A new analytical approach for limit cycles and quasi-periodic solutions of nonlinear oscillators: the example of the forced Van der Pol Duffing oscillator
TL;DR: In this article, the authors proposed a technique to obtain limit cycles and quasi-periodic solutions of forced nonlinear oscillators and applied it to the forced Van der Pol oscillator and the forced van der Pol Duffing oscillator.
Abstract: In this paper we propose a technique to obtain limit cycles and quasi-periodic solutions of forced nonlinear oscillators. We apply this technique to the forced Van der Pol oscillator and the forced Van der Pol Duffing oscillator and obtain for the first time their limit cycles (periodic) and quasi-periodic solutions analytically. We introduce a modification of the homotopy analysis method to obtain these solutions. We minimize the square residual error to obtain accurate approximations to these solutions. The obtained analytical solutions are convergent and agree well with numerical solutions even at large times. Time trajectories of the solution, its first derivative and phase plots are presented to confirm the validity of the proposed approach. We also provide rough criteria for the determination of parameter regimes which lead to limit cycle or quasi-periodic behaviour.
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TL;DR: An intelligent computing algorithm is developed for finding the approximate solution of heart model based on nonlinear Van der Pol (VdP)-type second-order ordinary differential equations (ODEs) using feed-forward artificial neural networks (FF-ANNs) optimized with genetic algorithms (GAs) hybrid through interior-point algorithm (IPA).
Abstract: In this work, an intelligent computing algorithm is developed for finding the approximate solution of heart model based on nonlinear Van der Pol (VdP)-type second-order ordinary differential equations (ODEs) using feed-forward artificial neural networks (FF-ANNs) optimized with genetic algorithms (GAs) hybrid through interior-point algorithm (IPA). The mathematical modeling of the system is constructed using FF-ANN models by defining an unsupervised error and unknown weights; the networks are tuned globally with GAs, and local refinement of the results is made with IPA. Design scheme is applied to study the VdP heart dynamics model by varying the pulse shape modification factor, damping coefficients and external forcing factor while keeping the fixed value of the ventricular contraction period. The results of the proposed algorithm are compared with reference numerical solutions of Adams method to establish its correctness. Multiple independent runs are performed for the scheme, and results of statistical analyses in terms of mean absolute deviation, root-mean-square error and Nash---Sutcliffe efficiency illustrate its applicability, effectiveness and reliability.
57 citations
Cites methods from "A new analytical approach for limit..."
...Few potential examples in this regard are homotopy analysis method (HAM) [5], Adomian decomposition methods [6], He’s parameter-expanding methods [7], Laplace decomposition method [8] and linearization method [9], etc....
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TL;DR: In this article, the authors investigated unsteady MHD convective flow in a vertical porous channel with viscous dissipation, and the governing coupled partial differential equations of the problem was reduced to ordinary differential equations and then solved by homotopy analysis method.
24 citations
TL;DR: Stable behaviour of solutions obtained by considering different cases demonstrates that the model under consideration is well-conditioned, and the accuracy of the novel procedure is established by getting the lowest residual errors in the solution for all cases.
Abstract: This paper aims at the analysis of the VdP heartbeat mathematical model. We have analysed the conditionality of a mathematical model which represents the oscillatory behaviour of the heart. A novel neuroevolutionary approach is chosen to analyse the mathematical model. The characteristics of the cardiac pulse of the heart are examined by considering two major scenarios with sixteen different cases. Artificial neural networks (ANNs) are constructed to obtain the best solutions for the heartbeat model. Unknown weights are finely tuned by a combination of a global search technique the Harris Hawks Optimizer (HHO) and a local search technique the Interior Point Algorithm (IPA). Stable behaviour of solutions obtained by considering different cases demonstrates that the model under consideration is well-conditioned. The accuracy of our novel procedure is established by getting the lowest residual errors in our solution for all cases. Graphical and statistical analysis are added to further elaborate the accuracy of our approach.
22 citations
Cites methods from "A new analytical approach for limit..."
...In the terms of synchronization, chaos and limited cycles, VdP equation is similar to biological systems and that is why VdP system based differential equations are frequently used in representations of theoretical heart oscillations [1], [2], [4], [5], [7], [9], [12]....
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...For example, the Adomian Decomposition Method (ADM) [7], [8], He’s parameter expanding method [9], Laplace Decomposition Method (LDM) [10], method of linearization [11] and Homotopy Analysis Method (HAM) [12], etc....
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TL;DR: In this paper, the influence of thermophoretic particle deposition on the MHD mixed convective heat and mass transfer flow in a vertical channel in the presence of radiative heat flux with thermal-diffusion and diffusion-thermo effects is discussed.
18 citations
TL;DR: In this article, an analytical study is conducted to present thermal radiation, Dufour, and Soret effects on viscous flow over a contracting cylinder, where the coupled nonlinear partial differential equations are transformed into a system of coupled non-linear ordinary differential equations by using a suitable similarity transformation.
Abstract: An analytical study is conducted to present thermal radiation, Dufour, and Soret effects on unsteady viscous flow over a contracting cylinder. The coupled nonlinear partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations by using a suitable similarity transformation. The homotopy analysis method (HAM) and HAM with a nonhomogeneous term are employed to obtain analytical solutions for the system of coupled nonlinear ordinary differential equations. A significant reduction in the averaged square residual error is obtained when the nonhomogeneous term is introduced. A comparison between analytical and numerical solutions is presented for validation. The effects of various emerging parameters on flow variables are discussed. It is found that the temperature distribution increases with an increase in Dufour number, but decreases with an increase in Soret number. The concentration distribution decreases for a given increase in the Dufour number, b...
17 citations
References
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TL;DR: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle, which qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve.
5,430 citations
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27 Oct 2003TL;DR: In this paper, a simple bifurcation of a nonlinear problem multiple solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free oscillations with Quadratic nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous flow Boundary-layer Flow Boundarylayer Flow with Exponential Property Boundary Layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGR
Abstract: PART I BASIC IDEAS Introduction Illustrative Description Systematic Description Relations to Some Previous Analytic Methods Advantages, Limitations, and Open Questions PART II APPLICATIONS Simple Bifurcation of a Nonlinear Problem Multiple Solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free Oscillation Systems with Quadratic Nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous Flow Boundary-layer Flow with Exponential Property Boundary-layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGRAPHY INDEX
2,831 citations
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10 Jul 2012
TL;DR: In this paper, a convergence series for Divergent Taylor Series is proposed to solve nonlinear initial value problems and nonlinear Eigenvalue problems with free or moving boundary in heat transfer.
Abstract: Basic Ideas.- Systematic Descriptions.- Advanced Approaches.- Convergent Series For Divergent Taylor Series.- Nonlinear Initial Value Problems.- Nonlinear Eigenvalue Problems.- Nonlinear Problems In Heat Transfer.- Nonlinear Problems With Free Or Moving Boundary.- Steady-State Similarity Boundary-Layer Flows.- Unsteady Similarity Boundary-Layer Flows.- Non-Similarity Boundary-Layer Flows.- Applications In Numerical Methods.
852 citations
TL;DR: In this article, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers, are described.
835 citations
TL;DR: In this paper, an optimal homotopy analysis approach is described by means of the nonlinear Blasius equation as an example, which can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.
822 citations