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INTEGRAL , vol. 5 no. 1 , April 2000 1

Johan Matheus Tuwankotta

Studies on Rayleigh Equation

Intisari

Dalam makalah ini dipelajari persamaan oscilator tak linear yaitu

persamaan Rayleigh : ( ) + = x x x x 12

( ) . Masalah kestabilan lokal

di sekitar titik tetap dan hubungannya dengan paramater di bahas. Jugaakan dibuktikan eksistensi selesaian periodik yang dalam hal ini juga

merupakan limit cycle dengan menggunakan teorema Poincare-Bendixson.

Selesaian periodik tersebut akan dihampiri dengan menggunakan metode

Poincare-Linsted. Sebagai perbandingan juga di hitung selesaian numerik

dan dibandingkan dengan selesaian asimtotik

Abstract

In this paper we consider a type of nonlinear oscillator known as Rayleigh

equation, i.e. ( ) + = x x x x 1 2( ) . We study local the stability of thefixed point in the presence of positive parameter . We are also looking atthe existence of the periodic solution which is also a limit cycle in this

equation. Using Poincare-Bendixsons theorem we proof that the limit cycle

exists. For small values of the parameter, we use asymptotic analysis to

approximate the periodic solution. The method we apply in this paper is

Poincare-Linsted method. We calculate also the numerical solution for the

system and compare the result with the asymptotic.


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2 INTEGRAL , vol. 5 no. 1 , April 2000

I. Intr oductionConsider a system of differential

equations( ) =u f t u, , ..(1)

where u R 2 , f is a vector valuedfunction in R

2and is a parameter. We

assume that solution exists and is unique

with given a initial value. This system

is also known as a planar system. The

system depended on the parameter .For a fixed , let u(t) = (t) be asolution of the system (the subscript

shows the dependency on the

parameter). This vector valued function

parameterize by t defines a flow inthe two dimensional plane u = (x , y).

The curve of points (x , y) generated by

(t) when we let t run from - to + iscalled the trajectory. A collection of

trajectories of different initial value is

calling the phase portrait of the system.

A simple question that arises is; how to

know the phase portrait of a system.

The best way to have the phase portrait

of a system is to calculate the generalexpression of the solution of the system.

Unfortunately, this is not an easy thing

to do especially when f is non-linear.

The second best way is to calculate the

integral of the system. An integral (in

this case) is a real valued, two variables

function F(x, y) such that

d

dtF x y( ( , ))

= 0 , which is the

derivative of F with respect to time

evaluated at a particular solution zero.Consequently, F(x, y) = C define a

curve in the (x, y) plane such that the

flow of system (1) will map the curve

to it self. Such a curve is called

invariant curve of the system. In

elementary courses on differential

equation such a curve is called implicit

solution. (Unfortunately not every

system has an integral).

We remark that if we can calculate one

of those (either the solution or the

integral) we will have a global picture of

the flow. It means that the analysis is

valid for all time (t) and everywhere inthe (x, y) plane. That is why these two

methods are also known as global

analysis technique. Naturally, when the

global picture cannot be achieved, we

can try to have a local picture. We can

restrict our analysis in a small area

around some interesting location. The

question is, where the interesting

location is.

Some of the locations, which are

considered to be interesting, are arounda fixed point or around a periodic orbit.

A fixed point (also known as a constant

of motion point, or critical point, or

singular point) is a point (x0, y0) in the

plane such that f(x0, y0) = 0. A periodic

orbit is a solution (t) where a realnumber T such that (t + T) = (t)exists. Now supposed we have a

periodic orbit that start at a particular

point. After T time it will be back at that

starting point. It means the trajectory

will be a closed curve. The converse is

also true if we assume that T is finite.

The most common tool to understand

locally the flow of the system is

linearization. We expand f to its Taylor

series around a particular solution, i.e.

( )( ) = +u f u ! (2)where the dots represent the higher

order terms and represent the partial

derivative with respect to spatialvariables. And then we can use linear

analysis to obtain the information about

the flow. See [1], [4] or [5] for

example.

Our goal in this paper is to analyze the

non-linear oscillator known as the

Rayleigh equation, i.e.

( ) + = x x x x 12

( ) .(3)

where is a positive parameter. We

give the analysis in the neighborhood of


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the fixed point. We will also show that

there exists a periodic orbit and that it is

stable. This stable periodic orbit is alsoknown as the stable limit cycle. Due to

the fact that the analytic calculation of

the limit cycle is complicated, we will

construct an approximation of its for a

small parameter using Poincare-Linsted

method. For a large parameter we can

construct the approximation using

boundary layer technique or singular

perturbation technique (see [3]). These

technique is mathematically non trivial

so we will skip them. In this paper we

will also construct the numericalcomparison of the analysis.

II. Fixed Point AnalysisTo calculate the fixed point of the

Rayleigh oscillator, we transform the

system into a system of first order

differential equations. This is done by

setting x = x and y = x'. The Rayleigh

equation then becomes

( )

=

= +

x y

y x y y 1 2..(4)

It is easy to see that the fixed point of

(4) is only (0,0). In general, for a

system like in (1) the fixed point is also

depended on . The linearized systemof (3) written in matrix form is:

=

y

x

1

10

y

x'

'

. ..(5)

From linear analysis we know that the

stability of the fixed point (0,0)dependeds on the eigen-values of the

linearized system (5). The information

on the stability also gives information

on the flow it-self around the point. In

this case we have the eigen values

1 2

24

2, =

. ..(6)

For =0 we see that the eigen-valuesare purely imaginary 1,2 = i . This isclear since if = 0 , what we have is

just a linear harmonic oscillator with all

of the solutions being 2-periodic.Thus, the origin (0,0) is a center point if

= 0. In general this statement is notalways true.

Figure 1 : Phase portrait for = 0. Thefixed point in this case is a center point

and all solutions are periodic of period

2.Obviously the eigen-value of a non

linear system should be non linearly

dependent on the system. Thus if we

linearize locally, we restrict the domainso that the nonlinear effect can be

considered as a small perturbation. If

the real part of the eigen value is non-

zero then we can choose the domain

small enough so that the non-linear

perturbance will be small enough. It

implies that the real part will still be the

same in sign (see [4],[5] for details).

This is not the case for zero real part of

the eigen-value.

In the case of all the real parts of the

eigen-values is zero we have to consider

a higher order term on the expanded

system. In the case that there is at least

one is non zero, we can use Center

Manifold Theorem (see [2] for details).

This case does not arise on this problem

so we omit it .

For 0 < < 2 we can write the eigen

values as

1 2 2, ,=

2, we have two different realeigen-values. Thus we will have two

linearly independent eigen-vectors.

Since the eigen values are different, then

the origin will also be improper node

sources. The phase flow is flowing out

of the origin along two direction only.

The picture in figure (1) until (4) are

drawn using Maple V. It is clear that

the analysis of the origin coincides with

the numerical result using Maple V.

Figure 4: Phase flow for = 2.5.

III. Existence of the PeriodicSolution

To proof the existence of the periodicorbit, we will apply a fundamental


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theorem of Poincare-Bendixson. This

theorem will be stated without proof.

Theorem: Poincare-Bendixson If D is a

closed bounded region of the (x, y)-

plane and a solution u(t) of a non-

singular system (1) is such that u(t) D,then the solution either a closed path,

approaches a closed path, or approaches

a fixed point.

The theorem in other word says that if

we have a closed and bounded region

which the flow of the system in (1) is

flowing into the region, then there is aperiodic solution or a fixed point in the

region. We will apply this theorem to a

general type of oscillator with damping

by constructing a region which is

invariant to the flow. After that we will

relate the Rayleigh equation to the

general equation we have.

Consider now an oscillator equation

known as Lienard equation, i.e.

+ + =x f x x x( ) 0 ..(7)

We define F x f s ds

x

( ) ( )= 0

and

assume it to be an odd function. Futher

assume that for x > (a real positivenumber) F is positive, goes to infinity

for x goes to infinity, and monotonically

increasing , while for 0 < x < F isnegative.

We transform (7) to a first order system

using transformation (x, x) (x, y =x+ F(x)). Using this transformation we

have

= =

x y F x

y x

( )

...(8)

Now define a transformation to polar

coordinate by ( )R x y= +1

2

2 2.

Consequently we have R = -xF(x). It

implies that for - < x < , R 0.Thus the flow on the circle domain with

radius is flowing out of the domain.

What we need to construct is a domain

where the flow is flowing into the

domain. See figure (5).

Before we start constructing the domain,

we first make some important remark on

the gradient field of the flow. From (8)

we can calculatedy

dx

x

y F x=

( ). It

implies that on the y axis the tangent is

horizontal and on the curve y = F(x) the

tangent is vertical. One can clarify

easily that if we start at an initial value

say A we will have a trajectory as infigure (5) (this can be easily done by

considering the negative or positiveness

of x or y at certain location). The

reflection symmetry of the system gives

the negative trajectory.

Our purpose is now to proof that R(A) >

R(D). This can be done by considering

the line integral

R D R A dRABCD

( ) ( ) . = We split thecurve ABCD into three segments AB,

curve ABCD into three segments AB,

BC, CD. We also have two expressions

for dR, i.e.

Figure 5: Constructing the invariant

domain.

dRxF x

y F xdx dR F x dy=

=

( )

( )( )or .


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We first consider segment AB and CD.

If A (= (0, yA )) is high enough then we

know that y-F(x) is large while -xF(x) isbounded. Thus for yA going to infinity,

the integrals dRAB

and dRCD

go tozero. It also means that the integral is

dominated by the line integral in the

segment CD. In segment CD we have

dR F x dyBC B

C

= ( ) . Since we assumethat for x > , F is positive and

monotonically increasing, we see thatthe integral will be negative and

monotonically decreasing (it is clear

since the integration is from positive

values to the negative values of y). Also

it is continuosly dependent on yA. Thus

if yA goes to infinity the integral goes to

infinity.

It means that we can choose yA large

enough such that R(A) > R(D). In the

same way we can show R(-A) < R(-D).

Now consider the domain in figure (5)

which is bounded by the circle withradius , the two trajectories and theline segment connecting the

trajectories. This domain defines an

invariant closed and bounded domain.

Moreover, the flow is going into the

domain. Hence the Poincare-

Bendixsons theorem applies. We know

that the only fixed point of the Lienards

system is the origin. The conclusion is

that we have at least one periodic

solution. Moreover, since F is

monotonically increasing andconsequently the integral in the segment

BC is monotonically decreasing, the

condition that if = then leads toprecisely one periodic solution.

The question arises is how to

apply this result to the Rayleigh

equation. If we differentiate The

Rayleigh equation, we have

( ) + =x x x x 1 3 02( ) . ....(9)

Define z x= 3 and transform (7)into

( ) + =z z z z 1 02 ....(10)The solution of (9) is just a solution of

ordinary Rayleigh equation with an

additional constant. Thus by setting the

constant to be zero, we have the original

solution. Since (10) satisfies the

assumption we now find that there exists

at least one periodic solution. We can

proof that in the case of (10), =. Thuswe have a unique periodic solution.

IV. Asymptotic Approximation

The next aim is to approximateasymptotically the periodic solution of

Rayleigh equation. We will apply the

Poincare-Linsted Method to

approximate the periodic solution. We

will first give a short introduction to the

method. Details of the method can be

found in [4].

Consider a second order differential

equation of the form

( ) + = x x f x x , , (11)It is easy to see that for = 0, allsolutions are 2-periodic. For 0 weassume that there exists a periodic

solution with period T() starting atinitial values x(0) = a() and x'(0) = 0.The fact that the period and the location

or the periodic orbit are dependent on is natural. Define a time transformation

= t such that in this new timevariable, the period of the periodic orbit

is 2. We write -2 = 1 - ().Obviously, the periodic solution is alsodependent continuesly on . Thus weassume that we can write the periodic

solution as

x x x x( ) . = + + +0 12

2 ! ..(12)

With the new time variable we can write

(11) in the form of (the dot represents

the derivative with respect to )


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( )"" ,"

,x x x f xx

+ = +

11

.

If we write the right hand side as

( ) g x x, ", and transform it to itsintegral form, then the periodicity

condition of the solution leads to two

equations, i.e.

( )

( )

F a g x x d

F a g x x d

1

0

2

2

0

2

0

0

( , ) sin( ) , ", ,

( , ) cos( ) , ", ,

= =

= =

..(13)

Implicit function theorem gives the

conditions for the existence of a non-

trivial solution of (13) in the

neighborhood of = 0, provided

( )( )

F F

a

1 20

,

, .(14)

For the case that the existence condition

is satisfied we will have a unique

periodic solution. This is in agreement

with the previous analysis (we have

proven that the Rayleigh system has a

unique periodic solution). Note that if

(14) fail to hold, it does not mean that

there is no periodic solution. In many

cases such as in hamiltonian system, the

periodic solution is not isolated so that

apriory we know that the condition fails

to hold.

It is instructive to apply the method.

Note that this is an asymptotic method

so that it is valid for 0. Thus wetake equal to small parameter . Thecalculations are rather routine and

lengthy. We write the result of the

calculation up to order 4. The

calculation is done using Maple release

V. The periodic solution is

approximated by (11) where

)cos(33

2)(x 0 =

)sin(312

1)3sin(336

1)(x 1 =

)cos(312

1

)5cos(3288

1)3cos(3

48

1)(x 2

+

=

)sin(32304

5)7sin(3

1728

1

)3sin(32304

13)5cos(3

216

1)(x 3

+

=

)cos(312288

23

)9cos(3552960

61)7cos(3

331776

365

)5cos(382944

241)3cos(3

27648

59)(x 4

+

+

+=

and = t. Obviously this is anontrivial thing to do but nevertheless,

as we noted above it is instructive to do

it. Furthermore, we also calculate the

period, i.e.

T O= + +21

8

5

1536

2 4 5 ( ) .

The location of the periodic solution is

a

O

= + +

+

2

33

29

2883

2743

16588803

2

4 5

( )

We will check on this result with a

numerical integration of the system

using MatLab 5.2.

V. Numerical ResultWe will now check the approximation

above using a numerical integration.

We plot both of the result in one picture

so that we can immediately compare theresult.

We use only the first approximation

x() = x0() and O(5) approximationfor . This has the advantage of a verylong time-scale. It means that the

approximation is valid for quite a long

time and in this case until -4. Thenumerical integration is done using

build in integrator in MatLab 5.2 for

non-stiff system, i.e. ODE45. This

integrator uses Runge-Kutta scheme for


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integration. We plot the result in figure

(6), (7), (8) and (9). The curve plotted

by symbol o represent the numericalsolution and the line curve is for the

asymptotic approximation.

In figure (6) and (7) we take =0.05.Thus the approximation is still good

until 16,000 second. For numerical

integration this long time integration is

not recommendable. We have to worry

also about the numerical error. Thus we

integrate with initial values x(0) = a and

x(0) =0 for 20 second only. Obviously

we can expect a very good result in thecomparison.

Figure 6: The comparison between

numerical result and asymptotic

approximation for =0.05. This picturerepresents the time evolution of the

periodic solution.

Figure 7: The comparison between

numerical result and asymptotic

approximation for =0.05 on the (x,

x)-plane.

In figure (8) and (9), we increase thevalue of to 0.25. We integrate withthe same initial values for 280 second.

This is already longer than the time-

scale of validity of the approximation.

Thus we expect to see an O() deviationon the picture. In figure (8) the

deviation is not very clear but in figure

(9) it is.

Figure 8: The comparison of the timeevolution for =0.25.

Figure 9: The comparison in the (x, x)-

plane for =0.25.

To make the error even clearer we plot

the error defined as the difference

between the numerical solution and the

asymptotic solution in figure (10) and

(11).


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Figure 10: The error on x.

Figure 11: The error on x.

VI. Remarks and AcknowledgmentWe have shown an example of

asymptotic approximation of a periodic

solution of a planar system. This

Poincare-Linsted method can also be

extended to higher dimension system.

We note that it is not trivial to do so.

Beside the method we have used here

there is also a simpler method to get theapproximation. The method is called

averaging method. This is a very natural

method introduced intuitively by

Lagrange et. al.. Unlike the Poincare-

Linsted method, this method is more

general since it can be applied to

approximate non-periodic solutions.

The disadvantage of the averaging

method is that , to get a higher order

approximation is non-trivial , while for

the Poincare-Linsted method it is

routine. Also in extending the time-

scale, using averaging method is rather

difficult. For a higher dimension case,

where everything is restricted, averagingis easier to apply.

We would like to express our gratitude

to Santi Goenarso for her diversified

contribution. We also like to thank our

student Maynerd Tambunan for

providing a comparison work for the

Maple calculation.

References:

[1] Boyce, W.E., Di Prima, R.C.,Elementary Differential Equations

and Boundary Value Problems, John

Wiley & Sons, Inc., New York et.

al., 1992.

[2] Carr, J., Applications of Center

Manifold Theorem, Applied

Mathematical Science 35, Springer-

Verlag, New York, 1981.

[3] O Mailley Jr., R. E., Singular

Perturbation Methods for Ordinary

Differential Equations, Applied

Mathematical Science 89, Springer-

Verlag, New York, 1991.

[4] Verhulst, F., Nonlinear Differential

Equations and Dynamical Systems

2nd

ed., Springer-Verlag, Berlin,

1996.

[5] Wiggins, S., Introduction to

Nonlinear Dynamical System, Text

on Applied Mathematics 2,

Springer-Verlag, New York, 1990.

[6] Guckenheimer, J., Holmes, P.,

Nonlinear Oscillations, DynamicalSystems and Bifurcations of Vector

Fields, Applied Mathematical

Science 42, Springer-Verlag, New

York, 1983.

Author Johan Matheus Tuwankotta is a lecturer

at the Mathematics Department ITB.

E-mail address: [email protected]

Received August 25, 1999; revised

September

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