Applied Mathematics and Mechanics (English Edition, Vol. 19, No, 12, Dec. 1998)

Published by SU, Shanghai, China

ON THE M U L T I P L E - A T T R A C T O R C O E X S T I N G S Y S T E M WITH P A R A M E T E R U N C E R T A I N T I E S .USING G E N E R A L I Z E D CELL M A P P I N G METHOD* Gong Pulin (~-t3~k)x

Xu Jianxue ( ~ , ~ - ~ ) '

(Received April 25, 1997; Revised Oct. 15, 1997; Communicated by Liu Zengrong) Abstract In this paper the generalized cell mapping (GCM) method is used to study multiple-attractor coe.\'isting s),stem with parameter uncert,thlties. The e/.'['ect.s"that the tolcertaht parameters htts on the global properties of the system are presented. Amt It is obtahwd that .the attractor with much smaller' v,~lue of protect thickness, will disappear firstly with the degree of the uncertain O, of parameter increashtg.

Key words

generalized cell mapping, multiple-attractor coexisting system, parameter uncertainties

I.

Introduction

One of the dominant characteristics of Non-linear systems is the non-uniqueness of the response for a given system parameter. Each steady state is an attractor having its own basin of attraction in pl!ase space. An attractor may be fixed point, limit cycle, invariant torus, strange attractor. In practical applications, only one of these multiple attrators is of interest and corresponds to the safe, operational region of the systems, and the othcrs are usually unwanted. However, the obtained solutions are always under the assumption of the system parameters a r e certain, in practical application, some systems parameters uncertainties can arise from human error or other experimental errors. Above all, the environment a b o u t t h e dynamical system has strong effects on the system, making parameter have uncertainties. Because the parameter uncertainties will have a strong effect on the global properties of the multiple-attractor systems, assumping the system parameter is certain, the domain of the attractor corresponding to safe operation could disappear, resulting in the system running in the other steady state and having catastrophic effect on the system. The analytic methods used to study parameter uncertain dynamical system are only in the linear systems ttnd the weak non-linear systems I*' :1. On the other hand, the direct numerical simulation has to calculatethe long term response of each orbit, so too much time is needed. Generalized cell mapping method as an effective numerical method used to study non-linear systcm, including strong non-linear system, was put lbrward by Hsu [3" "q. In this paper, we will use generalized cell mapping method to study the multiple-attractor coexisting non-linear system with parameter uncertainties. * Project supported by the National Natural Science Foundation of China (19672046) Department of Engineering Mechanics, Xian Jiaotong University, Xian 710049, P. R. China 1179

1180

Gong Pulin and Xu Jianxue

The domain of atti-action of an attractor were first viewed as protecting region by Hsu and Chiu 151, and the erosion of the domain of attraction was related to the engineering integrity salty 161, so the domain of attraction has an important means, and this corresponds to the protection thickness in generalized cell mapping method. II.

O v e r v i e w o f G e n e r a l i z e d Cell M a p p i n g

The basic idea of cell mapping is to consider the state space not as a continuum but rather as a collection of a large number of state cells with each cell being taken as a state entity. Up to now, two types of cell mapping have been investigated. One is simple cell mapping (SCM), and the other is generalized cell mapping (GCM). Under simple cell mapping each cell has only one image cell, while for a generalized cell mapping a cell may have several image cells with each cell having a certain fraction of the total probability. When a dynamical system is casted into a generalized cell mapping, then the generalized cell mapping corresponds to a finite stationary Markov chains. For a differetial dynamical system:

= f(x,t,pt),

x E R~,l-t E R ~

(2.1)

through poincare section, the system can be changed to a point mapping system:

x ( n + 1) = g [ x ( n ) , t z ] ,

x E R~,g E R"

(2.2)

in order to use cell mapping, firstly, we choose a domain D in R", and then discrete the domain D using some cells, each cell hasing uniform size hi in the direction of xl and the interval is defined to be one which contains all xl satisfying: (Zi - 1/2)h; ~< xl < (Zi + 1/2)hl,

(i = 1,2,'",N)

(2.3)

Here, by definition Zj is an integer. Each cell can be denoted by a N-tuple of integers. In generalized cell mapping method, using interior-sampling method, each cell may have more than one image cell. For each cell, say cell Z, divide it into M subcells o f equal size and calculate the mapping images from the M center points of these subcells, if M1 points m a p to cell Zt, then the transition probability is Pz~z = M1/M, for the same reason, Pzz = Mi/M, i = 1 , 2 , " " , m. m is the total number of the image cells of cell Zi, and ~i=1 1 g gP = , p~/ denotes the transition probability from cell j to cell i, and it is the element of the transition probability 9matrix P, the probability of all the cells at nth step is denoted by probability vectorp(rt), the evolution of the system is given by:

p ( n + 1) = Pp(n)

(2.4)

Much of the dynamics of the system can be readily extracted from the evolution equation (2,4). In terms of generalized cell mapping, if the transition probability from cell Zj to cell Zi is positive, then, say that cell Zj leads to cell Zi, if cell Zj leads to cell Zi, and cell Zi leads to celll Zj also, then cell Z~ communicates with cell Zj. If a cell starts from Zi, through the mapping, will come back to itself, then the cell Zi is called a persistent cell, if not, it is a transient cell. If some cells in a set communicate cells with each other, then the set is called persistent group. Persistent groups represent the possib;e long term stable m o t i o n s of the system. In general they are attractors. A set of cells that lead to one specific pcrsistent group are called single domicile cell. The single 9 cells leading to a persistent group constitute the basin of attraction for that persistent group. Cells which have more thafi one domicile are

Multiple-Attractor System with I'aramctcr Uncertainties called multiple domicile cells. Multiple domicile cells form the -basin of boundary. The basin of gth attractor Vg is defined as follows: V, = { Z I

Dra(Z) = 1, DraG(Z,1) = g } ,

(g = 1,2,'",Nps)

(2.5)

The multiple-domicile cells are defined as following: V0 = { Z I

Dra(Z) > 1}

(2.6)

where

Dra(Z) is the number of domiciles of cell Z ' DraG(Z, i) denotes the ith domicile of cell Z.

III.

Fractal Basin Boundary

and Protect Thickness

Dynamical system: x = f ( x , t , t z ) , x E RlV,,u E R• in which x is state variable, p is system parameter. When parameter p has some uncertainties, it is considered as a random variable with a prescribed probabilitic distribution, and the mean-square and the deviation are all known. When its mean-square is prescribed, as deviation increases, the uncertainty degree of the w~riable increases. Ill the system, multiple attractor may coexist, and each attractor has its own basin of attraction. So in this case, there are basin of boundary. In general, the basin boundary is not a smooth curve, but fractal, and its dimension is not an integer IL~ When the basin boundary is fractal, if an initial condition specified with error e is within the basin boundary, the attractor to which it will be attracted cannot be predicted with certainty. Furthermore, the location and the structure of a basin boundary depends on the system parameter, while uncertainty in system parameter values could affect the ability to predict tile final state from an initial condition, independent of the degree of precision with which the initial condition is Chosen. The basin boundary corresponding to mutiple-domicile cell of generalized cell mapping, through refining the multiple-domicile cells; can tell whether the basin boundary is fractal or not. In order to tell a state is stable or not, Liapunov method in convention can be applied to. Yet, when the basin boundary is fractal, the conclusion drawn from this method, is not reliable. On the other hand, when basin boundary is fractal, a little perturbation .that system parameters will result in the basin of attraction is eroded greatly. which will subsequently resuh in the unsteadiness of engineering. So when a basin bouudary is fractal, and in practical, parameter p has some uncertainty, to judge an attractor is stable or not must consider whether the attractor is enclosed by its basin of attraction. If the basin of an attractor is blurred and disappear, the attractor will be unstable. Generalized cell mapping method-can be used to give the gloabal picture effectively without too much time, but other methods cannot do so. In order to delineate whether an attractor is protected" by its basin of attraction, we use the protect thickness of generalized cell mapping. From the rate of computation, we define the protect thickness as follows:

d, = d(a~,, V0)

(g = 1,2,'",Np,)

(3.1)

dg denotes the distance between the set As and set V0, As = { Z I Z is cell of tile gth persistent group], and set V0 is defined in equation (2.6). The distance between the set of cells in cell space see [7]. When p is uncertain parameter, as the degree of uncertainty increases, the basin of coexisting system is blurred by the fractal basin boundry, so the basin disappears, and the attractor will disappear too. However, which attractor will disappear first. In the following example, we can see the attractor with much smaller value of protect thickness will disappear first. On the other hand, using generalized cell mapping method, we can obtain the global

1182

Gong Pulin and Xu Jianxue

picture, through comparing the global picture of system with certain parameter and the global picture of dynamical system with uncertain parameter, we can obtain the effects that the uncertain parameters have on global properties of the multiple-attractor coexisting system intuitively. If p a r a m e t e r / t has some uncertainties, we regard it as a random variable with prescribed mean-square and deviation, tn practical numerical simulation, we use the random number generator to gerterate N random numbers satisfying the prescribed mean-square and deviation. In order to use generalized cell mapping, for each cell, we choose M interior sample points, and for each sample points in a cell, in equation (2.2) letting p equate every generated random number respectively, and then from equation (2.2) to calculate the image cells of the cell. So a total of MX N sample trajectories from each cell are used to calculate the image cells and the transition probabilites. 3.1 E x a m p l e 1 We will consider Mathieu equation as an example. When there are cubic nonlincar elements in this equation. Mathieu equation has complex dynamical propcrtics. + 25x 3 + 0.173e + (2.62 - 0.456a + 0 . 4 5 6 a e o s 2 t ) z = 0.92a(1. - cos2t)

(3.2)

First, convert (3.2~ into a set of one order ordinary differential equation:

e2 = - 25x3t - 0.i73x2 - (2.62 - 0.456a + 0.456aeos20)x~ + 0 . 9 2 a ( I - cos20)

(3.3)

0=1 w h e r e ( x l , x 2 , 0 ) E R 2 x S 1. We pick a cross-section

~S = { ( x l , x 2 , 0 ) I 0 = 0}i being a

poincare section, and then to construct a poincare map p : ~ ---,- ~i. Suppose O = { - 1.5~< x < 1 . 3 , - 2.5 ~< y < 1.7} being the region of interest, region outside D is considered as a single cell called sink cell, casting region D into 150x200 cells. For parameter ~., when cz= 3.54, the system has three persistent groups. But in which PGo denotes t~ae sink cell, so there are two-coexisting attactor in region D. The two attractors are shown in figure 1. Basin boundary is shown in figure 2. After refining the persistent group and the basin boundary, two-coexisting attrators are shown in figure 3, and the basin boundary is shown in figure 4. Through comparing figure 1 and figure 3, we can see that the volume of PGI decrease greatly after refining, but that is not to PC~. So we can obtain that PGI corresponds to a p-I solution, PG2 corresponds to a strange attractor. Through comparing figure 2 and figure 4, we can see after refining the ,volume of the mutiple cells does not decrease greatly, so the basin boundary is fractal. The system's global picture is shown in figure 5. On the base of the concept of protect thickness of G C M , through calculating we obtain, 9 when the parameter is certain 7.=3.54, the protect thickness of attractor p-1 is dt = 4, the protect thickness of strange attractor is d2 = 25, d l < d2. When the parameter cz has some uncertainty, we can consider the uncertain parameter as random variable with prescribed mean-square and deviation. In order to compare with the parameter certain system, firstly, letting E(ct) = 3 . 5 4 , a ( a ) = 0.1301. 9points are uniformly sampled from each cell, and generate ~ random number with the prescribcd mean-square and; deviation by random number generator. For each sample points of each cell, in equation (3.2) letting ct be every generated random number respectively, and thcn from equation (3.2) using 4order Runge-kutta method to integrate one period n, initiating from the sample point, to.

M taltiple-Attractor System with Parameter Uncertainties

1.7 I.I

1.7;

1183

\

I

1.1-

'0.5

0.5

-0.1

-0.I

-0.7

-0.7

-1.3

-1.3

-1.9

-1.9 -2.5 -1.5

-2.5 -1.5

'' -0.7 '

Fig. 1

'

'

. .0.1 ..

0'9.

The two-coexisting attractors

I !

I

l i. -0.7

Fig. 2

0.1 '

'

0.9

T h e basin b o u n d a r y

1.7 ill

1.7 1.1

1.1

":

II

!

0.5 -0.1

-0.1

.

-

-0.7

-o,7i,

-1.3

-1.3

-1.9~ / -2.51 !!I,

-1.9 -2.5 -1.5

0 1i

...... -0.7

'

-1,5

0 i9

T h e t w o attractor after refining

Fig. 3

Fig. 4

the persistent g r o u p

-0.7

0.l

019

T h e basin b o u n d a r y after refining the cell space

12

!

0.! {

-0

-0..'

-0

-I -1. -I

- 1.5

(n

- I

-0.5

0

0.5

1

denotes the cells which arc mapped out

of domain D, O denotes the basin of p-l,, 9 dcnotcs the basin boundary, 4 denotes the persistent group, the blank is attraction basin of strange attractor)

-2 - 1 . 5 -1.1 - 0 . 7 - 0 . 3 0.1 0.5 0.9 (r dcnotcs the cctls which arc mappcd out of domain D, O dcnotcs the basin of p-l, 9 dcnotcs the basin b o u n d a r y , ,~ dcnotcs the pcrsistcnt group, thc blank is attraction basin of strange attractor. ) Fig. 6

F i g . 5 T h e g l o b a l p i c t u r e w h e n :t= 3.54

The global picture

a(a) = 0.001

when

E(a) =3.54-,

Gong Pulin and Xu Jianxue

1184

calculate image cells o f the cell. So a total of 72 sample trajectories from each cell are used to calculate the image cells and the transition probabilities. In this case we can obtain the global picture as is shown in Fig. 6. And through calculating, the protect thickness of p-1 is

dt

= 3, the protect thickness of

strange attractor is d~ = 23.

When the degree of the

uncertainty increases, that is to say that the deviation increases, letting

a(a)

= 0.018 in this

case there is only one strange attractor in figure 7. Through comparing the calculating results, and the picture of all these cases, we can see that with the uncertainty of the parameter increasing, the attractor p-I which protects thickness is much smaller in the system without parameter uncertainty, as the degree of uncertainty increases, will disappear firstly. It also appears that the uncertainty of the parameter makes more multiple-domicile cells near the deterministic boundary of the domain of attraction, because the basion boundary is fractal, the points in the multiple cells have the final state sensitivity. So the parameter uncertainties increase the uppredictability of the system.

3 2

1.7 1.1

t

0.5 -0.1

0

3

-1

-0.7

-2

-1.3

-3

1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2

-1.9 -2.5 . . . . . . . . . . ,.,r ,,..L. - 1.5

-0.7

0.1

(s denotes the cells which are mapped out or" domain D, O denotes the basin of p-3, @ denotes the basin boundary, a, denotes the persistent group, the blank is att,action

0.9

basin of strange att,-actor). Fig. 8 The global p i c t u r e w h e n ~.=0.021

Fig. 7 The only one strange attractor when E ( a ) =3.54,a(a) =0.018 3.2 E x a m p l e 2 Ill this example we will study Duffing equation: x + k~ + a x

(3.4)

+ x 3 = boost

convert (3.4) into a set of one order ordinary differential equation:

~i = x2 ~2 = -

kx2-axl-

} x 3 + bcosO

(3.5)

0=1 In this system when

k = 0 . 2 5 , b = 8.5 in [5] the writers point out that i r a

~<- 0.11,

1185

M ultiple-Attractor System with Parameter Uncertaintics

only p-3 solution exists. I f - 0 . ll 0.053, only one strange attractor exists. In [9], the writers consider the effects that random force have on the system. In this paper, we assume parameter has some uncertainties, and regard it as a random variable indepentent of time, and with prescribed mean-square and deviation. Fi,'st, let 7=0.021, the parameter is certai n, there are two,coexisting attractors, and the global picture about the system is shown in figure 8. Through calculating, we obtain in this case the protect thickness of p-3 solution is dl = 2 , the protect thickness of strange attractor is d 2 = 4 , d t < d2. In what follows, we consider the case of parameter having some uncertainties. Letting E ( a ) = 0 . 0 2 1 , a ( a ) = 0.003, 9 points are uniformly sampled from each cell, and generate 9 random number with the prescribed m e a n - s q u a r e and generator. F o r e a c h sample point of each cell, in equation

deviation (3.14)

by

random

number

letting c~ being

every

generated random number respectively, and then from equation (3.4)using 4-order RungeKutta method to integrate one period,

in poincare section to calculate image cells of the

cell, a total of 81 sample trajectories from each cell are used to calculate the image ceils and we can obtain the system's global picture shown in Fig. 9. In the case, the protect thickness of p-3 is d t =

1, while the protect thickness of strange attractor is d2 = 3. When a(ct) =

0.018., only one attractor exists as is shown in Fig 10.

3 2

6 5

I

4

0

3

-1

2

-2

l

-3 1.5 1.8 2.1 2.4 2.7 3

3.3 3.6 3.9 4.2

0 -I

(O denotes the cells which are mapped out of domain D, O denotes the basin of p-3, 9 denotes the basin boundary, A denotes the persistent group, the blank is attraction basin of strange attractor).

.))

-2 1.51.82,12.42.7 3 3.33.63.94.2

Fig. 10 T h e o n l y o n e a t t r a c t o r , w h e n E(a) = 0.021, a((~) = 0.018

Fig. 9 The global picture w h e n E(a) = 0.021,a(a) =0.003

Whe,~ the parameter is certain, dl < d2, if the degree of uncertaities of cz increases, the p-3 solution disappea,'s firstly, so is the conclusion we obtain in example 1. Comparing Fig. 8 and Fig. 9, we find that example 3.1.

uncertain

parameters

have the same effects on the system as the

G o n g Pulin a n d Xu Jianxue

1186

1.5 1.8 2.1 2.4 2.7

1.5 1.8 2.1 2.4 2.7

3 3.3 3.6 3.9 4.2

3 3.3 3.6 3.9 4.2

(ID denotes the cells which arc mapped out

(if] denotes the cells which are mapped out

of domain D, <> denotes the basin of p-3,

of domain D, O denotes the basin of p-3,

9 denotes the basin boundary, ~, denotes

9 denotes the basin boundary, h. denotes

the persistent group, the blank is attraction

the persistent group, the blank is attraction

basin of strange uttr~ictor).

basin of strange attractor). F i g . 12

Fig. 11

T h e g l o b a l p i c t u r e , w h e n a =0.052

The globalpietureE(a)= 0.052,a(a) = 0.01

W h e n cz.=0.052, there are two coexisting

6 5

attractors.

4

a t t r a c t o r . T h e global picture is shown in Fig. 11.

One is p-3 and the o t h e r is strange

3

In this case,

2

d t = 5 , and the protect

1

a t t r a c t o r is d2 = 2, d l When

0

the protect thickness o f p-3 is

E(a)

thickness >

d2.

= 0.052, a(a)

= 0.01,

the

global picture, as is shown in Fig. 12. We can calculate the protect thickness o f p-3 is dl = 4.

-1 -2

The protect thickness o f 1.5 1.8 2.1.

Fig. 13

o f strange

2.4 2.7 3 3.3 3 . 6 3.9 4.2

The only one attractor,

E(a) =O.052,a(a) =0.07 So when

E(a)

strange a t t r a c t o r

is

d 2 = 1. When a(a)

when

= 0 . 0 7 , th'ere are only

one

a t t r a c t o r p-3, is shown in Fig. 13.

= 0.052, through c o m p a r i n g all the cases with different deviations, we

can d r a w the same conclusion as the examples above. IV,

Conclusion W e use the generalized cell m a p p i n g m e t h o d to study m u l t i p l e - a t t r a c t o r coexisting system

with p a r a m e t e r uncertainties, a n d proved that generalized cell m a p p i n g can effectively give the global picture without too much theoretically analysis. W e i n t r o d u c e d the concept o f the protect thickness through c o m p a r i n g the protect thickness based on the same cell grid o f every

Multiple-Attractor System with Parameter Uncertainties

I187

attractor, and obtain that the attractor with the much smaller protech thickness will disappear first in the multiple attractor coexisting system. So protect thickness can act as a quatitat~ve value to tell which attractor will disappear as the uncertainties of system parameter increases. Through comparing the global picture of all cases, it is found out that system uncertainties generally increase the system unpredictability of the system by making more multiple-domicile cells near the boundary of domain of attraction, and the domain of attraction also changes. References [1 ]

R. Valery Roy Noise, Perturbations of a non-linear system with rnultiple steady states, htternat. J. Non-Lhzear Mech., 29, 5 (1994), 755~773. [2] Rsingh and C. Lee,,Frequency response of linear systems with parameter uncertainties, J. Sound Vibration, 168 (1993), 71 -- 92. [3] C . S . Hsu, A generalized theory of cell-to-cell mapping for nonlinear dynamical systems, J. Applied Mechanics, 48 (1981), 634--642. [4] C. S. Hsu, Cell-to-Cell Mapphtg, A Method of Global Analysis for Nonlhlear Systems, Springer-Ver!ag, New York (1987). [5] C. S. Hsu. and H. M. Chiu, Global analysis of a system with multiple responses including a strange attractors, J. Sound Vibration, 114 (1987), 203~218. [6] Mohamed S. Soliman and J. M. T. Thompson, Global dynamics underlying sharp basin erosion in nonlinear driven osccillators, Physical Review A, 45, 6 (1992), 3425--3431. [7] C. S. Hsu, Global analysis by cell mapping, lntermttional Journal of B(/itreation r Chaos, 2, 4 (1992), 727~771. [8] Wilfred D. Iwan and Ching-Tung Huang, On the dynamic response of non-linear systems with parameter uncertainties, Internat. J Non-Lbzear Mech., 311 5 (1996), 631 645. [9] J. Q. Sun and C. S. Hsu, Effects of small random uncertainties on non-linear systems studied by the generalized cell mapping method, J. Sound Vibration, 147, 2 (1991), 185~ 201. [10] Steven W. Mcdonald, Celgo Grebogi, Edward Ott and James A. Yorke, Fractal basin boundary, Physica D., 17 (1985), 125~153.