Multistage Homotopy Perturbation Method Research Articles

Path Tracking of dynamics of a Chaotic Memristor Circuit

Since its recent invention, the Memristor has proven to be a revolutionary device due to the versatility of its applications. Among these, the study of chaotic circuits using Memristors is a fertile area to be explored. However, despite several ways to track the chaotic paths for memristive circuits have been previously proposed, all of these provide merely numeric or qualitatively solutions. As it could be necessary for some applications to have all the possible information about their dynamics, a new methodology to path tracking the chaotic trajectories for memristive circuits using the Multistage Homotopy Perturbation Method is proposed. This methodology provides a straightforward semi-analytic expression to obtain the chaotic path in an iterative way, providing both, the numeric and qualitative behaviour of the circuit.

Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameterhby following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

Multistage homotopy perturbation method for nonlinear reaction networks

In this paper, we present the multistage homotopy perturbation method for finding the solution of the chemical kinetics with nonlinear reactions. We develop a general scheme for finding the analytic solution of chemical reaction networks and apply it to motivating chemical examples such as the enzyme kinetics model and the Brusselator model. We illustrate the numerical result for the models and show the accuracy of the method.

시분할구간 호모토피 섭동법을 이용한 공간 트러스의 비선형 동적 해석

This study aims to apply multistage homotopy perturbation method(MHPM) to space truss composed of discrete members to obtain a semi-analytical solution. For the purpose of this research, a nonlinear governing equation of the structures is formulated in consideration of geometrical nonlinearity, and homotopy equation is derived. The result of carrying out dynamic analysis on a simple model is compared to a numerical method of 4th order Runge-Kutta method(RK4), and the dynamic response by MHPM concurs with the numerical result. Besides, the displacement response and attractor in the phase space is able to delineate dynamic snapping properties under step excitations and the responses of damped system are reflected well the reduction effect of the displacement.

Characterization and stability analysis of a multivariable milling tool by the enhanced multistage homotopy perturbation method

This work deals with the characterization of a three stage multivariable tool used for high-performance milling and the modelling of its dynamic behaviour to predict the stability against chatter. The influence of the variable pitch, variable helix and variable rake angle on each cutting edge of the tool is modeled to calculate the stability lobes diagrams.The proposed approach includes a mechanistic milling forces model and a cutting coefficients characterization procedure along with a modification of the Enhanced Multistage Homotopy Perturbation Method (EMHPM) to capture the multivariable phenomena of the tool in stability prediction, resulting in an accurate method to reproduce the regenerative vibration problem in end milling. Experimental data is provided to validate the followed approach for this new tool concept.

Application of Multistage Homotopy Perturbation Method to the Chaotic Genesio System

Finding accurate solution of chaotic system by using efficient existing numerical methods is very hard for its complex dynamical behaviors. In this paper, the multistage homotopy‐perturbation method (MHPM) is applied to the Chaotic Genesio system. The MHPM is a simple reliable modification based on an adaptation of the standard homotopy‐perturbation method (HPM). The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chaotic Genesio system. Numerical comparisons between the MHPM and the classical fourth‐order Runge‐Kutta (RK4) solutions are made. The results reveal that the new technique is a promising tool for the nonlinear chaotic systems of ordinary differential equations.

Application of multistage homotopy-perturbation method in hybrid synchronization of chaotic systems

In this paper, the multistage homotopy-perturbation method (HPM) is applied to realize the synchronization between the Lorenz system and the Chen system using active control. The multistage HPM is only a simple modification of the standard HPM, in which it is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions. The simulation results reveal that the multistage method is an effective tool to give the approximate analytical solutions for chaotic systems.

Application of the multistage homotopy-perturbation method to solve a class of hyperchaotic systems

Due to the complex dynamical behaviors of hyperchaotic system, it is very difficult to gain its valid analytical solution by using many existing methods. In this paper, the multistage homotopy-perturbation method is first employed to solve a class of hyperchaotic systems. The method is only a simple modification of the standard homotopy-perturbation method, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding hyperchaotic systems. Finally, some numerical comparisons among the multistage homotopy-perturbation method, the standard homotopy-perturbation method and the Runge–Kutta method have been made, which manifest that the modified method is a very accurate and effective algorithm to solve the hyperchaotic systems.

Simulation of the predator-prey problem by the homotopy-perturbation method revised

In this paper, the predator-prey problem is revisited. Previous solution by homotopy-perturbation method (HPM) is improved by treating the homotopy-perturbation method as an algorithm in a sequence of intervals (i.e. time steps) called the multistage homotopy-perturbation method (shortly MHPM). Numerical results show that the multistage homotopy-perturbation method and the classical fourth-order Rungge-Kutta (RK4) methods are in complete agreement.

The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system

In this paper, a new reliable algorithm based on an adaptation of the standard homotopy-perturbation method (HPM) is presented. The HPM is treated as an algorithm in a sequence of intervals (i.e. time step) for finding accurate approximate solutions to the famous Lorenz system. Numerical comparisons between the multistage homotopy-perturbation method (MHPM) and the classical fourth-order Runge–Kutta (RK4) method reveal that the new technique is a promising tool for the nonlinear systems of ODEs.

On multistage homotopy-perturbation method applied to nonlinear biochemical reaction model

The total time evolution of the reactant concentrations in the basic enzyme-substrate reaction is simulated by an adaptation of the standard analytic homotopy-perturbation method (HPM). The standard HPM is converted into a hybrid numeric–analytic method called the multistage HPM (MHPM). The numerical results obtained from the MHPM and the classical fourth-order Runge–Kutta (RK4) method are in complete agreement.

Application of multistage homotopy-perturbation method for the solutions of the Chen system

In this paper, a new reliable algorithm based on an adaptation of the standard homotopy-perturbation method (HPM) is applied to the Chen system which is a three-dimensional system of ODEs with quadratic nonlinearities. The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chen system. We shall call this technique as the multistage HPM (for short MHPM). In particular we look at the accuracy of the HPM as the Chen system changes from a nonchaotic system to a chaotic one. Numerical comparisons between the MHPM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for the nonlinear chaotic and nonchaotic systems of ODEs.

Adaptation of homotopy-perturbation method for numeric–analytic solution of system of ODEs

Abstract A new reliable algorithm based on an adaptation of the standard Homotopy-Perturbation Method (HPM) is presented. The HPM is treated, for the first time, as an algorithm in a sequence of intervals (i.e., time step) for finding accurate approximate solutions of linear and nonlinear systems of ODEs. Numerical comparisons between the Multistage Homotopy-Perturbation Method (MHPM) and the available exact solution and the classical fourth-order Runge–Kutta (RK4) method reveal that the new technique is a promising tool for linear and nonlinear systems of ODEs.

Stability chart for the delayed Mathieu equation

In the space of system parameters, the closedform stability chart is determined for the delayed Mathieu equation defined as (t)(cost)x(t) bx(t2). This stability chart makes the connection between t...