Forced Van Der Pol Equation Research Articles

Stable Subharmonics of the Forced Van der Pol Equation

On etudie numeriquement l'equation de Van der Pol a terme de forcage periodique pour un domaine de valeurs des parametres. On integre l'equation par une methode de Runge Kutta d'ordre 4

Dynamics of the Van der Pol equation

This paper is a review of approaches to understanding dynamics in the forced Van der Pol equation. In addition, it discusses the phenomena of entrainment and phase locking from the point of view of dynamical systems theory. There are two principal regions of the parameter space where chaotic motion has been analyzed. The first occurs in a nearly linear system near resonance. Here one uses the method of averaging to initially reduce the problem to a two-dimensional one. This two-dimensional problem is analyzed by using bifurcation theory and topological methods. The appearance of homoclinic orbits in the averaged equations signals the presence of more complicated dynamics for the original problem. The second region of parameter space one examines is the one in which the Van der Pol equation describes a relaxation oscillation. In this situation one can approximate the dynamics by the iteration of a noninvertible one-dimensional mapping. This process is described together with the use of symbolic dynamics in parametrizing the resulting limit sets.

Symbolic dynamics and relaxation oscillations

This paper describes the application of qualitative methods of dynamical systems theory to a specific problem. It examines the forced Van der Pol equation as an example of a relaxation oscillation with aperiodic solutions. The technique of symbolic dynamics, particularly for one-dimensional mappings, is used to give a complete topological characterization of the set of these aperiodic solutions for parameter values for which the equation appears structurally stable. These results are a mathematical interpretation of numerical computations and are not the result of rigorous analysis.

An approximation approach to the identification of nonlinear systems based on frequency-response measurements

We present a method for determining a model of the form SM from measurements on the steady-state response of a non-linear system to co-sinusoidal inputs at a number of amplitudes and frequencies. In earlier papers the parameter identifiability of the SM structure for this type of input/output experiment was established using an interpolation approach. Here we use a harmonic analysis technique to choose linear subsystems in the model and then minimization of an integral square-error criterion to select certain free coefficients within these subsystems. The results are illustrated by application to a forced Duffing equation and a forced Van der Pol equation.

An analytical method for certain weakly nonlinear periodic differential systems

An analytical method is introduced for certain weakly nonlinear periodic differential systems by inserting the small parameter € of the given system in the coefficients of certain harmonics of the trigonometric polynomial for an approximate periodic solution. This method is based on the harmonic balance method and the perturbation method. Higher improved approximations are systematically obtained. The error of the equations of the system in the nth improved approximation is proportional to (FORMULE). These approximations are sufficiently refined to prove the existence of an exact isolated periodic solution in a small neighborhood of these approximations and to determine the error bound of these approximations. This method is illustrated by the forced van der Pol equation and the coupled forced Duffing equations with two degrees of freedom.

On Linear Ordinary Differential Equations with Periodic Coefficients

Previous article Next article On Linear Ordinary Differential Equations with Periodic CoefficientsKarl K. StevensKarl K. Stevenshttps://doi.org/10.1137/0114066PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] K. K. Stevens, Masters Thesis, Parametric excitation of a viscoelastic column, Ph.D. Dissertation, University of Illinois, Urbana, 1965 Google Scholar[2] Lamberto Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 16, Academic Press Inc., Publishers, New York, 1963viii+271 MR0151677 (27:1661) 0111.08701 CrossrefGoogle Scholar[3] V. M. Staržinskii˘, A survey of works on the conditions of stability of the trivial solution of a system of linear differential equations with periodic coefficients, Amer. Math. Soc. Transl. (2), 1 (1955), 189–237 MR0073774 (17,484c) CrossrefGoogle Scholar[4] K. G. Valeev, On the solution and characteristic exponents of solutions of some systems of linear differential equations with periodic coefficients, J. Appl. Math. Mech., 24 (1960), 877–902 10.1016/0021-8928(60)90068-X MR0148970 (26:6466) 0104.05902 CrossrefGoogle Scholar[5] I. G. Malkin, Some problems in the theory of nonlinear oscillation, AEC Transl. 3766, 1956 Google Scholar[6] Raimond A. Struble and , John E. Fletcher, General perturbational solution of the harmonically forced van der Pol equation, J. Mathematical Phys., 2 (1961), 880–891 10.1063/1.1724236 MR0136824 (25:285) 0111.28802 CrossrefISIGoogle Scholar[7] Raimond A. Struble and , John E. Fletcher, General perturbational solution of the Mathieu equation, J. Soc. Indust. Appl. Math., 10 (1962), 314–328 10.1137/0110023 MR0149032 (26:6528) 0109.09501 LinkISIGoogle Scholar[8] Raimond A. Struble and , Steve M. Yionoulis, General perturbational solution of the harmonically forced Duffing equation, Arch. Rational Mech. 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Math., 1 (1943), 215–236 MR0008982 (5,83d) 0063.03669 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Concepts, Methods, and ParadigmsNonlinear Structural Mechanics | 25 July 2012 Cross Ref The Nonlinear Theory of Arch-Supported StructuresNonlinear Structural Mechanics | 25 July 2012 Cross Ref Discretization MethodsNonlinear Structural Mechanics | 25 July 2012 Cross Ref Stability and Bifurcation of StructuresNonlinear Structural Mechanics | 25 July 2012 Cross Ref The Elastic Cable: From Formulation to ComputationNonlinear Structural Mechanics | 25 July 2012 Cross Ref Nonlinear Mechanics of Three-Dimensional SolidsNonlinear Structural Mechanics | 25 July 2012 Cross Ref The Nonlinear Theory of BeamsNonlinear Structural Mechanics | 25 July 2012 Cross Ref Elastic Instabilities of Slender StructuresNonlinear Structural Mechanics | 25 July 2012 Cross Ref The Nonlinear Theory of Curved Beams and Flexurally Stiff CablesNonlinear Structural Mechanics | 25 July 2012 Cross Ref The Nonlinear Theory of PlatesNonlinear Structural Mechanics | 25 July 2012 Cross Ref The Nonlinear Theory of Cable-Supported StructuresNonlinear Structural Mechanics | 25 July 2012 Cross Ref Parametric instabilities of the radial motions of non-linearly viscoelastic shells under pulsating pressuresInternational Journal of Non-Linear Mechanics, Vol. 47, No. 5 | 1 Jun 2012 Cross Ref Parametric vibrations of a viscoelastic beam (Maxwell model) under steady axial load and transverse displacement excitation at one endJournal of Sound and Vibration, Vol. 115, No. 2 | 1 Jun 1987 Cross Ref Stability of Limit Cycle Solutions of Reaction-Diffusion EquationsDavis CopeSIAM Journal on Applied Mathematics, Vol. 38, No. 3 | 12 July 2006AbstractPDF (2399 KB)Chapter 3 Properties of Ordinary Differential EquationsStability of Linear Systems: Some Aspects of Kinematic Similarity | 1 Jan 1980 Cross Ref Linear systems of ordinary differential equationsJournal of Soviet Mathematics, Vol. 5, No. 1 | 1 Jan 1976 Cross Ref Stochastic kinetic theory and turbulent heatingIl Nuovo Cimento B Series 10, Vol. 69, No. 1 | 1 Sep 1970 Cross Ref Parametric Excitation of a Non-Homogeneous Bernoulli-Euler BeamJournal of Mechanical Engineering Science, Vol. 10, No. 3 | 6 February 2006 Cross Ref Volume 14, Issue 4| 1966SIAM Journal on Applied Mathematics641-959 History Submitted:16 July 1965Published online:03 August 2006 InformationCopyright © 1966 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114066Article page range:pp. 782-795ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

General Perturbational Solution of the Harmonically Forced van der Pol Equation

Some formal techniques for the study of nonlinear oscillations are illustrated through a development of a general perturbational solution of the harmonically forced van der Pol equation. These techniques provide for the study of perturbations of nearly linear oscillations in almost complete generality. The intricate resonance problems associated with small divisors, and in this case leading to the entrainment phenomena, are treated with unusual ease and serve to illustrate both the versatility and the generality of the techniques.