Yn+l -L-l(d/dt)Yn L-I

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
Download