advertisement

Internat. J. Math. & Math. Sci. Vol. 9 No. 4 (1986) 731-732 731 DECOMPOSITION SOLUTION FOR DUFFING AND VAN DER POL OSCILLATORS G. ADOMIAN Center for Applied Mathematics University of Georgia Athens, Georgia 30602 U.S.A. (Received February 13, 1986) ABSTRACT. The decomposition method is applied to solve the Duffing and Van der Pol oscillators without customary restrictive assumptions [I-4] and without resort to perturbation methods. 1980 AMS MATHEMATICS SUBJECT CLASSIFICATION CODES. 36G20 KEY WORDS AND PHRASES. Duffing oscillator, Van der Pol oscillator, nonlinear, stochastic, decomposition. I. INTRODUCTION The Duffing equation is written + + By + yy3 x(t) (l.l) The Van der Pol equation can be written J} Y + -, (If L + BY + y(d/dt)y 3 d2/dt 2, B l, e(d/dt) + R y /3, B, Ny x(t) (I.2) we have the form usually given.) Write in (l.l) and y(d/dt)y 3 in (I.2) Thus yy3 both are written Ly + Ry + Ny x(t) (1.3) in the standard form for the decomposition method [I-3] where definite integral from 0 to t. Then, Ly x(t) Ry- Ny. Assuming initial conditions define by YO YO (l .4) y(O), y’(O) are specified, let y y(O) + ty’(O) + L-Ix(t). Then Yn+l for n>O. s the two-fol d -L-l(d/dt)Yn L-I BYn L-I [Ny] n=OYn and 732 G. ADOMIAN 2. SOLUTION OF THE PROBLEM To get computable solutions, we need only substitute for for the Duffing case and y(d/dt) A y Z A n=O n for the Van der Pol case where the A are n n=O n Adomian’s polynomials [I-5] generated for the nonlinear term it exactly in a rapidly converging series A0 Ny the sum y3 and representing [I-5]. yg 3yy A2 3yY2 + 3YoY A + + The deterministic problem is now solved.since all components of y are determined. n-i which, because of the rapid convergence, We use an n-term approximation n (say half a dozen or so terms) but is generally sufficient with a very easily carried as far as necessary since the integrals do not involve difficult Green’s functions. Convergence has been previously established by Adomian [2,5] and has been shown [2] to be quite rapid. For the stochastic case [2], none of the usual approximations of statistical inearization are necessary. The x(t) need not be stationary nor Gaussian nor delta-correlated. Further e,B,Y and the initial conditions can be stochastic. No "smallnesS’ assumptions are necessary for the stochastic processes and the nonlinearities. No linearization is used. We can allow <=> + {, B <B> + n, {(d/dt)y y y (R)<y> + o and write Ly x <=>(d/dt)y <B>y A and proceed as before with Y o n u result is a stochastic series which statistics are obtained without the problems of statistical separability of quantities such as <Ry> where R d/dt n which normally require closure approximations. ..^Yi n sVl Z n=OTe Z^ Yn" <y>noAn fro REFERENCES ADOMIAN, G. Stochastic Systems, Academic Press, 1983. ADOMIAN, G. Nonlinear Stochastic Operator Equations, Academic Press, 1986. BELLMAN, R.E., ADOMIAN, G. Nonlinear Partial Differential Equations New Methods for Their Treatment and Ap_pl’iations, Reidel, 1985. 4. ADOMIAN, G., RACH, R. Anharmonic Oscillator Systems, J. Math. Anal. and Applic. (January 1983), 229-236. 9__I, no. 5. ADOMIAN, G. Convergent Series Solution of Nonlinear Equations, J. Comput. and App. Math. ll, no. 2 (October 1984), 225-230. 6. BONZANI, I. On a Class of Nonlinear Stochastic Dynamical Systems" Analysis of the Transient Behavior, J. Math. Anal. and Applic., to appear. I. 2. 3. Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Diļ¬erential Equations,” allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: Manuscript Due February 1, 2009 First Round of Reviews May 1, 2009 Publication Date August 1, 2009 Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil; [email protected] Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; [email protected] Celso Grebogi, Department of Physics, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected] Hindawi Publishing Corporation http://www.hindawi.com