CHADS STUDY IN SYSTEM DYNAMICS
@ifan Wang
Fudan University & Shanghai Inst. of Mech. Eng.
Shanghai, China
Zhiping Yao
Shanghai TV University, Shanghai, China
Guangle Yan
Shanghai Inst. of Mech. Eng. Shanghai, China
ABSTRACT
Based on our study of synergetics and dissipative structure theory, by
means of mathematics, in terms of the viewpoint of system dynamics,
this paper concentrates on the study of chaos in system dynamics: we
analyze the nature of chaos phenomenon and the characteristics of
system dynamics, put forward the viewpoint that chaos testing should
be included in model testing of S.D.; we investgate several necessary
conditions of chaos; we creat a model to question a famous sufficient
condition of chaos; then, we shed some light on the way toward which
chaos will occur. at last, we successfully apply our theoretic study
to a standard nuclear spin generator model.
1. CHAGS AND SYSTEM DYNAMICS
System dynamics is a powerful tool to study the complex social and
economic system. This complexity results from nonlinearity, high-order
and multy loop of systems. One character of such complexity is that,
though the system is deterministic, at certain point, its behavior is
like stochastic. Apart from this, the systme is extremely sensitive to
the inital value. This strong sensitivity makes two trajectories which
are very clase in the beginning become quite different from each
other in time. That is to say, the system will show chaotic behavior.
To anonliear dynamicist, a chaos system is one on which long-term
predication of the system's state is impossible because the
omipresent uncertainty in determining its initial state and because
the error of initial value grows exponentially fast in time.
So, it is of great significance to study chaos characteristics of S.
D. models in order ta make the model be more reliable.
Fortunately, System Dynamics has paid lots of attention to model
testing. So, we recommend that chaos testing be included in
sensitivity analysis while we do model testing.
Now, let us see the mechnics af chaos.
First, we classify all macro states in the world into Table 1.
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Table i.
state
in-between state final state
type
a. equilibrium
stable b. periodical
€- quasiperiodical
deterministic d. tend to limitlessness
unstable e. sometimes limited
sometimes limitless
#. always limited
stochastic
What classcial theories study are the stable processes, namely,
process a, b, and cs; process d and e are divergent; process "f" is
what we call chaos:
Def. 1. A state which is always limited and which is not equilibrium,
not periodical, and not quasiperiodical is named chaos.
From the above discussion, we easily get:
Theores 1. unstable, apericdical, bounded and extremely sensitive to
inital value are the necessary conditions of chaos.
In the following, we will concrete theorem 1.
For generality and transfer ability, this paper facuses on system (1):
dX/dt = P(X, Y, Z?
dY/dt = Q(X, Y, Z) —-——----
dZ/dt = RO, Y, Z)
Set q = (, Y, Z), then, system (1) becomes
dq/dt = Niq> enn nnn
waaa=----- 1)
2. BOUNDED
Def. 2. If existing a costant M, qit) is the solution of system (2),
we have:
lim sup q(t) <™ —---- (3)
Then, we €afl system (2) is dissipative (£11).
Theorem 2. Suppose q(t) = (((t), V(t), Z(t)) is the solution of system
(1), if (q(t>, dq(ti/dt) < 0, then, system (1) is dissipative.
Proof: take
SeS = Xax + Yay + ZZ
then (q(t), dq(t)/dt) <0
<==? K # dX/dt + ¥ * GV/dt + Z * dZ/dt < 0
<sssss==> § # dS/dt <0 ( since S>0)
> ds/dt < 0
So, system (1) is dissipative.
Example 1.
dX/dt = - bX + ¥
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d¥/dt =-X-bY ¢1-kZ) oO 5)
dZ/dt = 6 Ca €1-Z) —k YeY 1
where a, b,k > 0. We can verify (q, dq/dt) < 0 «f2]).
Based on theorem 2, we can construct a kind of systems which are
dissipative.
Theorem 3. In system (1), if P,@,R satisfy:
a) dq/dt = - Alq) + Gq) +H
b) (q, Atq)? > 0
©) tq, Giq)) = 0
ad) 610) = 90
then (q, dq/dt) < 0. (When q is big enough)
UNSTABLE
In order to study unstableness, we introduce Lyapunov exponent.
Def. 3. Suppose dq(t)/dt = L(t) q(t) ———-—-— 6)
Ltt) = (aij(t)), if existing B > 05 sup /aij(t)/ < B
then, for the solution vectory qi(t) of system (6), we define:
Zi = Jig sup ((1/t) * In qi (ty) ——-————_ >
Zi are called generalized characteristic exponents. Gne special case
of generalized characteristic exponent is so-called Lyapunov exponent.
Suppose qo(t) is a solution of system (1), sq{t) is a disturbence,
then,
att) = qott) + sqit) im
Put (8) into (2):
dgo(t) /dt + dsqit)/dt = Niqo(t) + sq(t))
=====) dqgo(t)/dt + dsq(t)/dt = Niqo(t)) + (dNiqo(t) /dt*sq(t) +
otsq(t)?
> deq(t)/dt = CaN(qo(t) /dt)#sq(t)]) ———--——----_ (9)
<
Def. 4. The generalized characteristics exponents of equation (9) are
called Lyapunov exponents.
Haken in [3] asserts that, if the three Lyapunov exponents of system
(1) are (+, 0, -), then, system is chaotic. In our study, we find
this assertion is not correct.
Theorem 4. If qa(t) = qo then lyapunov exponents are the eigenvalues
of N (qo).
Proof: Since qo(t) = qo , Niqo) is a constant matrix. So, equation
(9) is a linear diffential equation.
Suppose A1,A2,A3, are the three eigenvalues of equation
/PE-N(qo)/ = 0, V1, V2, V3 are relative eigenvectors. since
ALEXPCALCIVISN (qO) EXP (Ait) Vi<===N (qO)Vi=ALEVi<===>(N (q0)-AiE) Vi=0
then, EXP(A1#t)V1,EXP(AZ#*t) V2,EXP(A3#t)V3 are solutions af (9).
From Def. 3,
Bi = lim sup {1/t * In /exp(Ai#t)Vi/ 3
dig slp C(Aixt) /t + In /Vi/s /t3 = Ai
Sa, Lypunov exponents = eigenvalues.
where /C/ is the absolute value of C.
Example 2
dX/dt = —- bX + Y¥
d¥/dt = - X -bY + BkYZ --—
dZ/dt = b — cZ — bkY*Y
b,cl , k > 0. We have proved that system (10) is dissipative.
aimemee 3.
Theorem 5S. Ci = (0, 0, b/C), 2,3 = C£Ci/bebek) #SORE (bebebek—
©1468) ) /b1,41 /b®) k*SORE (babebek—c (1+beb)) /b1, (1+b*b) /bebek)
,
are three equilibrium points of system (10).
Theorem 6. Take C = (bebebek)/(i+b*b) 9 -—————--—-—-—- (11)
==> AI=0_— And AxAt (b-1/b+c) A+ (bc-C/b) =0,A2,3=0.50 (1/b—b—
©) 4SORE (b-1/b+c).# (b-1/b+c)—4 (bc-c/b) 7
for b <1,==>A2#A3<0 then, Lyapunov exponents of system (10) are
(1/b-b y 0, -c) = (+, 0, -). So, system (10) satisfies Haken’s
conditions. It should be chaos. But our study finds that system (10)
has periodical solutions and it is not sensitive to intial values
(E29).
4. APPLICATION
Example 2 shows that it is of little possiblity to be chaotic directly
at equilibrium points. Chaos is a dynamic process. The probable way
toward which chaos may occur is:
a) change parameters to make equilibrium points be limited
cycles.
b) go on changing parameters to let limited cycles bifurcate
into cycle 2,4,8 ......
©) at last, go to chaos.
In step a, we have:
Theorem 7. Suppose Wis a open set, © beiongs to W, Wis included in
RERER,
#2 We(-ao,ao ) ——> ReRER
daX/dt = f(xya ) 9 ————-—---- 12)
#0, a) = 0, for any “a”
The eigenvalues of Df(O, 0,) are +ib(0),-ib(O) ,8(0) ,S(0)<0,
biO) > 0, when a=0, (0, 0, 0,) is stable, a>0.
{0,0,0) lost stable.
Then, for small enough a, equation (12) has a stable closed
trajectory near (0, 0, 0).
Go along this road, we delve into a nuclear spin generator model.
Example 3. (system 13)
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This is a three levels, seven loops, nonlinear 5.D. model. It comes
from nuclear physics ((41).
Theorem 8. Example 3 is dissipative.
Theorem 9. when b = 1, k = 22, example 3 is chaos.
Our result are:
1. In example 1, we have proved that system (S) is dissipative.
2. C1 = (, O, 125 C2,3 = (4SORE (k-1) #beb-11/kebeb, +SORE (k-1) #bsb—
11/keb, (itbeb) /kebeb >
are the equilibrium points of system (5).
3. At point C=(0,0,1)
—b 1 °
H= [- -bUi-k) o|
o 9 -b
=== A1=-b<0, A2,3=0.5Eb (k-2) 4SOR (beb¥kek—-4) 1
A1,A2,A3 are Lyapunov exponents, or eigenvalues.
a) k <2, (k — idbeb- 1 < 0 ===> Re(Ai) < 0, i = 1, 2, 3
At this, Ci is stable, C2,3 have not meaning yet.
bb) k < 2, tk -— 2) ~1>0= C have got meaning. A2*AS < 0.
Proof: notice Sun ibebeheehe 4 > So (k-207
(k-1) #b¥b-1 > 0
co) k > 2, tk svapab "ES O mere? AZVES > 0.
d) k > 2, (k-1)*b*b — 1 > 0 ====> AZHAS < 0.
Under the condition of (k-1)*b%b -1 > 0, €2,3 have meaning; the
eigenvalue of Ci are (-, +, -), (-y 7, +).
4. At C2,3,
Hb 1 0
M “fi 1/b yl
Oo y2 -b
yl = +SQR¢ )
y2 = -2S0R( )
====> ul = -b, u2,3=1/24(1/b —b)4S@RE (b-1/b) (b-1/b) -B#E (kK-1) *
beb-113
According to theorem 6, take b = 1, then
u2,3 = +SOR2*(SOR(k—-2))i, tk > 2)
At this time, at €1,A2,35 < 0; at €2,3, u2,3 = +{(S@RC2(k—-2)]}i
5. At b = 1,
dA/db = F04(k-1) 1/SORE2(k-2) I
Therefore, this is a bifurcation point. So, go on.
Based on the above analysis, we simulated system (13):
a) b = 0.7, k = 1.5 ====> JL LAr oy
b) b = 2/3, k =
©} b= 4.0, k= tet ==> NL =
d) b= 0.5, k
)
>
°
===>
548
When b = 1, k = 22 ===> Chaos.
In plactice, people do find that the nuclear spin generator system will
show irregular behavior. But they can not explain this phenomenon. now
our chaos study successfully shed light on it.
REFERENCES
C13. Li Bingxis Periodic orbits of Autonomous Urdinary Differntial
Equations: Theorem and Applications, published by shanghai Science
and Technology 1984.8.
[2] Zhiping Yao: Chaos Theory and its Applications Master Degree
paper SIME 1988.6.
[32 Haken: Synergetics (advanced) Springer-Verlay 1983.
[41 Schmelzer, S. Lectures on the theory and design of an
alternatiing-gradient-proton-synchroton, proc.CERN. 1953.
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