A New Approach for Solving Nonlinear Differential Equations with Poincare Map and Poincare Section

Majlesi Journal of Electrical Engineering

مقاله 4، دوره 10، شماره 3، آذر 2016   

نوع مقاله: Review Article

نویسندگان

Ghasem Sadeghi Bajestani^{* 1}؛ Ali Sheikhani²؛ Seyed Mohammad Reza Hashemi Golpayegani²؛ Farah Ashrafzadeh³؛ Paria Hebrani³

¹Imamreza University, Mashhad, Iran

²Amirkabir University, Tehran, Iran

³Mashhad University of Medical Sciences, Mashhad, Iran

چکیده

Solving many of the scientific problems in physics and engineering leads to differential equations, which in many cases no analytical answers can be found. The question to be considered is that if the available numerical methods for solving diferential equations (that are all done by computers) are reliable. Is the Lipschitz validity assumption on differential equations with nonlinear dynamics true? What is the reason for the contradictory outcomes of solving a simple equation using numerical methods? Do the outcomes show the reality of the dynamic system? What is the acceptable replacement for the current methods? All the phenomena of the world have bifurcations, singularity, dissociation, behavioral changes and interaction; and today, science with assumptions like neglecting interactions and singularities, consider the systems as a continues model, although we are in need of a model in which we can solve the problem without inserting the  changes in time approach to zero chain. In this article, accompanied by showing different and contradictory results –which are all wrong– numerical methods for solving a simple differential equation and comparing them with analytical method, we introduce Poincare as a substitution for overcoming this scientific derivation. In this article we solved a differential equation with common numerical methods in MATLAB, and showed that these methods produce conflicting outcomes, and then we solved it using Poincare. After showing the invalidity of common numerical methods and introduction of a simple decomposition method, we investigated Van der Pol  equations using Poincare, and showed the fact that Poincare can simply show the system dynamics like a flashlight.

کلیدواژه‌ها

en

مراجع

D.F. Griffith, D.J. Higham, Numerical Methods for Ordinary Differential Equations, (2010) Springer

S. P. Hastings and J. B. McLeod Classical Methods in Ordinary Differential Equations: With Applications to Boundary Value Problems, Amer. Math. Soc., Rhode Island, 2011

K. Atkinson, W. Han, Elementary Numerical Analysis, (2004) John Wiley, Sons, Inc.

J. Stoer, R. Bulirsch, Introduction to numerical Analysis,(2002), Springer-Verlag

Paul Blanchard, Robert L. Devaney, Glenn R. Hall, Differential Equations (Preliminary Edition),PWS Publishing, Boston, 1996.

V.I. Arnold, Ordinary Differential Equations, Springer, Berlin, 1992.

E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

P. Hartman, Ordinary Differential Equations, 2nd ed., SIAM, Philadelphia, 2002.

L. Perko, Differential Equations and Dynamical Systems, 2nd ed., Springer, New York, 1996

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2nd ed., Springer, Berlin, 2000

Differential and Integral Calculus, Vol. II Paperback – December 1, 1999 by Piskunov

Fundamentals of Differential Equations, 6th Ed, by Nagle, Saff& Snider, Addison-Wesley, 2004

Alexey N. Pavlov, Dmitry V. Dumsky ‘Return times dynamics: role of Pointcare Section in numerical Analysis’ Chaos, Solutions and Fractuls 18(2003) 795-801

J. Awrejcewicz, P. Hagedorn ‘Nonlinearity, Bifurcationand Chaos Theoryand Applications’ JanezaTrdine 9, 51000 Rijeka, Croatia (2012)

Hegger, R., Kantz, H., and T. Schreiber, Practicalimplementation of nonlinear time series methods: The TISEAN package. CHAOS, 1999. 9: p.413-419

Hilborn, R.C., Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Second Edition ed, 2000.

Schuster, H.G., Deterministic Chaos: an Introduction. Weinheim: VCH, 1989

Pikovsky, A.S., M.G. Rosenblum, and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences. 2001, Cambridge: Cambridge University Press

Ott, E., Chaos in Dynamical Systems. 1994, Cambridge: Cambridge University Press.