A multistage linearisation approach to a four-dimensional hyperchaotic system with cubic nonlinearity

Motsa, S. S.; Sibanda, Precious

doi:10.1007/s11071-012-0484-1

Cited by 10 publications

(3 citation statements)

References 25 publications

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“…It has recently been observed that the rather restrictive radius of convergence of the series solution may be appropriately handled by using a time multistepping procedure wherein the standard approximation methods are implemented on a sequence of subintervals whose union makes up the domain of the underlying problem. Examples of time multi-stepping methods that have been developed recently to solve IVPs for chaotic and non-chaotic systems include, homotopy analysis method with time multistepping [13,14], piecewise homotopy perturbation methods [15], multi-stepping Adomian decomposition method [16,17], multi-stepping differential transform method [18,19], multi-stepping variational iteration method [20] and successive linearization method (SLM) with time multi-stepping [21,22]. The fact that these methods attempt to obtain the solution in each sub-interval renders them time-consuming and tedious from the viewpoint of computational operations.…”

Section: Introductionmentioning

confidence: 99%

Solution of boundary eigenvalue problems and IBVP involving a system of PDEs using the successive linearization method

Narayana

Siddheshwar

2023

Z Angew Math Mech

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The paper illustrates a numerical technique to solve a system of three partial differential equations that govern the problem of Rayleigh‐Bénard‐Brinkman convection in a two‐dimensional porous rectangular box. As a result of linear and weakly nonlinear stability analyses of the system a boundary eigenvalue problem (BEVP) and an initial boundary value problem (IBVP) arise. Spatial information on the periodicity of the convection cells is first used in the system of PDEs to make it possible for the successive linearization method (SLM) to be applied. The resulting much‐simplified versions of BEVP and the IVP are then solved by direct and time multi‐stepping versions of SLM, respectively. The SLM solution of the BEVP is compared with that obtained through MATLAB routine bvp4c and the multi‐stepping‐SLM solution of the IVP is validated with that of the Runge‐Kutta‐Fehlberg (RKF45) method (using MATLAB routine ode45). The present numerical technique is found to have quadratic convergence for any desired accuracy.

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Section: Introductionmentioning

confidence: 99%

Solution of boundary eigenvalue problems and IBVP involving a system of PDEs using the successive linearization method

Narayana

Siddheshwar

2023

Z Angew Math Mech

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“…Traditionally, the study of the behaviours of nonlinear systems as shown in Fig. 1 about each equilibrium is often conducted by using a linearization procedure to represent the system by several linear models, with each linear model representing the original system over a small range of operation around one equilibrium [5,6]. Consequently, at each equilibrium, the nonlinear systems can be studied using a linear system approach [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear model standardization for the analysis and design of nonlinear systems with multiple equilibria

2021

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In engineering practice, a nonlinear system stable about several equilibria is often studied by linearizing the system over a small range of operation around each of these equilibria, and allowing the study of the system using linear system methods. Theoretically, for operations beyond a small range but still within the stable regime of an equilibrium, the system behaves nonlinearly, and can be described and investigated using the Volterra series approach. However, there is still no available approach that can systematically transform the model of a nonlinear system into a form that can be studied over the whole stable regime about an equilibrium so as to facilitate the system study using the Volterra series approach. This transformation is, in the present study, referred to as nonlinear model standardization, which is the extension of the well-known concept of linearization to the nonlinear case. In this paper, a novel approach to nonlinear model standardization is proposed for nonlinear systems that can be described by a Nonlinear AutoRegressive model with eXogeneous input (NARX) or a nonlinear differential equation (NDE) model. The proposed approach is then used in three case studies covering the applications in nonlinear system analysis, nonlinear system design, and nonlinearity compensation, respectively, demonstrating the significance of the proposed nonlinear model standardization in a wide range of engineering practices.

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“…The MDBSCM is based on decomposing the given domain of approximation in the time variable into smaller subintervals and then solving the PDE independently in each subinterval using the bivariate spectral collocation method. The multidomain approach has been applied to solve nonlinear ordinary differential equations that model chaotic systems described as 1st order systems of equations [19][20][21]. In this study the same idea is extended to solutions of nonlinear parabolic PDEs.…”

Section: Introductionmentioning

confidence: 99%

Solving Nonlinear Parabolic Partial Differential Equations Using Multidomain Bivariate Spectral Collocation Method

Sydney¹,

Mutua²,

Shateyi³

2016

Nonlinear Systems - Design, Analysis, Estimation and Control

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In this study we introduce the multidomain bivariate spectral collocation method for solving nonlinear parabolic partial differential equations (PDEs) that are defined over large time intervals. The main idea is to reduce the size of the computational domain at each subinterval to ensure that very accurate results are obtained within shorter computational time when the spectral collocation method is applied. The proposed method is based on applying the quasi-linearization technique to simplify the nonlinear partial differential equation (PDE) first. The time domain is decomposed into smaller nonoverlapping subintervals. Discretization is then performed on both time and space variables using spectral collocation. The approximate solution of the PDE is obtained by solving the resulting linear matrix system at each subinterval independently. When the solution in the first subinterval has been computed, the continuity condition is used to obtain the initial guess in subsequent subintervals. The solutions at different subintervals are matched together along a common boundary. The examples chosen for numerical experimentation include the Burger's-Fisher equation, the Fitzhugh-Nagumo equation and the Burger's-Huxley equation. To demonstrate the accuracy and the effectiveness of the proposed method, the computational time and the error analysis of the chosen illustrative examples are presented in the tables.

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A multistage linearisation approach to a four-dimensional hyperchaotic system with cubic nonlinearity

Cited by 10 publications

References 25 publications

Solution of boundary eigenvalue problems and IBVP involving a system of PDEs using the successive linearization method

Solution of boundary eigenvalue problems and IBVP involving a system of PDEs using the successive linearization method

Nonlinear model standardization for the analysis and design of nonlinear systems with multiple equilibria

Solving Nonlinear Parabolic Partial Differential Equations Using Multidomain Bivariate Spectral Collocation Method

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