Published by Shanghai University,
Shanghai, China
Applied Mathematics and Mechanics
(English Edition, Vol 22, No 11, Nov 2001)
Article ID: 0253-4827(2001) 11-1240-12
S T U D Y FOR THE B I F U R C A T I O N TOPOLOGICAL S T R U C T U R E
A N D THE GLOBAL COMPLICATED C H A R A C T E R O F
A K I N D OF N O N L I N E A R F I N A N C E S Y S T E M ( I ) ~,
MA Jun-hai ( z ~ g j ~ ) l ,
CHEN Yu-shu ( 1 ~ _ ~ , ) 2
( 1. School of Management, Tianjin University, Tianjin 300072, P R China;
2. Department of Mechanics, Tianjin University 300072, P R China)
(Paper from CHEN Yu-shu, Abstract
Member of Editorial committee, A/VIM)
The one-dimensional
the motionmodel
of a rigid
plate
under explosive
Abstract: Basedproblem
on the of
mathematical
of a flying
kind of
complicated
financialattack has
an analyticsystem,
solutionall only
when
the
polytropic
index
of
detonation
products
equals
to three. In
possible things that the model shows in the operation of our country' s
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
macro-financial system were analyzed, such as balance, stable periodic, fractal,
behavior of the reflection shock in the explosive products, and applying the small parameter purHopfbifurcation,
relationship
between parameters
Hopf-bifurcations,
and of flying
terbation method,
an analytic,thefirst-order
approximate
solution isand
obtained
for the problem
motionhigh
etc.explosives
By the changes
of parameters
of all
economic
meanings,
the to three.
plate drivenchaotic
by various
with polytropic
indices
other
than but
nearly equal
conditions
on which
complicated
such a financial
and
Final velocities
of flying
platetheobtained
agreebehaviors
very welloccur
withinnumerical
resultssystem,
by computers.
Thus
an analytic the
formula
withoftwo
of the
highmacro-economic
explosive (i.e. policies
detonation
and of
polytropic
influence
the parameters
adjustment of
andvelocity
adjustment
index) for estimation
of theonvelocity
of flying
platesystem
is established.
some parameter
the whole
financial
behavior were analyzed. This study
will deepen people ' s understanding of the lever function of all kinds of financial
policies.
1. Introduction
Key words : stable periodic; bifurcation chaotic; topological structure; global
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
complicated character; finance system
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
CLC
numbers
: O175.4 ; O241.81
code: A
cladding of metals. The method
of estimation of flyorDocument
velocity and
the way of raising it are questions
of common interest.
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
Introduction
approach of solving the problem of motion of flyor is to solve the following system of equations
governing
the flow
field
of detonation
products
behind
the system.
flyor (Fig.The
I): definiteness is caused by the
Chaotic
is the
inherent
randomnesS
in the
definite
system internals but not by the external disturbance ~I-4] , while the inherent randomness is the
=o, [5' 62 What is m o s t attractive of the
irregular and difficult-to-be-predicted--ff
behavior of the system
ap +u_~_xp+ au
chaotic theory is that it provides a kind of means by which the complicated things can be
au
au
y1 a certain
=0, structures and aims, but not as the
interpreted as the internal behavior in themselves with
(i.0
external and accidental behavior ET' 8]. Since the chaotic phenomenon in economics was first found
aS
as
in 1985, great impact has been imposed
prominent western economics at present, because
a--T on the =o,
the chaotic phenomenon' s occurringp =p(p,
in the s),
economic system means that the macro economic
operation has in itself the inherent indefiniteness. Although the government can adopt such macro
where
p, S, u areaspressure,
density, policies
specific entropy
particle policies
velocity of
products
controlp, measures
the financial
or the and
monetary
to detonation
interfere with,
the
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on
it are governed
by the flowRevised
field I ofdate
central
rarefaction wave behind the detonation wave
Received
date: 2000-08-30;
: 2001-04-26
D and by
initial
stage
of
motion
of
flyor
also;
the
position
of F and
state
parameters of products
foundation item: the National Natural Science Foundation
of the
China
(69874004)
Biography: MA Jun-hai ( 1 9 6 5 - ) , Professor, Doctor
293
1240
Bifurcation Structure of Finance System ( ]~ )
1241
effectiveness of the interference is very limited. The instability and complexity make the precise
economic prediction greatly limited,
and the reasonable prediction behavior has become
complicated as well. In the fields of finance, stocks and social economics, because of the
interaction between nonlinear factors, with all kinds of economic problems being more and more
complicated and with the evolution process from low dimensions to high dimensions, the diversity
and complexity have manifested themselves in the internal structure of the system and there exists
extremely complicated phenomenon and external characteristics in such a kind of system. So it
has become more and more important to make a systematic and deep study in the internal
structural characteristics in such a complicated economic system, and to reveal the reasons why
such complicated phenomena occur by studying the instability of the periodic solution,
bifurcation, multi periodic bifurcation, the position of the value of every bifurcation point, the
Abstract
way that the complicated system come into chaotic, and on this base, to provide theory basis and
practical
for the analysis,
and control
of flying
the complicated
economic
The ways
one-dimensional
problemprediction
of the motion
of a rigid
plate under continuous
explosive attack
has
system.
an
analytic solution only when the polytropic index of detonation products equals to three. In
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
1 The ofMthe
a t hreflection
e m a t i c a lshock
M o dine ltheof explosive
a Kind products,
of Financial
System
behavior
and applying
the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
When using the method of systematic dynamics to create and test a financial model
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
composed
of theofparts
product,
money,
bound,
and with
labornumerical
force, we'
ve found
that some Thus
longFinal velocities
flyingof plate
obtained
agree
very well
results
by computers.
term
behaviors
that with
the model
provides have
irregularity
the extreme
sensitivity
to the
an
analytic
formula
two parameters
of hightheexplosive
(i.e. and
detonation
velocity
and polytropic
index)
estimation
of theand
velocity
of flying
is established.
initial for
value
of the state
the changes
of plate
parameters.
The occurrence of these properties is not
entirely caused by the main circuit effect, therefore it is necessary to make a further study in the
reasons and mechanism of the structure1.aspect
of this kind of chaotic behavior model.
Introduction
To make the problem easily solved, the key part of the model has been purified and
Explosive
drivencareful
flying-plate
technique
ffmds
its importanti t 'use
in the study
behavior xoft o
simplified.
Through
analysis
and many
experiments,
s decided
to useof variable
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
represent the interest rate in the previous model, variable y to represent the investment demand,
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
variable z to represent the price exponent, and to regard the three state variables and the
of
common interest.
information
feedback
circuitof included
in themplane
as detonation
the key structure.
Because
Under the
assumptions
one-dimensional
and rigid flying
plate,sensitivity
the normalis
approach
the problem
motion of
to solve
the following
system
of about
equations
Concernedofinsolving
the being
discussed ofproblem,
the flyor
aboveis three
variables'
changing
rates
time
governing
the of
flow
detonation
productsnamely:
behind the
I):
are thought
as field
three of
new
state variables,
:r =flyor
d x / d(Fig.
t,
y = dy/dt,
~ = dz/dt.
The
following is the discussion of the configuration of the key structures with this group of new state
--ff
=o,
variables.
ap +u_~_xp+ au
The factors that influence the changes of x mainly come from two aspects: firstly, it is the
au
au
1
=0, between investment and savings;
contradiction from the investment market, i . e . ythe surplus
(i.0
secondly, it is the structure adjustment from goods prices. Therefore x function can be represented
aS
as
as : ~c = f l ( Y - S V ) x + f 2 z therein,
amount of saving, f l and ./:2 are constants.
a--TS V is the=o,
The changing rate of y is in proportion
with the rate of investment, and in proportion by
p =p(p, s),
inversion with the cost of investment and the interest rate. Suppose the benefit rate of investment
where
p, p, S, uin are
pressure,
density,
specific
products
is a constant
a certain
period
of time,
thus entropy
we can and
get: particle
y = f3 (velocity
B E N - of
a y detonation
- / 3 x 2 ) therein
BEN
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
is the benefit rate of investment, f 3 , a and t3 are all constants.
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paraof z, by
on the
oneflow
hand,
by the contradiction
between
supply
and
metersThe
on itchanges
are governed
fieldare
I ofcontrolled
central rarefaction
wave behind the
detonation
wave
Ddemand
and by of
initial
of motion
of flyor
of F and
the state parameters
of products
the stage
commercial
market,
andalso;
on the position
other hand,
are influenced
by the inflation
rate.
Here we suppose that the amount of supplies and demands of commercials is constant in a certain
293
1242
MA Jun-hai and CHEN Yu-shu
period of time, and that the amount of supplies and demands of commercials is in proportion by
inversion with the prices. However, the Changes of the inflation rate can in fact be represented by
the changes of the real interest rate and the inflation rate equals the norminal interest rate subtracts
the real interest rate. Therefore we get: 2 = - f4 z - f5 x , therein, f4 and f5 are both constants.
Thus we can get the key structural model composed of x , y , z, the three state variables and
the feedback circuit concerned. Compared with the previous model, the key structural model has
been to a great degree condensed. But it can be found that there still exist nine independent
parameters to be adjusted, so the model needs to be further simplified. As for the problem to be
solved, what is most concerned is not the absolute amount value of every parameter in the key
structure model, but the combination relationship between the parameters and their relative
changes' influence on the system behavior.Abstract
Therefore, by choosing the appropriate coordinate
system and setting an appropriate dimension to every state variable, we can get the following
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has
more simplified model with only three most important parameters : x , y , z. Ref. [ 2 ] offered the
an
analytic solution only when the polytropic index of detonation products equals to three. In
differential
equation group
in the chaotic
system. by
Seeutilizing the "weak" shock
general,
a numerical
analysismodel
is required.
In thisfinancial
paper, however,
behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
1 - by - x 2 ,
(1)
plate driven by various high explosives with polytropic
indices other than but nearly equal to three.
Final velocities of flying plate obtained agree very
with numerical results by computers. Thus
X - - well
Cfl,
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
therein, a ( I> 0) is the saving amount, b( >I 0) is the per-investment cost, c ( / > 0) is the
index) for estimation of the velocity of flying plate is established.
elasticity of demands of commercials. The following is the study of the local bifurcation
i = z+(~-a)~,
topological structure in model (1) (limited by the length).
1.
Introduction
Suppose Ap is the attraction set of the system. To make the geometrical structure of Ap clear,
driven
ffmds its important
use in point
the study
first Explosive
we study the
localflying-plate
bifurcationtechnique
in the neighborhood
of the balance
of ( 1of
) , behavior
note: of
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
U = ( interest.
x,y,z),
f(U)
=
x2by+
,
Df(U)
=
-2x
- b
0
.
of common
Under the assumptions of one-dimensional
plane
detonation
and
rigid
flying
plate,
the
normal
-x-cz
-1
0
-c
approach of solving the problem of motion of flyor is to solve the following system of equations
governing the flow field of detonation products behind the flyor (Fig. I):
Suppose U = ( x , y , z) is the balance point of ( i ) , then f ( U ) = 0, i . e .
(2)
--ff
=o,O,
!+ (ya)x =
ap +u_~_xp+ au
au
-
au
x12 = O,
~
----.0.
by.
CZ
y
(3)
=0,
(i.0
aS balance
a s point P = ( 0 , I / b , 0 ) ,
Then the system (1) has the only
system ( 1 )
has
a--T
=o,
three balance'points q• = ( +
p =p(p, s),
~/(c
-
b -
ifc - b -abc
abc)/c,
(1
<< 0;
+
the
ac)/c,
- T - 1 / c ~ / ( c - b - a b c ) / c ) and P = ( 0 , 1 / b , 0 ) i f c - b - abc >~ O.
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
2 S t u d y with
of t hthe
e L
o c a l T oRp oof
l o reflected
g i c a l S tshock
r u c t uof
r edetonation
a n d B i f uwave
r c a tDi oas
n a boundary and the
respectively,
trajectory
trajectory
F of flyor as another boundary. Both are unknown; the position of R and the state paraThe following study is the geometrical structure and the stability of the system at the balance
meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
o i n tby
P initial
= ( 0 , 1stage
/ b , 0 of
) motion
i f c - bof- aflyor
b c also;
~ O.theSuppose:
Dp and
position of F and the state parameters of products
X = x, Y = y1/b, Z = z.
2(4)
93
Bifurcation Structure of Finance System ( I )
{~
= (1/b
Then ( 1 ) becomes :
-
a)X
1243
+ Z + XY,
= - bY - X2 ,
Z
(5)
-X-cZ.
Through transformation ( 4 ) ,
the balance point P becomes balance point ( 0 , 0 , 0 ) .
The
characteristic equation of the ( 5 ) ' s corresponding linearized system is
(2
+ b
2~2 +
c + a -
2 + 1 + ac -
Then the corresponding three characteristic value of ( 6 ) :
= 0.
~1
(6)
=-
b,
X2 and X3 arc
determined by ( 7 ) :
,~2 +
( c + a - Abstract
2 + 1 + ac
c - 0.
b
(7)
F o r cone-dimensional
- b - abc < problem
O, t h e n of
l +theacmotion
- c / b of.>a rigid
0. The
solutions
to theexplosive
two corresponding
The
flying
plate under
attack has
an
analytic
solution
only
when
the
polytropic
index
of
detonation
products
equals
to
three. In
characteristic solutions of ( 7 ) 2 2, 2 3 can be divided into three cases:
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
Case 1
c - b - abc < O, c + a - 1 / b > 0 , then 2 2 < 0, 2 3 < 0 simultaneous with
behavior of the reflection shock in the explosive products, and applying the small parameter pur21 = - bmethod,
< 0 , then
the balance
point P approximate
= ( 0 , 1 / b , 0solution
) of system
( 1 ) under
conditions
case
terbation
an analytic,
first-order
is obtained
for the problem
of of
flying
plate
driven
by
various
high
explosives
with
polytropic
indices
other
than
but
nearly
equal
to
three.
1 i s t h e stable convergence, at this t i m e Ap = { p } .
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
Case2
c - b - abc < O, c + a 1 / b < 0, then,~2 > 0 , A3 > 0 s i m u l t a n e o u s w i t h
an analytic
formula
with two parameters of high
explosive (i.e. detonation velocity and polytropic
,~1
=
b
<
0,
then
the
balance
point
P
=
(
0
,
1
/
b
, 0 ) o f system ( 1 ) under the conditions of case
index) for estimation of the velocity of flying plate is established.
2 is saddle, at this time Ap = {p }.
Case 3
-
c - b -abc
= 0 , then1.'~ 2 Introduction
= O, .~ 3 = - ( c + a - 1 / b ) simultaneous with ,~1 =
b < 0 , at this t i m e , it can be divided into two cases:
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
;t3 = - ( c + a - 1 / b )
= (1 - c 2 ) / c
> 0, i . e . 0 < c < 1, then according to
materialsi )under
intense impulsive loading,
shock synthesis
of diamonds, and explosive welding and
Ref. [ 3 ] ofp169
theorem,
the balance
point Pof =flyor
( 0 , velocity
1 / b , 0 ) and
is the
unstable
balance
of the
cladding
metals.
The method
of estimation
the way
of raising
it arepoint
questions
ofn ocommon
interest.
n-hyperbola.
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
ii ) ;t3 = - ( - c + a - 1 / b ) = (1 - c 2 ) / c < 0 , i . e . c > 1, we use central manifold
approach of solving the problem of motion of flyor is to solve the following system of equations
theorem to
o o m field
the stability
of theproducts
balance behind
point: the
At flyor
this time
governing
thedflow
of detonation
(Fig. the
I): corresponding characteristic
vectors of the three characteristic values respectively are: for ;tl
characteristic vector is:
(0,1,0)
(1,0,
=-
b the corresponding
for /~2 = 0 the=o,
corresponding characteristic vector is:
--ff
ap +u_~_xp+ au
T,
- 1 / c ) T , for .~3 = (1 - c 2 ) / c < 0 the corresponding characteristic vector is: ( - 1 / c , 0 ,
au
au
1
=0,into the stable subspace E ~ , and the
1 ) T . The characteristic vectors of 21 and ,~3 are y
extended
(i.0
characteristic vector of ~2 is extended
aSinto the
a s central subspace E ~ .
For the characteristic basis (a--T
X , Y, z ) T , =o,
make the linear transformation
T ( u , v , w)T
p =p(p, s),
namely :
C 2 / ( C2 -- 1 )
0
c/(c 2 - 1)
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
T -1
c 2 / ( 1 -and
c 2 )the
c / ( 1 wave
- c 2D) as a0 boundary
0 R of reflected
0
,
=
respectively, with theT trajectory
shock
of detonation
trajectory F of flyor as another
boundary.
Both
are
unknown;
the
position
of
R
and
the
state
para- 1/c
1
1
0
0
meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
(8)
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
Put ( 8 ) into ( 5 ) ,
we can get
293
1244
MA Jun-hai and CHEN Yu-shu
t! li o o llil lc2 clvwJc211
=
(c 2 - 1)/c
+
0
c(u
C-Iv)w/(1
c 2)
-
9
(9)
( U -- C-1 V) 2
b
As for the new variable ( u , v , w)T its three characteristic values do not change, and their
corresponding characteristic vectors respectively are: for ;t i = - b the corresponding characteristic
vector is: ( 0 , 0 , 1 ) T , f o r '~2 = 0 the corresponding characteristic vector is: ( 1 , 0 , 0 ) w , for 23 =
(1
-
< 0 the corresponding characteristic vector is: ( 0 , 1 , 0 ) T , therefore E c = u axis,
c2))c
E s = s p a n { ( u , v , w)W}, E L = 4 . The linear part of equation group ( 9 ) has been uncoupled:
C2
a -
c2_ l(U - c-iv)w,
(10a)
0)(v)
/
Abstract
(;)
The one-dimensional
attack( 1has
= problem of the motion of a+rigid flying plate under explosive
.
0b)
- b index of detonation
- ( u - cproducts
- I v ) 2 equals to three. In
an analytic solution only when 0the polytropic
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
Central manifold W c is the curve which is tangent with the central subspace E c , then we try to get
behavior
of the reflection shock in the explosive products, and applying the small parameter purthe equation
of Wan
c . Now
we need
to getapproximate
the functionssolution
hi ( u ) is
and
h2 ( u ) for
which
are deviated
from
terbation
method,
analytic,
first-order
obtained
the problem
of flying
plate
driven
by
various
high
explosives
with
polytropic
indices
other
than
but
nearly
equal
to
three.
u-axis to v-axis and w-axis after having made tangent at the balance point ( 0 , 0 , 0 ) in the vicinity
Final
velocities
of flying
obtained
very well
with numerical results by computers. Thus
W~ and
E c of the
balanceplate
point
( 0 , 0 , 0 )agree
. Central
manifold:
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
index) for estimation of the velocity of flying plate is established.
Wc =
= h(U)
= (h2(u)
1.
'
h(O)
= O,
Dh(O)
= 0,
(11)
Introduction
h ( U ) : R --~ R 2 is the differential in the vicinity of point ( 0 , 0 , 0 ) .
Now we use the method of
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
form power series to get h ( U ) .
materials
under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
Suppose:
cladding
of metals. The method of estimation of flyor velocity and the way of raising it are questions
of common interest.
Under the assumptions
( : ) of
= one-dimensional
h ( U ) = I alu2
+ a u2plane
2 + blU3+ClU4+""
I detonation
u3b+ c22u and
4 + rigid .flying plate, the normal
(12)
approach of solving the problem of motion of flyor is to solve the following system of equations
governing
detonation
products
flyor (Fig. I):
Put ( 1 2 ) the
intoflow
( 1 0field
b ) , of
therein
u is put
into asbehind
( 1 0 a )the
, then:
hi(u)
h2(u)
(
c2 - lhl(U ) +
C2
c_~_ l ( U
_ c-lv)h2(u)
)
--ff
ap +u_~_xp+
=
c
au
au
aS
as
au
y1
=o,
c
l-c"
- bhz(u)
-
c
-1 h l ( u ) ) h 2 ( u )
- (u - c-lhl(u))
=0,
2
(i.0
(13)
therein h
= 3hi( u )/Ou
= 2aiu +
3 b i n 2 + 4ci
u3 + " " . Put h and h into ( 1 3 ) as the form of
a--T
=o,
( 1 2 ) . After comparing coefficients, then v-equation:
p =p(p, s),
c2 - 1
O(U2):
a I = 0,
c specific entropy and particle velocity of detonation products
where p, p, S, u are pressure, density,
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
c2 - lbi
c
( u 3 ) : boundary.+ Both area2unknown;
= 0,
(14a)
trajectory F of flyor asOanother
the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
c
2ala2c
O(u4):
c2 - 1
-
c -1
- - cc i
+
(
b2-
ala2)
c
9
293
Bifurcation Structure of Finance System ( ]~
1245
w- equation:
O(/Z2):C
2 -- 1
C1 +
C
2~ 1
O(/t3):
--
C
C
1--C
b
2
2=
0,
(14b)
b i b 2 = O,
2a22c 2
o ( u 4 ) : c-~ _-~
.
2
2bl
_
-
-
c
. ct 1
_
bc
2
-
7
.
Combining (14a) and ( 1 4 b ) , then:
C2
a 1 = 0,
b1 _
1
a2 = - ~ - ,
b ( c 2 - 1) 2,
C 1 = 07
(15)
Abstract - 2 c
b2 =
O,
C2 -- b 3 ( c 2 _ 1 ) 2 ( b + c ( c 2 - 1 ) ) .
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has
The central manifold precise to the power of four orders is:
an
analytic solution only when the polytropic index of detonation products equals to three. In
general, a numerical analysis is required. In this paper, however,
by utilizing the "weak" shock
__
u3 + O ( u 5)
behavior of the reflection shock in the explosive products,
and
applying
the small parameter pur[ ( C 2 -- 1 ) 2
WC = method, an analytic, first-order approximate solution is obtained for the problem of flying
terbation
plate driven by various high explosives with
indices other+c(c
than 2but
- b polytropic
u2 - b3i-j-]5~<
- 1nearly
) 2 4 +equal
o ( .to5)three.
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
(16)
index)
of the
velocity
of flying
plate
is established.
Puttingforh 1estimation
and h 2 into
(10a)
we can
get the
equation
onto the manifold:
/ 84 2 b
(;):
-- C 2
a
-
b(c2
_ 1)
U3
/1
2C 3
- b 31.( c 2Introduction
- 1) 3 ( b + c ( c 2 - 1 ) ) u S + O ( u 6 ) "
(17)
Because
the coefficient
of u 3 in (technique
1 7 ) is negative,
the manifold
central
manifoldofis
Explosive
driven flying-plate
ffmds its important
use onto
in thethestudy
of behavior
graduallyunder
inclined
to beimpulsive
stable. loading,
According
to the
centralof manifold
the balance
materials
intense
shock
synthesis
diamonds,theorem,
and explosive
welding point
and
cladding
metals.
The method of
estimation
raising it are
questions
( 0 , 0 , 0 ) ofand
its corresponding
balance
pointofP flyor
( 0 , 1 /velocity
b , 0) inand
( 1the
) isway
alsoofgradually
inclined
to be
ofstable.
common interest.
Under
the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
Synthesizing i ) and H ) , we can conclude that bifurcation occurs at the balance point
approach of solving the problem of motion of flyor is to solve the following system of equations
P(O,1/b)
,0) in system ( 1 ) if c = 1, under the conditions of Case 3.
governing the flow field of detonation products behind the flyor (Fig. I):
Case4
c-
b-
abc
<0,
= 0, t h e n w e can get: c 2 < 1, /!2 andA3 are
c+a,1/b
purely imaginary roots, with21 = - b < 0, g i v e n a = - ( c +
a-1/b),thenaa/Oa
l a=ao =--ff
=o,
ap +u_~_xp+ au
- 1 # 0 satisfying the intersection conditions of Hopf-bifurcation. This shows that when a passes
au
au
a passes, a0 the orbit line of the system surpasses
y1 the=0,imaginary axis, namely, under the
conditions of Case 4 at the balance point P = ( 0 , 1 / b , 0) Hopf-bifurcation occurs and there (i.0
exists
aS
as
periodic solution group. At t h i s t i m ea--T
: 21 = -
3
=o,22 = i ~ / 1 - c 2, 2 3 = - i ~ / 1
b,
- c 2.
p =p(p, s),
T h e R e s u l t s of C a l c u l a t i o n s
where Cp,a sp,e l S, u are
density,
entropy
products
c - b -pressure,
abc
< 0 , c +specific
a-1/b
> 0, iand
f g i vparticle
e n a = 4velocity
. 5 , b of
= detonation
0 . 2 , then coefficient
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
c should satisfy, 0 . 5 < c < 2. We take c = 0 . 6 and take ( x 0 , y o , z o ) = ( 0 . 0 0 0 0 1 1 , 4 . 9 9 8
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para655, -on0 .it0 0are
0 0governed
1 1 ) , At by
= the
0 . 0 1flow
, at field
the same
time, rarefaction
and the x - ywave
phase
diagram
of its first wave
4096
meters
I of central
behind
the detonation
is initial
sketched
in of
Fig.
1 ( a ) , ofthe
y - zalso;
phase
in FFig.
b ) , state
theparameters
x - z phaseofdiagram
Dpoints
and by
stage
motion
flyor
the diagram
position of
andl (the
productsin
Fig. 1 ( c ) , the x - y - z phase diagram in Fig. 1 (d) ; The calculation results show: the balance point
293
1246
MA Jun-hai and CHEN Yu-shu
of the system under the conditions of Case 1 is the stable convergence, which is completely
consistent with results of theoretic deduction.
Case 2
c
b - abc < O, c + a - 1 / b
-
cshouldsatisfy, c < 2andc
< 0, if given a = 4 . 5 , b = 0 . 2 , then coefficient
< 0 . 5 . We t a k e c = 0 . 4 and t a k e ( x o , Y o , Z o )
= (0.000003,
4.998 635, - 0.001 040), At = 0 . 0 1 , at the same time, and the x - y phase diagram of its first
4 096 points is sketched in F i g . 2 ( a ) , the y - z phase diagram in F i g . 2 ( b ) , the x - z phase diagram
in F i g . 2 ( c ) ,
the x - y - z phase diagram in F i g . 2 ( d ) ;
The calculation results show: the balance
point of the system under the conditions of Case 2 is saddle, which is completely consistent with
the results of theoretic deduction.
Abstract
to
4.999.8
'1
9 ooo
ooo,
.,t
.---%
1 E - 005
-
5E- 006
-i
"
, : ,of
I the motion of a rigid flying plate under explosive attack has
The one-dimensional
problem
::
.|l
o so~
teo~
z of detonation
0
an analytic
solution
only
when
the
polytropic index
products
J equals to three. In
4.9994
o~
I
general, a numerical
analysis is required. In this paper, however, by utilizing the "weak" shock
4.~2
'o
-5E-~
9 g
II,
behavior of the reflection
shock in the explosive products, and applying the small parameter
--. . pur4.9990
terbation method, an analytic,
first-order
approximate
solution
is
obtained
for
the
problem
of flying
"'',
- 1E-005
4.9988
' - o o ~,
plate driven
by various high explosives
with polytropic indices other than but nearly equal to three.
.998 6
. 5 E - 005
Final 4velocities
obtained
agree very well- 1with
numerical
results by
computers.
-1.5E-005.of -flying
5 E - 0 0 6plate
0 5E
-006
1.5E-005
4.9986
4.9990
:4.999
4
4.999 Thus
8
- 1E-005with two parameters
1E- 005of high explosive (i.e. detonation
4.998 8
4.9992
4.9996
an analytic formula
velocity and
polytropic
Y
index) for estimation of the velocity of flying plate is established.
Fig. l ( b )
Fig. l(a)
4.9996
lieu
~f
1.
1E- 005
Introduction
999
~ . 00'3~
I E - 005
Explosive driven flying-plate technique ffmds its ximportant
use in the study of behavior
of
5E- 006
.......
0 diamonds,
0
6
0 explosive
~
5E- 0O6and
materials under intense impulsive loading, shock synthesis of
and
welding
5~"
0
cladding of ometals.7: The
of flyor velocity and the way of raising it are questions
" . 7 ~method
i . ' . ' ~ . ' ~ of estimation
',
:.
0O6'
-5E-006
of common
interest. " : . r.:.:. :. : z : :j:: : , :
-5E-~
-5~- ~ and
~ rigid
- flying
I
E plate, the normal
005
Under the assumptions of one-dimensional plane detonation
- I E - 005
5Eapproach
of solving the problem of motion of flyor is to solve the following system of- Iequations
governing
the flow field of detonation products behind the flyor (Fig. I):
1.5E- 005
9
9
9
o...
-'~
* o . . ~ 1 7 6
i
,
.
~176
~
.
.
.
.
~
.~
"~176
- 1E-005
oo~
I E - 005
~f
--ff
ap +u_~_xp+
au
Fig. l ( c )
au
au
=o,
y1
Fig. 1 (d)
=0,
(i.0
Case3
c
-
b -
abc
= O, aS
then22
it can be divided into two cases:
i ) 2~3 = -
(c
+ a-
1/b)
a=s O, 23
a--T
=o,
p =p(p, s),
= (1-
c2)/c
=-
c + a-
> 0, i . e . 0
1/b)with21
< c < 1,iftakea
= - b < 0,
= 4.5, b =
0 . 2 , p,c p,
= S,
0.5u and
take a = density,
4 . 5 , b specific
= 0 . 2 , entropy
c = 0.5and
at the
samevelocity
time, (Xo,
Yo, Zo) products
= (2, 1,
where
are pressure,
particle
of detonation
respectively,
with
the
trajectory
R
of
reflected
shock
of
detonation
wave
D
as
a
boundary
and point
the
2 ) , At = 0 . 0 1 , and the x - y phase diagram of its first 4 0 9 6 points [starting
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para( x , , y o , z o ) = ( 0 . 5 4 8 0 9 1 , 3.649 7, 0.232 718)] is sketched in Fig. 3 ( a ) , the y - z phase
meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
in F i g .stage
3 ( b ) ,of the
x - z phase
diagram
in Fposition
i g . 3 ( c )of
, Fx - yand
- z phase
diagram
in F i gof. 3products
( d ) ; the
Ddiagram
and by initial
motion
of flyor
also; the
the state
parameters
x - y phase diagram of its last 4 096 points is sketched in Fig. 3 ( e ) ,
the y - z phase diagram
2 9 3in
Bifurcation Structure of Finance System ( ]- )
1247
5'
4.9998
4.9996
I"
m
4
)
4.9994
t/
4.9992
3
f
z
2
1
4.999O
0
4.998 8
4.998'6
- 0.010- 0,006 - 0.002 0.002 0.006 0.010
0.004 0.008 0.012
-0.008-0.004
0
I
--1
'
0
1
2
3
4
5
8
9
Y
Fig.2(b)
Fig.2(a)
Abstract
0.010
i ~ ""~
0.0138
"
The one-dimensional
problem of the motion of aO.Ot{
rigid
attack
~
1 0.010has
! flying plate under explosive
0.008
0.006 solution only when the polytropic indexO.OtY
an analytic
of
detonation
products
equals
to
three.
0.006 In
0.~
0.004
0.~ however, by utilizing the "weak"
general, a numerical analysis is required. In this paper,
shock
t
o.OC;
0=
0.~
behavior 0of theN reflection shock in the\ explosive products, and applying the small parameter purz O.Ol ~.
_ 0 = method, an analytic, first-order approximate-0.0~
terbation
solution is obtained for the problem of flying
-0.004
~
-O._oo6
plate driven by various high explosives with polytropic-0.{
indices other than but nearly equal "0.oo~
to three.
-0.006
~
" ~
'~"
]
-0.{
Final velocities of flying plate obtained agree very well with numerical results by computers.
Thus
.Olo'O.ol
-o.~
I
I "
O.a.i~polytropic
~ O t~_
0.010 formula with two parameters of high explosive (i.e. detonation velocity
an analytic
and
~4
-0.010-0.006-0.002 0.002 0.006 0.010
b,
index) for estimation
of the0 velocity
flying0.012
plate is established. ~. ~.~s
-0.008-0.004
0.004 of
0.008
"o.~
"0.oo4
~0~ -0"~
-
x
1.
Introduction
Fig.2(c)
Fig.2(d)
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
materials
impulsive
and explosive
F i g . 3 ( f ) ,under
the intense
x - z phase
diagramloading,
in F i gshock
. 3 ( g ) ,synthesis
the x - yof
- z diamonds,
phase diagram
in Fig. welding
3 ( h ) . and
The
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
calculation results show: the balance point of the system under the conditions of Case 3 is the
of common interest.
unstable
balance
point of theof non-hyperbola,
is completely
Under
the assumptions
one-dimensional which
plane detonation
and consistent
rigid flyingwith
plate,thetheresults
normalof
theoretic of
deduction.
approach
solving the problem of motion of flyor is to solve the following system of equations
4
5.0
.
.
.
.
.
governing the flow field of detonation products behind the flyor (Fig.
I):
4.5
4.0
3.5
3.0
Y 2.5
2.0
1.5
1.0
0.5
0
- 1.5 - 1.0 -0.5
3
--ff
ap +u_~_xp+
au
au
au
1
aS
as
i
]~.-0
0,5
1.0
a--T
y
=o,
2
1l
=0,
(i.0
-1
=o,
1.5 2.0
p =p(p, s),
--2
: . . . .
0
1
2
3
5
4
6
7
8
9
y
g.3(a)
.3(b)
where p, p, S, u areF ipressure,
density, specific entropy and particle velocity Fofi g detonation
products
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
trajectory
as another boundary.
Both
are unknown; the position of R and the state paraii ) F23of =flyor
- (c + a -1/b)
= (1
C 2 ) / C < O, i . e . c > 1, i f t a k e a = 4 . 5 , b = 0 . 2 ,
meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
= 2by[ininitial
fact stage
we can
it from
( 4 2 )also;
, Hopf-bifurcation
likely
occur
in the system
when
Dc and
of get
motion
of flyor
the position of Fis and
the to
state
parameters
of products
-
a = 3, b = 0 . 2 ,
c < 2. According to ( 4 2 ) ,
c has only four imaginary roots: cl,22 9 3=
1248
MA Jun-hai and CHEN Yu-shu
2.0
,.5
j
~
/
2"0
1.5
1.0
.5
~00
1.5-
,.0
0.5 k ~kk 7 ~
o ' ~ " ~ L ~ ~. ' ~
-0.5
"-~.~ ~5~'
o.
~ _ 0 . 5 i:~
-1.0
1.5
_~.
- 1.0
~,.5
P
- 1.5
-1.5-,.0-0.5
0
0.5
1.0
1.5
b,.O 3.3t3
2.0
Y
.
x.u 13. k~ -
"0
Fig.3(d)
Fig.3(c)
Abstract
4.01
3.65
3.51 1
3.60
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has
3.0
3.55
an analytic
solution
only
when the polytropic index of detonation
products equals to three. In
2.5
r
"
,%,,
/
3.50 a numerical analysis is required. In this paper, however,
general,
by
utilizing the "weak" shock
2.0
3.45 of the reflection shock in the explosive products,
z 1.5
behavior
and applying the small parameter pur3.40
1.0 is obtained for the problem of flying
terbation method, an analytic, first-order approximate solution
0.5
3.35
plate3.30
driven
k various high explosives with polytropic indices0 other than but nearly equal to three.
~ by
%J
Final3.25
velocities of flying plate obtained agree very well with
-0.5numerical results by computers. Thus
1.0 detonation velocity and polytropic
3.20
an analytic
formula with two parameters of high explosive- (i.e.
3
'4 ' 5 6
7
8-9
0 1 2
- 0 . 2 0 0.2
0.6
1.0
"0.6
-1.0
index) for-0.8
estimation
flying plate is established.
0.4 of0.8
-0.4 of the velocity
Y
f~
)
,I"'~-YJ
x
Fig.3(e)
1.
Fig.3(f)
Introduction
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
0.8
materials
loading, shock synthesis of diamonds, and explosive welding0.8
and
p, intense impulsive
-.~
0.6 under
0.6 and the way of raising it are questions
'6
cladding
of
metals.
The
method
of
estimation
of
flyor
velocity
0.4
0.4
0.4
of common
interest.
0.2
.2
O.
\the assumptions of one-dimensional plane detonation
z
and rigid flying plate, the normal
z Under
0
0.2 _ ~
approach
the problem of motion of flyor is -to
~ i -0.4
0"2
-0.2 of solving
_0.4solve the following system of equations
|
\
-0.6
governing
flyor (Fig. I):
-~0.4 the flow field of detonation~rproducts behind the -0.6
.
I
-0.6
i
, ~
-0.8
0.6 --ff1.0
-0.2 0 0.2
- 1 .0 -0.6
ap +u_~_xp+
0.8
:-0.8 -0.4
0.4
x
Fig.3(g)
au
au
0.8
0 0.6 0.#
o.4
-O.g
,~~
au
y1
:
=o,
Y %~.
1000804 x
=0,
Fig. 3 ( h )
aS
(i.0
as
a--T
=o,
p =p(p, s),
- 1.209 246 42 _+0. 861 608 97i, c3, 4 ---- 1. 109 246 42 + 0. 361 018 69i, so when a = 3, b = 0 . 2 ,
for any c, Hopf-bifurcation will not Occur] at the same time, and the x - y phase diagram of its
first 4p,
096
( x0, specific
Yo, Zo ) entropy
= (0, 4.998
694, 0)velocity
] is sketched
in Fig. 4products
( a ) , the
where
p, points
S, u are[ starting
pressure,point
density,
and particle
of detonation
respectively,
with
the
trajectory
R
of
reflected
shock
of
detonation
wave
D
as
a
boundary
and
thein
y - z phase diagram in F i g . 4 ( b ) , the x - z phase diagram in F i g . 4 ( c ) , the x - y - z phase diagram
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paraFig . 4 ( d ) ; The calculation results show: the balance point of the system under the conditions of
meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
stable,stage
which
is completely
with theofresults
deduction.
DCase
and 4byis initial
of motion
of flyorconsistent
also; the position
F and of
thetheoretic
state parameters
of products
Case4
c - b - abc <0,
c + a -1/b
= O, t h u s w e c a n g e t : c2 < 1 . I f t a k e a = 42. 59,3
Bifurcation Structure of Finance System ( I )
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
-0.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
1249
/
I
, ~ 1 7. .6. . .
~
-05
0
0.5
1.0~
1.5
0
2.0
1
2
3
4
5
x
y
Fig. 4 ( a )
Fig. 4 ( b )
6
7
8
9
Abstract
2.0
The one-dimensional problem
of the motion of a rigid
attack 12"0.5
has
9.-"" ...-'"'"
2 flying
.
0plate~ under
1 explosive
.
5
an analytic solution only when the polytropic index of detonation products equals to three. In
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
behavior
of the reflection shock in the explosive products,o.,and applying the small parameter pur0.5
terbation method, an analytic, first-order approximate solution
0 ~ is obtained for the problem of flying
-0
o! by various high explosives with polytropic indices other than but nearly equal to three.
plate driven
Final-0.5
velocities of flying plate obtained agree very well with numerical results by computers. Thus
-0.5 formula
an analytic
velocity and polytropic
o
0with
5 9 two
l o parameters
1.5 2 . o of high explosive (i.e. detonation
r
,~"~"
index) for estimation of the velocity of flying plate is established.
Fig.4(c)
Fig.4(d)
1.
Introduction
b = 0Explosive
. 2 , c = 0driven
.5andtake(xo,Yo,Zo)
= ( ffmds
2 , 1 , 2its
) , important
At = 0 . 0 1use
, atinthe
time,
and the x-y
flying-plate technique
thesame
study
of behavior
of
phase diagram
its first
4 096 loading,
points is
sketched
in of
Fig.diamonds,
5 ( a ) , and
the explosive
y-z phasewelding
diagram
materials
under of
intense
impulsive
shock
synthesis
andin
cladding
of
metals.
The
method
of
estimation
of
flyor
velocity
and
the
way
of
raising
it
are
questions
F i g . 5 ( b ) , the x-z phase diagram in F i g . 5 ( c ) , the x-y-z phase diagram in F i g . 5 ( d ) ; the x-y
of
common
interest.
phase
diagram
of its last 4 096 points [ starting point (Xo, Yo, Zo) = ( 0 . 0 0 0 058, 4. 998 648,
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
0.000 12)] sketched in F i g . 5 ( e ) , the y-z phase diagram in F i g . 5 ( f ) , the x-z phase diagram
approach of solving the problem of motion of flyor is to solve the following system of equations
in F i g . 5 (the
g ) , flow
thefield
x-y-z
phase diagram
in Fbehind
i g . 5 ( hthe
) . flyor
From
( 2 3I):) , we can calculate: al =
governing
of detonation
products
(Fig.
5.0
4.5
4.0
3.5
3.0
Y 2.5
2.0
1.5
1.0
0.5
-0.5
5
--ff
ap +u_~_xp+
I'
au
i
=o, 4
I
3
au
au
aS
a--T
=o,
p =p(p, s),
.....
0
,,
0.5
1.0
1.5
as
y1
=0,2
(
1
(i.0
0
~176176
2.0
-1
0
1
2
3
4
5
6
7
8
9
Y
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
respectively, with theF itrajectory
R of reflected shock of detonation wave D Fas
and the
g.5(a)
i g .a5 (boundary
b)
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
2 . 2 by
3 6 8initial
4 2 105,
0.227 of
901flyor
422,also;
cl the
= - position
2.763 157
From
7 ) , we can
calcualte:
D- and
stageblof =motion
of F 895.
and the
state( 2parameters
of products
R = - 1.236 023 976. At this time, the limit cycle produced by Hopf-bifurcation at the balance
293
1250
MA Jun-hai and CHEN Yu-shu
point is stable. The calculation results show: on the conditions of Case 4, Hoph-bifurcation
occurs at the balance point in the system, which is completely consistent with the results of
theoretic deduction.
40
35
30
25
20
15
10
5
0
-5
2.0 ~
2.0
z 1.0 ~
~.0
0.5
.5
-0.5
-0.5
....
-0.5
0
0.5
1.0
1.5'
9~',v\: ~
2.0
Abstract
X
Y
o
.o _
% %.~.% . O s
x
The one-dimensional
of the motion of a rigid flying plate under
attack has
F i g . 5 ( cproblem
)
F i g .explosive
5(d)
an analytic solution only when the polytropic index of detonation products equals to three. In
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
5:
behavior of the reflection shock in the explosive products, and applying the small parameter pur4.9998
-..__
4
terbation method, an analytic, first-order approximate solution is obtained for the problem of flying
4.9996
3
plate driven by various high explosives with polytropic indices other
than but nearly equal to three.
4.9994
~
~
Final velocities of flying ~plate
obtained
agree very well with numerical
results by computers. Thus
2
4.9992 formula
(
an yanalytic
with two parameters of high explosive (i.e. detonation velocity and polytropic
1
index)4.9990
for estimation of the velocity of flying plate is established.
4.~S
0
~ - ~ .
4.9986 . . . . . . .
-0.00015 -5E-005 0 5E-005
-0.0001
X
-1
1.o.ooo~5
Introduction
0
1
2
3
0.0001
4
5
6
7
Y
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
F i g impulsive
.5(e)
g.5(f)
materials under intense
loading, shock synthesis of diamonds, andF iexplosive
welding and
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
of common
interest.
40
35
Under the assumptions of one-dimensional planeo.O,o
detonation and rigid flying plate, the0.000,5
normal
3O
approach
of solving the problem of motion of flyor is to solve the following system of equations
25
governing
the flow field of detonation products behind the flyor (Fig. I):
2O
15
10
5
0
-5
- 0.00015 - 5E - 005 0
- 0.0001
--ff
ap +u_~_xp+
au
5E - 005 . . . . t}.00015
0.0001
aS
Fig.5(g)
au
au
=o,
_0.000~5~
1
y
~
.
~
o
~
~176176176
=0,
(i.0
as
a--T
=o,
p =p(p, s),
Fig.5(h)
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
4 T h e R ewith
s u l t sthe trajectory R of reflected shock of detonation wave D as a boundary and the
respectively,
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paraof the
simulation,
we know
that: first,
thebehind
inappropriate
combination
metersFrom
on itthe
areresults
governed
by above
the flow
field I of central
rarefaction
wave
the detonation
wave
the by
parameters
in the
system is
source
causes chaotic
occur
the economic
system.
Dof and
initial stage
of motion
of the
flyor
also; that
the position
of F andtothe
stateinparameters
of products
It is likely to make the system inclined to chaotic and lose control, and make the system plunged
293
Bifurcation Structure of Finance System ( ][ )
1251
into the stagnant and ossified state. Therefore no matter when it is in the serious inflation stage or
in the economic slack stage, the transformation of mechanic or the adjustment of structure will
forever be the first task in the reform of financial system. Secondly, the elasticity deficiency of
variables will cause the lagging down of the information feedback and it can only provide the
strategic institute with distorted and wrong signals. The bad results caused by this can be
imagined. What is more serious is that these bad results will continuously extend or expand as the
form of a vicious cycle. Therefore, strengthening the elasticity of the variables appropriately will
help stabilize economy and help the normal operation of the financial system. In addition, the
saving amount variable a in the system must be kept in an appropriate level. The above study
results show that the smaller a is, the greater the fluctuation of the system is. When a is too much
small, it will cause chaotic situation/conjuncture. However, a can' t be too large, or it will cause
Abstract
economy to lack vigor.
problem of the motion of a rigid flying plate under explosive attack has
R e f e rThe
e n c e one-dimensional
s:
an analytic solution only when the polytropic index of detonation products equals to three. In
[ 1 ] CHENG
Si-wei.analysis
Complicated
Science In
andthis
Management[A].
In:CHENG
Si-wei
Article
Colgeneral,
a numerical
is required.
paper, however,
by utilizing
the Ed.
"weak"
shock
Beijing Xiangshan
[ C]. Beijing:
Press, 1998,1
- 9. parameter
(in Chinese)
behaviorlection
of theof reflection
shock inConference
the explosive
products,Science
and applying
the small
purterbation
method,Deng-shi,
an analytic,
first-order approximate
solutionofis the
obtained
for the
problem Publishing
of flying
[ 2 ] HUANG
LI Hong-qing.
Theory and Method
Nonlinear
Economics
plate driven
by
various
high
explosives
with
polytropic
indices
other
than
but
nearly
equal
to three.
[ M]. Chengdu: House of Sichuan University, 1993. (in Chinese)
Final
velocities
of
flying
plate
obtained
agree
very
well
with
numerical
results
by
computers.
Thus
[ 3 ] LU Qi-shao. Bifurcation and Queerness[M]. Shanghai: Shanghai Sciecne and Technology Educaan analytic
formula
with
two
parameters
of
high
explosive
(i.e.
detonation
velocity
and
polytropic
tion Publishing House, 1995. (in Chinese)
index) for estimation of the velocity of flying plate is established.
[ 4 ] LI Jing-wen. Chaotic theory and economics [ J ]. Quantitative Economic Technology Economy
Study, 1991, (24) : 19 - 26. (in Chinese)
Introduction
[ 5 ] Brunella M, Miarim. Topological 1.
equivalence
of a place vector field with its principal past defined
through Newton polyhedral J]. J Differential Equations, 1990,85(6):338- 366.
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
[ 6 ] Cima A, Llibre. Algebraic and topological classification of the homogeneous cubic vector fields in
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
the plane[J]. J Math Anal Appl, !990,47(4) :420 -448,
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
[ 7 ] Omer Morgul. Necessary condition for observer-based chaos synchronization[J]. Phys Rev Len,
of common interest.
1999,82(9)
:77 - 80. of one-dimensional plane detonation and rigid flying plate, the normal
Under
the assumptions
[ 8 ] YANG
Xiao-jing.
The localof
phase
diagram
of a kind[
System
Science and
Mathematics,
1999,
approach
of solving
the problem
motion
of flyor
is toJ].solve
the following
system
of equations
19(4)
:150156.
(in
Chinese)
governing the flow field of detonation products behind the flyor (Fig. I):
[ 9 ] M Clerc, P CouUet, E Tirapegui. Lorenz bifurcation instabilities in quasireversible systems[J].
Phys Rev Lett, 1999,19(11) : 3820
- 3823.
--ff
=o,
ap +u_~_xp
+ dynamics
au
[ 101 Jati K Sengupta, Raymond E. Sfeir
Nonlinear
in foreign exchange markets[J]. Internaau
tional Journal of Systems Science,au1998,129(11):
- 1224.
y1 1213 =0,
[ 11 ] Freedman H I, Singh M, Easton A K, et al. Mathematical models of population distribution (i.0
within
a culture group[J]. Mathematical
and
Commuter
Modelling,
1999,29(6)
:257
267.
aS
as
a--T
=o,
[ 121 Alexander Lipton-Lifschitz. Predictability
and
unpredictability in financial markets[ J]. Phys D,
1999,133(12) : 321 - 347.
p =p(p, s),
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293