hidden properties of fractional systems. Moreover, it is a helpful and necessary technique for the analysis of nonlinear and chaotic systems and represents a powerful characterization tool.

Order Reduction using Basic Characteristics and Factor Division Method

Abstract: The authors propose a mixed method for reducing the order of the high order dynamic systems. In this method, the denominator polynomial of the reduced order model is obtained by using the basic characteristics of the higher order system which are maintained in the reduced model while the coefficients of the numerator are obtained by using factor division method. This method is fundamentally simple and generates stable reduced models if the original high-order system is stable. The proposed method is illustrated with the help of the numerical example taken from the literature.

Successive lag synchronization of heterogeneous distributed-order coupled neural networks with unbounded delayed coupling

The problem of successive lag synchronization (SLS) for heterogeneous distributed-order coupled neural networks (HDOCNNs) with unbounded delayed coupling is investigated in this paper. Firstly, a model for HDOCNNs under the framework of distributed order Caputo derivative is proposed which is the generalization of integer order and fractional order cases. Secondly, a new Razumikhin-type stability theorem for distributed order systems is proposed. On this basis, the well-known Halanay type inequality is generalized to analyze the convergence of solutions for a class of distributed order dynamical systems. Thirdly, the SLS criteria for HDOCNNs with delayed coupling by designing a novel adaptive controller and a valid feedback controller are presented herein, respectively. Finally, a numerical example is presented to illustrate the reliability of the proposed scheme.

Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis

With multiple time delays, we investigated a Caputo fractional order dynamical system involving susceptible, exposed, infected, and recovered individuals. Positivity and boundedness are also theoretically demonstrated using Laplace transform and Mittag-Leffler function. The stability of the disease-free and epidemic equilibrium points has been studied for both delayed and non-delayed model. For generating numerical solutions to the model system, we used the Adam-Bashforth-Moulton predictor-corrector technique. With the help of MATLAB (2018a), we were able to conduct graphical demonstrations and numerical simulations. The system displays Hopf bifurcation and the solutions are no longer periodic beyond a certain threshold value of the time delay parameters.

Multi-scroll Attractors and Multi-stability Generated Via Fractional-order Switching Linear Systems

This work presents a method for fractional order dynamical systems design that exhibits multi-scrolls by means of a fractional-order switching piecewise linear system. This method is based on the equilibrium points positioning and the region sizes in which the system switches. Numerical results of the proposed method are also presented and the multi-stability phenomenon is exhibited.

A robust and generalized effective DAE framework encompassing different methods, algorithms, and model order reduction for linear and nonlinear second order dynamical systems

For linear and nonlinear dynamical problems, we propose the novel development and advancement of the well-known Generalized Single Step Single Solve (GS4) family of second order time-accurate algorithms encompassing the entire class of LMS methods developed over the past 50 years or so with/without controllable numerical dissipation in conjunction with the Differential Algebraic Equation (DAE) framework and Proper Orthogonal Decomposition (POD). Unlike traditional practices which have severe limitations in the ability to provide generality and flexibility and a wide variety of choices to the analyst without losing order of time accuracy, the proposed implementation uniquely not only allows altogether different numerical time integration algorithms within the GS4 family in each of the different subdomains in a single body, but also additionally allows the selection of different space discretized methods such as the Finite Element Method (FEM) and particle methods, and other spatial methods as well in a single analysis. Moreover, the addition of the POD further provides reduction in computational times. Furthermore, unlike existing state of the art, the present framework readily permits a wide array of implicit–implicit, implicit–explicit, and explicit–explicit couplings and the integration of such a technology is of interest here. Consequently, the present DAE-GS4-POD framework has the flexibility of using different spatial methods and different time integration schemes in different subdomains in a selective manner together with Reduced Order Modeling (ROM) to optimize capturing the local and global features of the representative physics. The ROM also employs an iterative convergence check in acquiring sufficient snapshot data to adequately capture the physics to the prescribed accuracy requirements. Such a novelty, and computational features, are not possible with existing state of art and numerical illustrations validate the claim.

Thermodynamics of Quantum Trajectories on a Quantum Computer.

Quantum computers have recently become available as noisy intermediate-scale quantum devices. Already these machines yield a useful environment for research on quantum systems and dynamics. Building on this opportunity, we investigate open-system dynamics that are simulated on a quantum computer by coupling a system of interest to an ancilla. After each interaction the ancilla is measured, and the sequence of measurements defines a quantum trajectory. Using a thermodynamic analogy, which identifies trajectories as microstates, we show how to bias the dynamics of the open system in order to enhance the probability of quantum trajectories with desired properties, e.g., particular measurement patterns or temporal correlations. We discuss how such a biased-generally non-Markovian-dynamics can be implemented on a unitary, gate-based quantum computer and show proof-of-principle results on the publicly accessible ibmq_jakarta machine. While our analysis is solely conducted on small systems, it highlights the challenges in controlling complex aspects of open-system dynamics on digital quantum computers.

Asymptotic Behavior of an Adapted Implicit Discretization of Slowly Damped Second Order Dynamical Systems

Asymptotic Behavior of an Adapted Implicit Discretization of Slowly Damped Second Order Dynamical Systems