Intelligent Information Management, 2011, 3, 17-21
doi:10.4236/iim.2011.31002 Published Online January 2011 (http://www.SciRP.org/journal/iim)
A Nonlinear Dynamic Model of the Financial Crises
Contagions*
Ke Chen1,2, Yirong Ying1
1College of Management, Shanghai University, Shanghai, China
2College of Finance and Economics, Chongqing Jiaoting University, Chongqing, China
Received December 19, 2010; revised January 7, 2011; accepted January 10, 2011
Abstract
Employing the Differential Dynamics Method, a nonlinear dynamic model is set up to describe the interna-
tional financial crises contagion within a short time between two countries. The two countries’ control force
depending on the timely financial assistance, the positive attitude and actions to rescue other infected coun-
tries, and investor confidence aggregation, and the immunity ability of the infected country are considered as
the major reasons to drive the nonlinear fluctuations of the stock return rates in both countries during the cri-
sis. According to the Ordinary Differential Equations Qualitative Theory, we found that there are three cases
of financial crises contagion within a brief time between two countries: weak contagion with instability but
inhibition, contagion with limit and controllable oscillation, and strong contagion without control in a brief
time.
Keywords: Financial Crises Contagion, Stock Return Rate, Nonlinear Dynamics Model, Limit Cycle
1. Introduction
Since 1990s, the international financial crises have fre-
quently broken out, and their contagions also greatly
increased. The crises like the infectious diseases often
spread quickly from the earliest crises country or region
to others after their outbreak. For instance, the 1994 “te-
quila crisis” in Latin American countries, the 1998 “Rus-
sian virus” and the 1997 Southeast Asian financial crises
all are the regional financial crises with strong contagion.
However, the crisis trigged by 2007 American sub-prime
crisis has more strong contagion, and has developed into
a global financial crisis. It has leaded some countries into
economic crisis and given a profound impact on the or-
der of global economic development. Therefore, we pay
particularly our attention to the dynamic transmission
problems of the financial crisis within a short time after
its outbreak.
Many researchers have been focused on the financial
crisis and its contagion test. But they show the different
empirical conclusions under the different definitions of
financial crisis contagion. Sometimes the different results
may be found even under the same definition framework.
For example, the financial crisis contagion is defined as
the significantly enhanced correlation among the financial
markets after the crisis outbreak. Reference [1-3] find
there are the significantly enhanced correlations in cross-
border capital markets when they study some major fi-
nancial crises. But [4] strictly separates the contagion
and mutual dependent degree, and taking account of the
different conditions variance, they don’t find the evi-
dence that the correlations between the various markets
are destroyed during those major crises. For the reasons,
[5] have pointed out that the correlation coefficient is
only used to describe a linear relationship between the
two financial markets, and it is not suitable for studying
the non-linear changes. Reference [6] also put forward
that without considering the conditional heteroskedastic-
ity, the results of correlation test are biased. Furthermore,
[7] suggest that these differences in definitions and tests
are small, even the same under certain conditions. And
after classifying the empirical research methods of finan-
cial contagion and analyzing their similarities and dif-
ferences, they explain that some models reflect all of the
information while others take part of the information.
Therefore, we follow the strict distinction on the infec-
tious and mutual dependence in [4], but don’t distinguish
the amount of information in the model. Then we inves-
* This work has been supported by the Research Fund of Program
Foundation of Education Ministry of China (10YJA790233)

K. CHEN ET AL.
18
tigate the financial crises contagions between two finan-
cial markets by analyzing their mutual impact changes
within a short time during the crises. Whatever nonlinear
effect is between two financial markets, the contagion
may occur if those changes are in an unstable state. Oth-
erwise the markets keep the mutual dependence rela-
tionship.
As we have the different angle from the previous lit-
erature to analyze financial crisis contagion, it may not
be rational for us to employ directly those non-linear
research methods based on the previous definition. Al-
though those methods can capture more nonlinear char-
acteristics of the financial crises contagions such as the
minimum spanning tree method, Copula functions, see-
mingly unrelated Probit techniques, GARCH model,
symbolic time series analysis, dynamic factor model and
other methods [8-14], it has been an outstanding issue
how to reduce the test errors of financial crisis contagion
between two markets. So we try to introduce the differ-
ential dynamics methods to set up a non-linear model of
the financial crisis contagion between two markets. Ad-
ditionally, we use the Ordinary Differential Equations
Qualitative Theory to describe the mutual impact state
between two countries or two financial markets within a
short time during the crisis, then to discuss the infectious
state or path.
The remainder of this paper is organized as follows.
Section 2 focuses on setting to the nonlinear contagion
model by using the differential dynamics methods. Sec-
tion 3 shows the major theorems based on Ordinary Dif-
ferential Equations Qualitative Theory and their proofs,
and suggests the financial crises contagion cases between
two markets. The last section summarizes our studies.
2. Modeling
The financial crisis contagion firstly affects the financial
security of a country. It may lead to the great volatility of
the financial asset prices (such as stocks, bonds, curren-
cies, real estate, etc.), deteriorate the operating conditions
of the financial institutions, cause the capital flight, de-
cline the foreign exchange reserves, and increase the
foreign debt. And one of the first performances during
crisis is the great volatility of the stock price matched
with time very well, so that it becomes one of the most
common means to use the stock prices or the volatility of
the stock return rate for analyzing the financial crisis
contagion. Thus, we put forward the nonlinear dynamics
model of financial crisis contagion between two coun-
tries by examining the dynamic change trends of the
stock return rate after the crisis outbreak.
On the one hand, observing directly the changes of the
stock return rate in the market, we found that when one
country has a financial crisis, its stock returns may drop
drastically and quickly affect other countries’ stock
market in a short time. Then there is a crisis contagion
among the crisis country and its neighboring countries or
countries with a close contact. On the other hand, sum-
marizing the previous studies, we recognize that the de-
gree or efficiency the crisis country affecting on other
countries usually depends on the affected country’s im-
munity ability, control force and investor confidence
aggregation. The immunity ability is determined by the
fundamental factors such as economic strength, eco-
nomic structure, financial system safety, financial market
openness, management level exchange system, etc. The
control force depends on the timely financial assistance,
the positive attitude and actions to rescue other infected
countries. And the growth of investor confidence aggre-
gation comes largely from the increase of first two capa-
bilities. Furthermore, we suggest that before the crisis,
one country’s average changing rate of stock returns de-
pends on its immunity ability, and reflects the interde-
pendence relationship with the crisis country. However,
during the crisis, affected by itself control force and in-
vestor confidence aggregation as well as the crisis coun-
try’s stock return changes, one country’s stock returns
may move away from its original average rate and have
nonlinear fluctuation following the change of the crisis
country’s stock market in a short time. Meanwhile, the
crisis country’s stock returns keep dropping sharply
within a short time, and also having the nonlinear volatil-
ity impacted by the control force and investor confidence
aggregation of itself and other countries.
Thus, we assume that the nonlinear function of the
stock return rate in two markets can be conducted to re-
flect the above nonlinear volatility affected by the con-
trol force, investor confidence aggregation and other
hidden factors in the two countries. And we adopt the
power function to describe that the impact on the other
countries is far greater than that on the crisis countries.
As a preliminary discussion, we constructed a nonlinear
dynamic model with a minimum power law, as follows:

2
112
2212
drdtar r
drdtrbr r





(1)
where a, bare positive constants, a is the increasing
rate of the average stock returns of A country under the
normal situation, and b is the decreasing rate of the
stock returns of B country. 1 r, 2 r are respectively the
stock return rates of A and B country. At the dynamic
angle, (1) show that:
1) Within a short time after the crisis outbreak, the
stock market of A country can’t develop with a
constant speed a due to the nonlinear effect of B
country, the crisis one. Here 2
2
r is similar to the

K. CHEN ET AL.
19
variable coefficient of stock returns in A country.
When 2 r is small, the financial crisis has a small-
er impact on the average stock returns of A country.
But when 2 r is becoming larger, the impact of fi-
nancial crisis on the average stock returns of A
country is going to grow up at the square speed of
the larger 2 r.
2) At the first period of the crisis, the increase rate of
the average stock returns in B country changes
with the speed 2 br. That means its return rate
may decrease gradually in accordance with expo-
nential law. When the contagion develops to a cer-
tain extent, this dropping speed becomes 12 brr


under the inter-effect of the stock returns in two
countries. Additionally, there are: a) When B
country increases its control force or the others rise
up their control force and investor confidence ag-
gregation, the dropping speed of 2
r may be con-
trolled, namely120rr . b) When those measures
don’t make any sense, the stock returns of B coun-
try may have an accelerative decreasing trend,
namely 12 0rr .
3. Analysis
At first, one can easily find the singular point in (1) is

2,obaab.
Exchanging 1 rand 2 rin (1), and moving the singular
point to the origin of coordinate, (1) can be transformed
into the following form.
22
11211 212
22222 2
2121211
(2 )
2(2)


 


drdtbrarbrbraarbr r
drdtbra rbb rarrarb
(2)
According to the Ordinary Differential Equations Qu-
alitative Theory, it is easy to find the conclusions as fol-
low.
Theorem 1 The characteristics of the singular point

2,obaab in (1) are:
1) if 23 ab, o is a stable node or focus;
2) if 23 ab, o is an unstable node or focus;
3) if 23 ab, o is a center point or focus.
Proof: The characteristic equation of the correspond-
ing linear part of (1) at point o is:

222 22
0ab bab  
Notice that22
pabb and 22 0qab. Thus, if
0b, ois the saddle point. If 0p, namely 23 ab,
ois the stable node or focus. And if 0p, namely,
ois the unstable node or focus. And if 0p, namely
23
ab, o may be a center point or focus.
Lemma Equation (1) may be described as the Lienard
Equation. That is:


dx dtyFx
dy dtgx
 




(3)
where




222 22
2
F
xxbxbxaab axbx ,






42 22
2
g
xaxbaabxbx
Proof: Let
2
12
,
x
rbayrab
Then it is easy to obtain:
2222 2
2222 2
22
2
dx dtax bbyaxybbyaxy
dydta x bbyaxy bbyaxy
 




(4)
Exchanging x and y in (4), and still mark x and y, then
one can get


 
01
01
dx dtAxAxy
dy dtBxBxy








where


22
0
A
xbxbxa ,

22
02Bx bxbxa 




22 2
11 2
A
xBxabaxbx

 
Thus it can be proved in accordance with Lemma 6.3
proposed by [15].
Theorem 2 If 0b and 23 ab, the singular point
of (1) is an unstable focus.
Proof: Introduce the following binary function,




22
1211223 124 12
,2,,
H
rrrrr rrrrr   
where 3

, 4

are respectively three homogenous po-
lynomial and four homogenous polynomial. When 0 cis
a sufficient smaller positive number,

12 0
,
H
rr c

in-
dicates a cluster of closed curves containing the origin.
Considering the differential dynamical system as follow:
22
1121 1212
22
212 112 12
2
22
drdtbrbrrbbrr rr
drdtbrbrrbbr rr r
 




(5)
After simple calculation, one can obtain



2
22
12 512
5,dH dtb rrhrr
Here


512
,hrr is a polynomial with less than 5 de-
gree.
Denote


34
2913 8
s
bbb ,

324
13 4tbb
And give the specific forms of 3

, 4

in (5) as fol-
low:


32 23
3121121231242
,r rmrmrrmrrmr   


432234
41211212 31241252
,rrnr nrrnrr nrrnr  
where the coefficients are respectively

K. CHEN ET AL.
20


322322
12
125 3,63mbbmb b ,
 
32 2322
34
42, 423mb bmbb
5
12 2
110 8,2020nstn st 
95
345
416
1818,44,  nstnstnst
Then, due to 0b, it can be proved from (5) that the
singular point o in (1) is an unstable focus.
Theorem 3 If 23 ab, there is no limit cycle in (1).
Proof: Denote
 
2
,2
x
yy Gx  , where
 

43
0
42
ln
x
Gxgzdza ba bxa
ab xabx



  
2
22 1
a
b
x
Fxx bxxbxa 

 
if,0 0xabxFx x 
And there is
 
2
4
g
xab abxx
.
Then




400,ddtgxFxxx ab
  
So (4) does not possess any limit cycles. Then there is
no limit cycle in (1).
Theorem 4 If 23 ab, there is an unique limit cycle
in (1).
Proof: According to our Lemma and the Theorem 6.6
proposed by [15], one can get
1) There is0100a

. Then32 0ba, namely
23
ab;
2) There is 00

. Then 20a, namely 0a

;
3) There is 21 40a

. Then 0b;
4) There is 30

. Then 23 20ab
Summarizing the above four conclusions, one can find
that Theorem 4 is true.
According to Theorem 2-4, we conclude that there are
three cases of financial crises contagions between two
countries: weak contagion with instability but inhibition,
contagion with limit and controllable oscillation, and
strong contagion without control in a brief time.
Case 1, when 23 ab (0b), Theorem 4 indicates
that the (1) has unique limit cycle. Its phase plane shows
that there occurs an alternating oscillation of stock re-
turns between two countries. However, this oscillation
may not enlarge without any limitation due to there ex-
ists a limit cycle. The immunity ability and self-repair
capacity of the economy system in both two countries
may limit the oscillation magnitude within some con-
trollable size, which depends on the size of the limit cy-
cle. So it is a contagion case with limit and controllable
oscillation.
Case 2, when 23 ab (0b), Theorem 2 suggests
that the singular point of (1) is not a center but an unsta-
ble focus. Due to there is a little difference between cen-
ter point and focus, it show that during the first period of
financial crises contagions, the stock return rates of two
countries increase gradually their oscillation magnitude
by a imperceptible way. Thus this stage should be the
best time to control the financial crises contagions. We
call it as the weak contagion with instability but inhibi-
tion.
Case 3, when 23 ab (0b), Theorem 3 dominates
that there is no limit cycle in (1). In this case, the finan-
cial crises contagions have evolved into a disaster and
both the two countries have to endure more and more
strong impacts. The governments must take the firmest
monetary policies and fiscal policies to curb its spread.
4. Conclusions
Within a short time after the crisis outbroke, the stock
returns of crisis country would be to drop abruptly and
cause quickly the stock returns volatility of other coun-
tries. Then there is a nonlinear contagion of the financial
crisis. To investigate the above phenomenon, we intro-
duce the differential dynamics methods to construct a
simple nonlinear dynamic model and derive four theo-
rems in accordance with the qualitative theory of differ-
ential equations. Furthermore, based on the discussions
about the stability of focus and the existence of limit
cycle in the nonlinear volatility equations of stock re-
turns, we analyze the financial crisis contagion situation
between two countries during the crisis, and find there
are three cases of financial crises contagions within a
short time after the crisis outbroke. First one is that if the
average increasing rate square of stock returns in the
infected country is more than the decreasing rate cube of
stock returns in the crisis country, there is no limit cycle
in the nonlinear dynamic model of financial crisis conta-
gion. Thus there is a strong contagion between the two
countries to be controlled difficultly within a short time.
Second one is that if the former is less than the latter,
there is a unique stable limit cycle in this model. Thus
there is an oscillation contagion between the two coun-
tries, whose oscillation magnitude depends on the size of
limit cycle. Third one is that if the former equals to the
latter, this model has an unstable focus. Thus there is a
weak contagion between the two countries. It is easy to
be adjusted to the interdependence state before the crisis.
Our analysis results are closer to the actual state of fi-
nancial crisis contagion within a short term between the
two countries. For example, Hong Kong Monetary Au-
thority expected the U. S. subprime mortgage crisis will
not create systemic impact on Hong Kong banking sys-
tem in its paper submitting to Hong Kong SAR Legisla-
tive Council in January of 2008. Later this point was also
proved. Using Hong Kong’s Hang Seng Index and U. S.
S & P500 stock index, we calculated the stock returns by

K. CHEN ET AL.
21
the logarithmical return method and their increasing or
decreasing rates. Then we found that before the outbroke
of the subprime crisis, the average growth rate of Hong
Kong stock returns was about 4% from 2006 to July of
2007; and that the average decreasing rate of U.S. stock
returns was 600% or more from July to August in 2007.
It implies there may be “23 ab” in a shorter time after
the subprime crisis outbroke. So the crisis contagion be-
tween U. S. and Hong Kong could be controlled by tak-
ing some measurements even if the prices of Asia-Pacific
stock markets greatly dropped down in August of 2007.
It indicates that the differential dynamics method is a
better way to investigate the financial crisis contagion
state or path between the two countries. However, as a
preliminary discussion, this paper introduced a power
function with the lowest power times into a simple non-
linear dynamic model of crisis contagion. The simple
form may be extended to others for the further study of
variable contagions among three or more countries. In
addition, another further work may be doing some em-
pirical research such as discussing the calculating meth-
ods of those two indicators in our simple nonlinear
model, testing the contagion states by employing the
time series data of stock return, trying to contract some
early-warning indicators to monitor timely the volatility
of financial markets, and so on.
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