A Comprehensive Review on Meta-heuristic Algorithms for Solving Nonlinear Equation Systems

The mathematical numerical methods can consistently serve the economical and managerial phenomena, including those dedicated to zootechnical and agricultural areas. The paper fully turns to good account a series of numerical methods implemented on the computer toward the author, using it in the implementation of improvements of classical mathematical methods. The author has built his own algorithms, improved in terms of convergence of the implemented numerical methods. The mathematical numerical methods whose implementation is to be used in this paper refers to solving non-linear equations, non-algebraic (transcendent) equations, and also to solving non-linear systems of equations. Enough to mention the connection between the size of economical activity (production capacity, capital, etc.) on the one hand, and economical performance (unit cost, unit of income or profit, etc.) on the other hand. There are corresponding phenomena, mathematically, through a nonlinear dependence – parabolic, exponential, hyperbolic one. Mathematically treatment of these phenomena is done through specific functions, such as the Gompertz function: (exponential), then the logistic function: (of exponential and hyperbolic kind). In problems of technical performance optimization of a production process, with or without financial restrictions, with one or more inputs, the rate of technical substitution, and the rate of economic substitution, both of these rates, are obtained by solving a system of differential equations with partial derivatives. Numerical mathematical methods meant to solve these differential systems categories, involve solving the equations and systems of equations, both linear and non linear. We also point out the Cobb-Douglass nonlinear equation focused on throughout almost all the paper, through which the author wants to shape up economical issues in the field of animal husbandry and agriculture. Unlike algebraic equations and linear systems, where the solving methods are direct, the non-linear equations and systems do not allow resolution through direct methods. Here, the solving methods are iterative, but even if it offers approximate solutions, they are sufficiently precise. We also should say that, if in solving non-linear equations, the insurance of the convergence is a simple issue, the solving of the non-linear systems the convergence conditions are more severe. Here, the author, after a detailed study of the issue, starts the implementation with an initial approximation closely related to the exactly solution (mathematical solution). Otherwise, the majority of the methods can follow the unwanted way, the divergence one, due to the rather weak global convergence. Regarding the used methods in the implementation of solving non-linear equations, besides the classical methods (the chord method, the tangent method, etc.), the author also followed more convergent methods, such as Bailey's Method, the Lagrange or Rational Interpolation Method, Jarrat’s Method, Wegstein’s Method, the method of Steffensen and Aiken’s one, as well as his own generical method. Regarding the methods used in the implementation, concerning the solving of the non-linear systems of equations, we mention the classical unperformant Gauss-Seidel Method, but mostly the Gauss-Newton-Raphson Method, adapted by the author. We must mention that the author is the owner of the computer implementation in an acknowledged programming environment, of all the methods listed above, and more. The implementation has been organized in an Integrated environment for numerical methods in mathematics and statistics. The mentioned integrated environment has been the object of a scientific research contract funded by the former Minister of Science and Technology (MCT), the contract drawn up between the parties: ITC (Institute of Technical and Scientifical Computer) Cluj-Napoca, and the mentioned ministry. From all the implemented numerical methods in the integrated environment, in order to serve the economical and managerial phenomena of Animal Husbandry and Agriculture, the paper appeals to the non-linear equations and the non-linear systems of equations.

Non-linear equations system is a collection of some non-linear equations that will find the best solution. Finding the solution of non-linear equations system usually by using analytical method, but there are some complex cases that cannot be solved analytically, so new methods are needed to solve them. One method that can be used to solve non-linear equations system is by using metaheuristic algorithm. This research aims to solve non-linear equations system with complex roots by using metaheuristic algorithm. The metaheuristic algorithm used in this research is Particle Swarm Optimization (PSO), Firefly Algorithm (FA) and Cuckoo Search (CS). The input of this research is non-linear equations system that will be tested and parameters of the PSO, FA, and CS algorithm. Non-linear equations system which is the object of problem is polynomial function and transcendent function, which include logarithmic function, first degree trigonometric function and exponential function. The resulting output is an approximation of the complex roots and function value. Then the obtained solution of non-linear equations system compared with the result of accuracy by finding the value of the function f(x) which is closer to zero. The result of this research obtained by comparing the value of the function produced by each algorithm showed that Particle Swarm Optimization (PSO) algorithm is better at solving non-linear equations system with complex roots because the value of the resulting function is close to zero.

Solving nonlinear systems with fractional derivative damping is often challenging, particularly in cases of strong damping and excitation. To derive solutions for such strong nonlinear systems more concisely, this manuscript presents an approximate method for analyzing the random responses of nonlinear systems with fractional derivative damping. By representing the system responses as generalized harmonic functions, the impact of fractional derivative damping is effectively transformed into a quasilinear damping and quasilinear stiffness with amplitude‐dependent coefficients. Consequently, the nonlinear system with fractional derivative damping is approximately replaced by a modified nonlinear system that excludes the fractional derivative term. The equivalent nonlinear system of this modified nonlinear system is established through a careful selection of the equivalent system family and by minimizing the discrepancies between them. This process leads to an iterative determination of the equivalent nonlinear system, allowing the statistical properties of the original system with fractional derivative damping to be approximated using those of the equivalent system. The consistency of the proposed results with those obtained from Monte Carlo simulations demonstrates the method’s effectiveness, while its simplicity highlights its advantages over conventional stochastic averaging techniques. Furthermore, the proposed approach can be extended to strong nonlinear damping systems, such as hysteresis systems and viscoelastic systems subjected to Gaussian white noise.

Several numerical methods for solving nonlinear systems of equations assume that derivative information is available. Furthermore, these approaches usually do not consider the problem of finding all solutions to a nonlinear system. Rather, most methods output a single solution. In this paper, we address the problem of finding all roots of a system of equations. Our method makes use of a biased random-key genetic algorithm (BRKGA). Given a nonlinear system, we construct a corresponding optimization problem, which we solve multiple times, making use of a BRKGA, with areas of repulsion around roots that have already been found. The heuristic makes no use of derivative information. We illustrate the approach on seven nonlinear equations systems with multiple roots from the literature.

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

Special issue on ‘new results on neuro‐fuzzy adaptive control systems’

The article considers a scientific problem of investigating mathematical models with polynomial nonlinearity, which are represented by systems of nonlinear differential equations. There is presented a numericall-analytical transformation method for investigating nonlinear dynamical systems with many degrees of freedom of polynomial structure. As opposed to the analogues, this method allows solving a wide range of problems for nonlinear systems 
 of general polynomial structure while reducing the computational resource intensity. An algorithm of the method 
 of transformations for investigating the nonlinear dynamic systems with m degrees of freedom is given. The research of stability of solutions of nonlinear dynamic systems with m degrees of freedom is carried out. Algorithmic formulas of transformation method for the solution of systems with nonlinearity of the fourth degree are given. Computational experiments on solving systems of differential equations with a small nonlinear part in the form of a multinomial 
 of the sixth degree using the transformation method show fourth order accuracy in computation. A nonlinear mathematical model of the vibration protection system is investigated by the presented method of transformations. The theorem on determination of the stationary state for systems of differential equations of polynomial structure by transformation method is proved. Algorithmic formulas for computation are presented. A general matrix form for vector indices is presented. Formulas for economical computation of right-hand sides of polynomial structure are presented and it is proposed to apply Pan's scheme with coefficient preprocessing. This method allows studying the different modes 
 of nonlinear models dynamics, for example, to determine such extreme modes as resonance, sub-harmonic, and polyharmonic modes. As an example of the method application, the vibration protection problem of the tower from external periodic influences was solved. The transformed solution takes into account all nonlinear polynomial components. The method allows to investigate the dynamics of a wide range of nonlinear systems with the necessary accuracy

This paper presents a semi-analytical method for periodic flows in continuous nonlinear dynamical systems. For the semi-analytical approach, differential equations of nonlinear dynamical systems are discretized to obtain implicit maps, and a mapping structure based on the implicit maps is employed for a periodic flow. From mapping structures, periodic flows in nonlinear dynamical systems are predicted analytically and the corresponding stability and bifurcations of the periodic flows are determined through the eigenvalue analysis. The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods.

The Simpson’s 3/8 rule is used to solve the nonlinear Volterra integral equations system. Using this rule the system is converted to a nonlinear block system and then by solving this nonlinear system we find approximate solution of nonlinear Volterra integral equations system. One of the advantages of the proposed method is its simplicity in application. Further, we investigate the convergence of the proposed method and it is shown that its convergence is of order O(h4). Numerical examples are given to show abilities of the proposed method for solving linear as well as nonlinear systems. Our results show that the proposed method is simple and effective.

1R11. Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions. Pure and Applied Mathematics, Vol 246. - AA Martynyuk (Stability of Processes Dept, Inst of Mech, Natl Acad of Sci, Kiev, Ukraine). Marcel Dekker, New York. 2002. 301 pp. ISBN 0-8247-0735-4. $150.00. Reviewed by RA Ibrahim (Dept of Mech Eng, Wayne State Univ, 5050 Anthony Wayne Dr, Rm 2119 Engineering Bldg, Detroit MI 48202).This addition to the series of pure and applied mathematics monographs deals with the modern theory of dynamics of continuous, discrete-time, and impulsive nonlinear systems using Liapunov matrix-valued functions. It is known that this theory is originally rooted in the developments of Poincare´’s and Liapunov’s ideas for treating nonlinear systems of differential equations. The book is devoted to introduce mathematical theorems for analyzing Liapunov matrix-valued functions in five chapters. The first chapter introduces the mathematical statements of qualitative methods of the general equations of continuous nonlinear systems. The definitions of various types of stability are introduced for nonlinear non-autonomous systems. Scalar, vector, and matrix-valued Liapunov functions, and the comparison principle were introduced to allow the estimation of the distance from every point of the system integral curve to the origin when the time changes from the fixed value. Other stability theorems, based on the work of the author and others, are stated with their proofs. Some methods for analyzing continuous nonlinear systems of hierarchical structure are presented in Chapter 2. These methods are supported by an example of third-order systems. Some stability theorems of systems with regular hierarchy subsystems, large systems, and their extension to overlapping decomposition are discussed. The problem of poly-stability of nonlinear systems with separable motion is analyzed as an application of the matrix-valued function. Chapter 2 includes the concepts of integral and Lipschitz stability based on the use of the principle of comparison with a matrix-valued Liapunov function. Chapter 3 presents the qualitative analysis of discrete-time systems that model mechanical systems with impulse control, digital computing devices, population dynamics, chaotic dynamics of economical systems, and many others. These systems are usually described in terms of difference equations whose stability conditions are defined in terms of the matrix-valued functions method. Chapter 4 introduces the stability of nonlinear dynamical systems subjected to impulsive perturbations. The impulsive system of differential equations are stated for general class of dynamical systems. The stability definitions presented in Chapter 2 for ordinary differential equations are adapted for the impulsive systems. Conditions and definitions of uniqueness, continuity, boundedness, and stability of solutions of impulsive systems are presented. Chapter 5 culminates the theorems and general results presented in the first four chapters by introducing some applications. They include numerical algorithms of constructing a point network supported by illustrative examples. The oscillations and stability of coupled mechanical systems are demonstrated for three pendulums through elastic springs and coupled two non-autonomous parametric oscillators. Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions is recommended to researchers who are studying the mathematical stability theory of dynamical systems. The author is commended for introducing illustrative examples from different applications to support the idea of Liapunov’s matrix functions.

For the numerical solution of non-linear two-point boundary value problems: y″ + f(x y) = 0, 0≧x≧1y(0) = α,y(1) = β, the well-known Numerov's method leads to a non-linear system of finite difference equations for the approximate solution. In [1], Chawla and Shivakumar considered in detail the application of Newton's method for the solution of the resulting non-linear system. But Newton's method requires the setting up of the Jacobian matrix for the non-linear system. As an alternative to the use of Newton's method, in the present paper we present an iteration scheme which provides a monotonically decreasing (or increasing) sequence of approximations which converges to the solution of the Numerov discretization equations. In contrast with Newton's method the present iteration scheme does not require the setting up of the Jacobian matrix of the non-linear system of equations, but instead it requires sup f and inf f over a suitably determined interval [V 0,U 0] containing the true solution of the non-linear discretized system. The present iteration method for solving the Numerov discretization equations is illustrated by considering two non-linear examples; it compares quite favourably with the Newton's solution of the non-linear systems. In the following we assume familiarity with the notation and discussion in Chawla and Shivakumar [1].

Control of a reduced size model of US navy crane using only motor position sensors.- Algorithms for identification of continuous time nonlinear systems: a passivity approach. Part I: Identification in open-loop operation Part II: Identification in llosed-loop operation.- Flatness-based boundary control of a nonlinear parabolic equation modelling a tubular reactor.- Dynamic feedback transformations of controllable linear time-varying systems.- Asymptotic controllability implies continuous-discrete time feedback stabilizability.- Stabilisation of nonlinear systems by discontinuous dynamic state feedback.- On the stabilization of a class of uncertain systems by bounded control.- Adaptive nonlinear excitation control of synchronous generators with unknown mechanical power.- Nonlinear observers of time derivatives from noisy measurements of periodic signals.- Hamiltonian representation of distributed parameter systems with boundary energy flow.- Differentiable lyapunov function and center manifold theory.- Controlling self-similar traffic and shaping techniques.- Diffusive representation for pseudo-differentially damped nonlinear systems.- Euler's discretization and dynamic equivalence of nonlinear control systems.- Singular systems in dimension 3: Cuspidal case and tangent elliptic flat case.- Flatness of nonlinear control systems and exterior differential systems.- Motion planning for heavy chain systems.- Control of an industrial polymerization reactor using flatness.- Controllability of nonlinear multidimensional control systems.- Stabilization of a series DC motor by dynamic output feedback.- Stabilization of nonlinear systems via forwarding mod{L g V}.- A robust globally asymptotically stabilizing feedback: The example of the artstein's circles.- Robust stabilization for the nonlinear benchmark problem (TORA) using neural nets and evolution strategies.- On convexity in stabilization of nonlinear systems.- Extended goursat normal form: a geometric characterization.- Trajectory tracking for ?-flat nonlinear delay systems with a motor example.- Neuro-genetic robust regulation design for nonlinear parameter dependent systems.- Stability criteria for time-periodic systems via high-order averaging techniques.- Control of nonlinear descriptor systems, a computer algebra based approach.- Vibrational control of singularly perturbed systems.- Recent advances in output regulation of nonlinear systems.- Sliding mode control of the prismatic-prismatic-revolute mobile robot with a flexible joint.- The ISS philosophy as a unifying framework for stability-like behavior.- Control design of a crane for offshore lifting operations.- New theories of set-valued differentials and new versions of the maximum principle of optimal control theory.- Transforming a single-input nonlinear system to a strict feedforward form via feedback.- Extended active-passive decomposition of chaotic systems with application to the modelling and control of synchronous motors.- On canonical decomposition of nonlinear dynamic systems.- New developments in dynamical adaptive backstepping control.

The article deals with the decomposition of nonlinear differential equations based on the group-theoretic approach. At the beginning, the decomposition of differential equations of linear systems using a transition matrix of state is presented, and then, based on the theory of continuous groups (Lie groups), the process of decomposition of differential equations of nonlinear systems is shown. The decomposition approach is based on the isomorphism theorem of the space of vector fields and Lie derivatives, which allows us to consider vector fields as differential operators of smooth functions. A formula is derived about the adjoin representation of a Lie group in its Lie algebra, which actually determines the finding of a vector field that characterizes the interaction of two or more vector fields. The Lie algebra of derivatives makes it possible to determine the infinitesimal action of the Lie group, i.e. the linearization of this action is carried out (transformation of the points of the trajectory space of the original system in a small neighborhood). Decomposition allows, as in the linear case, to separate the finding of an action (only locally) of a group of transformations from the transformed points themselves. For linear systems, this separation is global. It is also shown that the decomposition of linear equations is a particular case of the decomposition of nonlinear equations. An algorithm of the method of model predictive control with Gramian weighting using this decomposition is presented. A practical example of decomposition and application of the model predictive control for stabilization of a nonstationary nonlinear system is considered.

Linear and non-linear equations and equation systems

In this article, the problem of the event-triggered optimal control for a class of strict-feedback nonlinear systems with external disturbances is investigated. First, by introducing proper low-pass filters to the steps of backstepping design, the problem of "jump of the error surfaces" is avoided. What is more, it facilitates the design of a fuzzy hybrid-mode-triggered feedforward controller, which aims to transform the original controlled nonlinear strict-feedback system into an equivalent nonlinear system in affine form. Second, by utilizing the adaptive dynamic programming theory, an event-triggered feedback controller is developed for the equivalent affine nonlinear system. Differing from the existing methods, the real control input consists of two event-triggered terms, which can be updated synchronously under the proposed composite error-based triggering condition. Moreover, it is proven that all the closed-loop signals are bounded. At last, by taking a second-order nonlinear system and missile integrated guidance and control system as examples, simulation results demonstrate the effectiveness of the proposed approach.