SM13(Cuong) Sdarticle

8/11/2019 SM13(Cuong) Sdarticle

1/12

Journal of Sound and Vibration (1991) 144(2), 293-304

QU ASI-PERIODIC SOLUTION S CALCULATED WITH THE

SIMPLE SHOOTING TECHNIQUE

F. H. LlNGt

Department of

Engi neeri ng M echanics, Shanghai Jiao Tong U ni versit y, Shanghai 200 030,

Peopl es Republ i c of China

Recei ved 24 Apri l 1989, and in revi sed,form 22 M arch 1990)

A shooting-type numerical metho d for calculating quasi-periodic solutions in a multi-

excited system by using derivatives and with an improved interpolation technique is

presented. Three examp les of ordinary differential equations and iterative m aps show the

efficiency of the new algorithm. This algorithm can also be effectively applied to a periodic

system.

1. INTRODUCTION

The calculation of quasi-periodic solutions is very important from a practical point of

view. A great number of regular motions in a multi-excited system should be regarded

as quasi-periodic rather than as periodic, since the ratio of two or more exciting frequencies

will usually not be a simple rational number, and if the denominator of this number

is not very small, then the period of the solution will be so large that it is difficult to

reveal the periodicity even within a pretty long observation time. In a real computation,

it is also more convenient to treat them as quasi-periodic solutions as stated below .

On the other hand, there is also a need for calculating quasi-periodic solutions of a

non-linear system in chaotic dynamics. Firstly, a route to chaos through quasi-periodic

motions has been intensively studied. Secondly, one often wants to know the distribution

of different motion patterns in the param eter space, but it is rather time-consuming to

identify a chaotic motion. How ever, since there exist only four kinds of spatially bounded

motion in non-linear systems-equilibrium, periodic, quasi-periodic and chaotic motion-

and the domain of equilibrium in the param eter space is usually of zero measure, one

can perform the task by determining the distribution of periodic and quasi-periodic

motions.

An efficient way for calculating quasi-periodic solutions is to extract a discrete time 7

map from the continuous flow and its derivative starting from an initial point, w here

T

is a reference period. This map is usually called a PoincarC map, and the iterative points

of the PoincarC map of quasi-periodic solutions are located on a closed curve in the phase

plane. For a single-excited system, i.e., a periodic system w ith period T, this curve is

invariant, but for a multi-excited system with non-commensurable frequencies, i.e., a

non-periodic system , a lthough this curve is still close d it is no longer invariant. In other

word s, th ere is a time-independent map for the periodic system, and a time-dependent

map for the non-periodic system.

There are quite a number of publications on the topic of the numerical computation

of the invariant curve of a periodic system. In earlier work s [l-4] one tried to locate a

t Present address: Departm ent of Physics and Engineering Physics, Stevens Institute o f Technology. Castle

Point, Hoboken, New Jersey 07030, U.S.A.

293

0022-460X/91/020293+12$03.00/0

$Q 1991 Academ ic Press Limited


2/12

294

F. H. LING

single point on the invariant curve and henceforth the corresponding quasi-periodic

solution. In some recent pub lications [5-71, one searches for many p oints on the invariant

curve simultaneously with a collocation method. The collocation method is efficient for

a periodic system, but it is not applicable to a non-periodic system: e.g., most multi-excited

systems.

The method suggested by Kaas-Petersen [&IO] can be regarded as an improvement

of the above-mentioned single point method, in which an interpolation technique is

incorporated. This method is emp hasized in the present application to a multi-excited

system, and w ill be further improved in two aspects in this paper: namely, by using

derivatives instead of the difference approximation and by using all iterative points in

the interpolation. The corresponding new algorithm is described in the next section, and

three ex amples in section 3 illustrate that the new algorithm is more robu st and time-sav ing.

Moreover, the method is also very good at finding invariant curves in the periodic system,

as shown in section 4.

2 IMPROVED SIMPLE SHOOTING METHOD

In this method, the condition that the Poincar6 map of quasi-periodic solutions of a

non-linear ordinary differential equation should be a closed curve is used for determining

a quasi-periodic solution. For example, a bi-periodic solution x(t) with two incomm ensur-

able periods T, and

Tz

can be written as x(r) =

X(t, , fJ,

where t, = t/

T, , tz = t/ T,,

and

the quasi-periodicity requires that Z( 1,)

f2) = a( f, +

1, f?) = %(

, tz+

1). By defining

To =

kT,

mod

TJ/ T,,

1)

where

k

is an integer, one obtains the PoincarC m ap of the quasi-periodic solution as

P 7k) = x kT,) = ri k, kT,/ TJ = a 0, auk),

and therefore

P(1) =X(0,1) = Z(O, 0) =x(O)

and

P(0) = Z(O, 0) =x(O).

In this way, the calculation of quasi-periodic solutions is converted to a problem

determining a proper x(O), so that the following condition holds:

P( 1) -x(O) = P(0) -x(O) = 0.

I of

2)

This is simply a problem of finding zeroes of non-linear functions. Since the non-linear

function itself is obtained through numerical integration, the problem has to be solved

iteratively: e.g., with a simple shooting method.

Before considering the shooting method, one can note that from equation (1) one

knows ?k -s will be distributed randomly in the interval TVE (0,l) and never reach Tr,= 1

or 0, so one h as to use an extrapolation. Moreover, since rk = 1 and TV 0 are in fact the

same thing, one can redefine rk = TV 1 when Tk< I/2, so that then the Tks are in the

interval (l/2, 3/2), w hich is convenient for estimating P(1) by using an interpolation.

The application of the simple shoo ting method to this problem is illustrated schemati-

cally in Figure 1. Suppose one has a linear bi-excited system w ith the natural frequency

w, the steady state solution of which is x*(t) = cos t + cos

fi, which is being sought.

Since there is usually a transient process, if one starts from an arbitrary initial point, one

will obtain a solution of the form

x(t) = C eey

cOs(wr+e)+x*(t).

The iterative points of the PoincarC m ap P(Tk) of x*(f) are located on a bold circle in

the middle of the picture, and the successive iterative points of the PoincarC m ap P(Tk)

of Z(t) starting from an arbitrary initial point, say

A,

are located on a spiral which ex tends


3/12

MULTI-EXCITED QUASI-PERIODIC RESPONSES

295

.5)---x kT,)

Figure 1. Poincare series of a quasi-periodic solution; P r, 1 ransient; bold circle, steady state.

towards the bold circle. The task is to adjust the initial po int so that it will be located

on the bold circle. In the shooting process, one moves the initial point in a systematic

way, so that it approaches the bold circle rapidly. In order not to be misled, readers

should notice that Figure 1 does not exhaust all possibilities. For example, the spiral may

also start from a point inside approaching the bold circle from w ithin. M oreover, if the

solution is unstable, the spiral starting n ear the circle will go away. N evertheless, all these

cases can be treated uniformly with this method. As in a shooting for periodic solutions,

the distance between the point A and an intersection point of the spiral with the radial

?-k= l(0) (e.g., the first one) should be estimated and then gradually eliminated during

the shooting process. Since the iterative points are sparsely distributed on the spiral, it

is impossible to reconstruct the spiral by using an interpolation and henceforth to

determine a real intersection point. H owever, one may overcome this difficulty by assum ing

there is an average value of the distances between the point

A

and several intersection

points and the average value of the derivatives of these distances with respect to the

initial conditions. These ave rage values can be easily evaluated by using the interpola-

tion with rk as the argum ent. Fortunately, the shooting process runs smoothly with these

average amounts.

To perform the interpolation, Kaas-Pe tersen picks up a few points from the calculated

sequence P(rk), k = I, 2,.

. .

,

which meet the following requirements: (1) by arranging

them in the order of an increasing index

k ,

they form an increasing sequence of 7k; (2)

the va lue of Tk is located in the interval

1 -

E, 1 + E), where E is a sma ll positive number

representing the given error bound.

According to these requirem ents the points numbered 4, 7, 10 and 13 in Figure 1 are

used to interpolate the average intersection point and the corresponding derivatives.

In Figure 1 a curve through these points represents the interpolation curve, and the

intersection point of this curve with the radial T,, = l(0) can then be regarded as an average

value of P(1). It is evident that only a tiny part of the whole sequence is used, an d this

may mean a waste of information gathered in the calculation.


4/12

296

F. H. LING

In fact, the essential factor for a smoo th run of the shooting proces s is to evaluate a

proper average value of the difference P( 1) -x(O) and its derivatives with respect to

the shooting param eters. The accuracy of the interpolation in a normal sense is not

important. Therefore, it will be better if one uses all calculated P( TV valuest and their

derivatives to perform the interpolation, which are rearranged in the order of the increasing

rks. The interpolation curve o f this new schem e is illustrated with a dashe d line, and that

of the old one with a solid line, in Figure 2. The new interpolation schem e wor ks very

well in practical calculations, as is shown in section 3. The new schem e enables one to

save a great d eal of comp uter time, since, for a given accurac y, far fewe r iterative points

are required than those in the old schem e. From intuitive considerations, we have also

tested other interpolation sets by excepting several bad points, such as 3, 8 and 14, or

the first few points, such as 1, 2 and 3, in Figure 2. But all these variants do not work

better. Since the calculation work of a numerical integration is dominant, to mak e use

of all calculated points means a great saving of comp uter time.

:

\

\

2 ;

\

/ \

\

/

\

ig

I

1.1

\

45

\

I

\

012

2

C

Figure 2. Interpolation curve. -, Kaas-Petersen [8-IO]; - - -, this paper.

For ordinary differential equations, this means that the integration interval can be

greatly reduced , and therefore the integration error is smaller. As is well known, the error

in initial conditions gro ws exponen tially with the length of the integration interval; see,

e.g., reference [ 111.

The second point is to calculate the derivatives of P(rk) with respect to the shooting

param eters directly instead of approximating them with a difference quotient. The main

advantages of this substitution are the following. Firstly, the New ton iteration when using

the derivative is of a quadratic convergence, and that when using the difference quotient

is of a (super)linear convergence. Secondly, it is not very easy to choos e a prope r increment

of the shooting param eters when using a difference approximation, since too large an

t Although this can be regarded as meaning that the error bou nd F equals 0.5 in the above-m entioned

requirement, nevertheless, since E = 0.5 means there is no error co ntrol at all, the idea behind the new algorithm

is different from that of the old one.


5/12


297

increment would cause a bad approxima tion, and too sma ll an increment would cause a

severe round-off error, also leading to a poor app roximation.

The new algorithm works especially well if the quasi-periodic solution is unstable. In

this case the absolute values of the derivatives are often very large, and the difference

quotient usually fails to be a reasonable approxima tion. This is, however, much less

serious if one calculates the derivative directly. The practical limitation for evaluating an

unstable solution with the improved method is that the instability should not cause an

overflow. T his situation will, however, seldom appear in practical calculations.

The above-mentioned derivatives can be calculated as follows. For ordinary differential

equations (see references [ll, 121, for example)

i =f(x, t),

.YE 5%,

(3)

the derivative matrix ~P(T ~)/~s = [ax( f)/d~],_~~,

is evaluated from an initial value problem

of the ordinary differential equations

ax t)

1 l

afb, t) ax t)

ax t)

=~_

as

ax as

[

1

ax0

- =-

as

,=o

as

(4)

where ax, / ass determined according to the meaning of the parame ter vector s. Although

one often takes s =x0 (then ax, / as Z, , uni t matrix), many other quantities, e.g., the

coefficients in equations, can also be taken as the comp onents of s. In the latter case,

there w ill be some zero column s corresponding to these quantities in the derivative matrix.

The main calculation work of evaluating derivatives is to integrate

n x n

first order

ordinary differential equations (4). On the other hand, if one approximates derivatives

with difference quotients, one needs to integrate n first order ordinary differential equations

(3) n times with different initial cond itions to find all the difference quo tients. Therefore,

the amount of calculation in both sch emes is almost the same. This statement is also true

for an iterative map.

The treatment of the iterative map y,,, = f( y,), y, E $?I, = 1,2, . . . , where the index

denotes the iteration number, is similar but simpler. The derivative matrix af

~ ~ ) / a s

d,vi+klas is evaluated by using

aYi l afay,

=--

~,

i=1,2 ,..., k-l,

as

dyi a s

and the starting value

ay, / as

s determined according to the meaning of s.

Various interpolation formulas can be used. For a bi-excited system, this is performed

by a one-dimensional interpolation. Lagrang ian interpolation has been used in this work,

but a spline interpolation could be an alternative. If there are more than two exciting

frequencies, one can no longer use a Lagrang ian or spline interpolation due to the random

distribution of the iterative points. A polynomial of two or more argum ents has been

tried for this purpose, but the calculation was quite tedious.

After the initial point of a steady state quasi-periodic solution is decided, a time series

is generated for obtaining the Fourier coefficients by using the Fast Fourier transform or

a triangular function fitting. The latter one has been used in the work reported in this paper.

The stability of the solution is decided by the spectral radius of

[ aP ~~) / ax, ] . , =,

The

solution is stable if this radius is smaller than one, a nd is unstable if it is larger than one.

3. EXAMPLES

Several examples have been calculated. Although the purpose of developing this

algorithm is to apply it to non-linear problems, two linear examples are discussed first


6/12

298

F. H.

LING

in this section. Since the aim is to show the advantage of the new algorithm, in the linear

case the exact solution and hence a convincing comparison is available. In the third

examp le results a re presented for a non-linear oscillator with both stable and unstable

solutions. The non-linearity of this oscillator seems to be weak; how ever, the excitation

is strong enough so that the non-linear effect is apparent.

3.1. EXAMPLE 1: LINEAR OSCILLATOR

The oscillator equation is

jr+2a~++x=(/3-2)cosJZt-2JZasinJZt+(~-1)cost-2~sinf,

(5)

with T, =fin and T2 = 27r. For initial conditions x(O) =2,1(O ) =O, the exact quasi-

periodic solution of equation (6) is known as

x=cos~t+cos t.

(6)

To illustrate the metho d equation (6) is rewritten in the form of equation (3) as

x1=x2,

~,=-2~~x~-~x,+(~-2)cos~t-2~~sinJZt+(~-l)cost-2~sint.

With s = {x,~, x*,,}~ the initial value problem (4) for evaluating the derivatives is

ax, ax,

8 x 2 a x 2

ax x

-

--

x10x 2 0

8 x 1 0

i

I[-

a x z o

o 8 x 2 0

1 0

ax, ax2 =

--

axlo axzo

_&.33-

10

IO

-I[ 1

2q33f5

,

2 ax,

=o 1

1

l o axzo t= .

In reference [8], the calculation was carried out for Q = 2, p = 1 in the interval

[0,

K T ,] , K =

13. For E = 0.2, only four points 4,7,10 and 13 were used in the interpolation.

The resulting error is A x (O ) = 7.38E - 03 and A O ) = 3.79E -04. The results obtained by

using the new algo rithm are given in Table 1. It can be seen that to reach a similar

TABLE 1

A ccu r a c y o f t he num er i c a l l y ca l c u l a t ed

quasi - pe r i o d i c so l u t i o n o f t h e o r d i na r y

d i f er en t i a l equa t i o n ( 6) w i t h a = 2 ,

p=1

K

Ax(O)

Ai(0)

20 -1 20E-10

19 -3.76E-10

18 -6 32E-10

17 4.78E-11

16 7.33E-10

15

1.38E-09

14

7.67E-10

13

4.25E-10

12

8*98E-08

11

7*06E-08

10

2.96E-06

9 5.00E-06

8

-2.83E-04

7

1.38E-04

6

1.85E-02

5

3.21 E-02

4.57E-10

1.40E-09

2.34E-09

-1.79E-10

-2.708-09

-5.28E-09

-1.38E-09

l.lSE-08

4.24E-10

-3.66E-07

1.78E-07

-6.37E-05

-8.33E-05

-9*23E-04

2*98E-03

-6*95E-02


7/12


299

accuracy, one needs only seven points; this means that almost 50% of the computer work

is saved. It can also be seen that with a too large K

(K >

17 in this example), the accuracy

will be decreased due to the accum ulation of the round-off error. This example shows

clearly that this algorithm works very well for a system w ith large damping.

3.2. EXAMPLE 2: DIFFERENCE EQUATION

The difference equation is

u(t+2)+u(t+1)+0~75u(t)=cos~~t,

(7)

or, rewritten in the iterative map form,

u,(t+ 1) = u*(t),

u*(t+1)=u,(t)-0~75u,(t)+cosGrt,

(8)

and possesses a steady state quasi-periodic solution

u(t)=Ccosfi&+Dsin& rrt,

(9)

since the frequency of the solution fir is incomm ensurable with the basic frequency of

the map. After deciding on the constants C and B , one obtains

u,(O) = C = -1.09085 67916 34538,

u,(O) = C cos fin-+

D

sin fir = 1.55572 71093 23350.

437 points were required in reference [ 81 for obtaining an accuracy of Au, 0) = 1990E - 09

and Au, O) = 2*62E - 09. The present results are show n in Table 2. Only 14 points were

required for reaching the same accuracy-only 3% of the original calculation amount.

3.3.

EXAMPLE

3:

NON-LINEAR OSCILLATOR

The non-linear oscillator equation is

TABLE 2

Accuracy of the numericall y calculated

quasi-periodic solution of the dif erence

equation 8) with a = 2, p =

1

K

Ax(O)

AX(O)

20

19

18

17

16

15

14

13

12

11

10

9

8

6

5

-1.35E-14 5.19E-15

-5*47E-15 -2.16E-15

-1.20E-14 7.048-15

8.93E-14 4.788-14

-3*75E-12

5.67E-12

-5.33E-11

-5.44E-12

-2.06E-10

2.96E-10

3.868-09 4.75E-09

-9.01 E-08 9.84E-08

-3.27E-07 -2.36E-07

-1.75E-06

2.88E-06

-9.79E-05 -4.88E-06

-1.228-05 -2.82E-04

- 1.39E-03 -1.99E-04

-2.12E-02

4.57E-02

-1.42E-01

-2.8OE-03

(10)


8/12

300 F.

H. LING

with T, = 2rr and Tz = 27rlO.1 15 when 0 = 1. This equation possesse s a periodic solution,

but its period is too large to deal with by a standard shooting metho d for evaluating a

periodic solution [12, 131. In the linear case, the solution has the form

x(t) = (0.15/a) sin t+A cos (0*11 5t)+ B sin (0.115t),

where A and B are determined by

(11)

l-o.l15z

-0.23

1_~2l315~][~]=[01~

(12)

With cr = 0.025, 44 points were required in reference [9] for obtaining an accuracy of

Ax(O) = 6*70E -05 and A.%(O)= 1.96E -07. The present results are shown in Table 3. Only

ten points are required for reaching the same accuracy-about 80% computational work

is saved. The results with a smaller damping cr = O-00 005 and a negative damping (Y= -0.25

are also shown in Table 3. It is clearly seen from these results that this method is also

applicable to a system with very small, even negative, damping: i.e., for calculating

unstable solutions. If there is no damping at all, then one will have a tri-periodic solution

which depe nds on initial conditions.

TABLE 3

Accuracy of the numerically calculated quasi-periodic solution of the ordinary di erential

equation

( 11)

Y = 0.025

Y = 0*00005

ff = -0.025

K

Ax(O)

A 01

Ax(O)

A1(0)

Ax(O)

Ai(0)

20

-7*61E-16

19

-1.48E-16

18 6+51E-15

17

4.39E-14

16

-8.18E-12

15

-1.6lE-11

14

3.93E-10

13

7.22E-10

12 l.O5E-08

11

-2.14E-08

10

-1*08E-06

9

1.79E-06

8

1.27E-03

7 7.48E-03

6

8.26E-01

5

8.86E-01

6.208-16

1.43E-16

6.51E-15

3.11E-14

-4.99E- 12

-2.98E-11

4.13E-10

1.51E-09

9*77E-09

-3.13E-08

-4.09E-07

2.44E-06

-3.52E-04

1.33E-03

3.46E-01

3.27E-01

-6*37E-14

-5*5OE-15

-4.72E-13

9.22E-12

-1.79E-09

-3.49E-09

8.38E-08

1,43E-07

2.30E-06

-4.94E-06

-2.68E-04

5.06E-04

3.00E-01

2.07E-01

1.32E-00

8.52E-01

1.95E-13

2*80E- 14

2.20E- 12

4.84E-12

-1.47E-09

-7.27E-09

9.97E-08

3*45E-07

2.39E-06

-7.78E-06

-l.l2E-04

6.54E-01

-6.86E-02

-1.97E-01

4.79E-01

-6.71E-01

-2*67E-16

-5*OOE-18

-2.52E-16

-4.268-15

1*05E-12

2.558-12

-5.98E-11

-9.36E-11

-1*69E-09

3.85E-09

2.27E-07

-4.84E-07

-3.57E-04

-2.81 E-03

-9.79E-03

-1.63E-01

-l.l4E-16

-l*OOE-16

-5.74E-16

-1.37E-15

1.42E-12

5.97E-12

-8.OlE-11

-2.64E-10

-1.95E-09

6.50E-09

l.O3E-07

-6.03E-07

1.24E-04

9.87E-04

-2.23E-01

-6.99E-01

Nex t, one can fix c = 0.01, (Y= 0.025 and vary n for obtaining the resonance curve in

the non-linear case . Afte r finding the quasi-periodic solution, a triangular function fitting

of the steady state time series is perform ed to obtain the resonance curves show n in

Figure 3 with solid lines. Apart from the basic harmonics with frequency 0 and O-115 0

and amplitudes A, and A0.,,5

respectively, the two largest difference harmonics with

frequency 1.230 = 1 + 2 * 0.115)0 and 0-77fl= 1 - 2 * 0.115)fl and the corresponding

amplitudes A, .23

and A,.,, respectively are also shown in the figure. A resonance peak

can be seen near R = 1, which has a clear physical meaning.


9/12


301

a

Figure 3. Resonance curves of a non-linear oscillator, i+O~05i+x+O~Olx3 =0.3 cos Rr+ 1.5 cos O.llSR1).

0, Harmonic balance; -, shooting method. A

o ,,5

is the amplitude of the harmonic with frequency 0~115R.

and so on.

For comparison the resonance has been calculated by the harmon ic balance method

with the trial function

x=A,.,,5~~~(0~115t+~,)+A,cos(t+~2),

(13)

and the corresponding results are plotted in Figure 3 with dots. It is apparent that for

these harmo nics the harmo nic balance method gives pretty good results. However, the

error will increase when the amplitude becomes larger. M oreover, if one tries to calculate

other harm onics by taking more terms in the trial function, the derivation will be quite

tedious. It also can be noted that in order to determine the coefficients Ao. 15 and A

one has to solve some non-linear algebraic equations numerically. In view of the large

amount of derivation work when using the harmon ic balance method, even in relatively

simple cases, the harmo nic balance method does not have obvious ad vantages.

4. PERIODIC SYSTEM

The ma in purpose of developing this simple shooting algorithm is to calculate quasi-

periodic solutions in a multi-excited system, but it turns ou t that this method is also very

effective for calculating quasi-periodic solutions of a periodic system. In this case one

has an extra unkno wn (the winding number), and in order to find a physically mean ingful

solution of the crucial equation (2), one solves the problem

I~P(O)-X(O)[[~= Min,

where

11 [I2

denotes the Euclidean norm, by using an optimization algorithm (IMSL

routine UM INF). Our experience shows that if we have a proper guess of the winding

number (e.g., with a deviation less than 10% of the correct value), the shooting process

converges very rapidly.


10/12

302

The example

F.

H. LING

i-p l-x2)i+x=Fcoswt+F

15)

is treated here. The P oincare section (with y = i) of the case with parameter values

CL O-2, F=0* 2, w = 1.1, Fo= 0.5 has been studied [2] and w ill be referred to as the

reference case shown in the center of Figure 4. The change of parame ter values induces

different solutions as plotted in the other eight parts of Figure 4; each of them differs

from the reference case only by one parameter value as is indicated above the correspond-

ing part. Several phase-locking regions such a s F0 3 0.75 and

F 3

O-3 are found, where

there are only periodic solutions. The case of

F = 0

has a trivial solution, x = 0, but it

possesses also a quasi-periodic solutions as shown in Figure 4. Moreover, the p = 0 case

represents a linear oscillator which does not have quasi-periodic solutions. In Figure 4

one sees 11 discrete p oints representing a

P

11 response as referred to the natural

frequency unity of the system. However, it is in fact a

P

1 response referred to the

excitation frequency w = 1.1.

Fio=oa

Wind ing = 0 06491

Reference w=I-3

Wind ing = 0 .06761

Wlnd ing=0~06710

, , -.. ,,,.

/

,.

/

..-..

i

i

lJ

:*

\

c

. .

.

. . .

- .

,

\

I

I

\

,

I

\

I

I

\

,

---

Fo=0.7

Wind ing = 0 .09233

-Y

c,

II

-3

3 3

3.

x

Figure 4. Quasi-periodic solution o f the periodic sys tem

i-~(l-u~)i+x=Fcoswr+Fb,

with the reference parameter values of p = 0.2, F = 0.2, w = 1. I and F0 = 0.5.

i

1

(16)


11/12

MULTI EXCITED QUASI PERIODIC RESPONSES

303

The winding number is very important especially in studying the transition from a torus

solution to chaos, and its precise value has been obtained here, being evaluated during

the iterative process. T his is an obvious advantage in comp arison with other methods

[ l-7,13].

5. CONCLUDING REMARKS

There are two efficient methods-shoo ting and collocation-for the boundary-value

problem of ordinary differential equations, and they are both useful for evaluating

quasi-periodic solutions. The shooting-type method can be applied to both a periodic

and a non-periodic system, w hile the collocation method can be used only for a periodic

system.

A new shooting-type algorithm in which all calculated iterative points of the PoincarC

map are used in the interpolation procedu re and derivatives of the map with respect to

the shooting param eters

are used has been presented in this paper. Usually, a rather

accurate bi-periodic solution is obtained by integrating over an interval of lo-15 shorter

periods, so that a great d eal of computer time will be saved. A system with very small or

even negative dam ping, which results in an unstable quasi-periodic solution, can also be

treated with the algorithm. Moreover, this algorithm can also be successfully adopted for

calculating quasi-periodic solutions of a periodic system.

It is also possible to obtain the quasi-periodic solution boundary in the parameter

space as has been done for periodic solutions [ 141, but some problems can appear, since

it is not very easy to determine the spectral radius of the matrix corresponding to the

quasi-periodic solutions with a high enough accuracy.

There are also other methods in engineering applications for estimating a @asi-periodic

solution of a multi-excited system, e.g., a Galerkin type method [15] and the technique

in which the estimation of the quasi-periodic solution is converted to a boundary-value

problem of a partial differential equation, which is then solved by a finite difference

method [16]. But these m ethods do not seem to be able to compete with the shooting

method described in the paper. The cell-to-cell mapping method [ 171 is another useful

tool of studying quasi-periodic solutions; see reference [ 181

for an examp le of a study

of a circle map by using this method.

ACKNOWLEDGMENTS

This work has been partially supported by the Natura l Science Fo undation of China

and a grant of the Ministry of Education of China . The revised versions were produced

during the authors visits to Institute B of Mechanics of the University of Stuttgart, and

to the Departm ent of Physics and Engineering Physics, Stevens Institute of Technology.

Thank s are due to W . 0. Sch iehlen, E. Kreuzer and G. Schm idt for their hospitality. The

author also appreciates many sug gestions mad e by G. Schm idt to improve the presentation

of the paper. The financial support for the visits was provided by the Ministry of Science

and Art of the State, Baden -Wiittemberg, West Germany , and the U.S. Departm ent of

Energy, Contract N o. DE-AC 02-84ER 13146.

1.

2.

REFERENCES

M. HEAN 1980

nternat i onal Journal of Non-l i near M echanics 15,367-376. SW

la methode des

sections

pour

la recherc he de certaines solutions presque periodiques de systemes forces

periodiquement.

E. THOULOUZE PRAY 1981 Nonl inear A nalyt ical Theory and M athemati cal Appli cati ons 5,

195-202. Existence theorem

of an invariant torus of solutions to a periodic differential system.


12/12

304

F. H LINC

3. E. THOULO UZE-PRAT-I- and M. JEAN 1982 I nternati onal Journal of Non-Li near M echanics 17,

319-326. Analyse numerique du compo rtement dune solution presque periodique dune

equation differentielle periodique par une methode des sections.

4. E. THOULOUZE-PRAY-~ 1983 Lect ure Not es in Biomathemat ics 49. Berlin: Springer-Verlag.

Num erical analysis of the behaviour of an almost periodic solution to a periodic differential tori.

5. T. N. CHAN 1983

Master of Comput er Science Thesis, Depart ment of Comput er Science, Concordia

Unioersi t y, M ontr eal, Canada.

Num erical bifurcation analysis of simple dynamical systems.

6. I. G. KEVREKIDIS, R. ARIS, L. D. SCHMIDT and S. PELIKAN lo85 Phy sica D 16, 243-251.

Num erical computation of invariant circles of maps.

7. M . VAN VELDHUIZEN 1987 SIAM Journal of Scient if ic and St at i sti c Computat ions 8, 951-962.

A new algorithm for the numerical approximation of an invariant curve.

8.

CHR. KAAS-PETERSEN 1985

Journal of Comput at i onal Physics 58, 395-403.

Computation of

quasi-periodic solutions of forced dissipative system s.

9. CHR. KAAS-PETERSEN 1985 Journal of Comput at i onal Physics 64, 433-442. Computation of

quasi-periodic solutions of forced dissipative systems II.

10. CHR. KAAS-PETERSEN 1987 Phy sica D 25, 288-306. Comp utation, continuation, and bifurca-

tion of torus solutions for dissipative maps and ordinary differential equations.

11. J. STOER and R. BULIRSCH 1973

Einfuhrung in die Numeri sche Mat hemat ik I I .

Berlin: Springer-

Verlag.

12. F. H. LING 1981 Di ssert ati on, Uni uersit i i t St utt gart , West Germany. Numerische Berechnung

period& her Liisungen einiger nichtlinearer Schwingun gssysteme.

13. F. H. LING 1982

Zeit schrzft fTi r angew andte M athemat ik und M echanik 62,

TSS-T58. Numerische

Berechnung periodisch er Liisungen nichtlinearer Schwingun gssysteme.

14. F. H. LING 1989 Zeit schrzf t ur Physi k B 75, 539-545. A numerical study of the distribution of

different attractors in the parameter space.

15. L. 0. CHUA and A. USHIDA 1981

I nsti t ut e of Elect ri cal and El ectroni c Engi neers Transact ions,

Ci rcui t s and Syst ems CAS-28, 953-971. Algorithms for computing almost periodic steady-state

response of nonline ar systems to multip le input frequencies.

16. B.

WEYH 1982

VDI-Bericht

Nr. 456, 113-12 0. Berechnung nichtlinearer quasiperiodischer

Schwingungenen in Maschinenskzen mit Schlupfkupplungen.

17. C. S. HSU 1987 Cell -t o-Cell M apping. New York: Springer-Verlag.

18. F. H.

LING

and Z. R. LIU 199 0 (Preprint). Limiting probability density of a quasi-periodic orbit.

SM13(Cuong) Sdarticle

Documents

Transcript of SM13(Cuong) Sdarticle