SOME NOTE ABOUT THE
EXISTENCE OF CYCLES AND
CHAOTIC SOLUTIONS IN
ECONOMIC MODELS
by
Beatrice Venturi
Department of Economics
University of Cagliari
Italy
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EXISTENCE OF CYCLE AND CAOTIC
SOLUTIONS IN
ECONOMIC-FINANCIAL MODELS
We analyze the global structure of
a tree-dimensional abstract
continuous time stationary
economic model
that includes some
determinates parameters.
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THE MODEL
We shall consider a generic
non-linear first order system with some
structural parameters:



x  f  ,  , x , x , x ,  ,   , i  1,2...n
1
i 1 2 3
i
2  g   , i , x1, x2 , x3 
x
3  h  , i , x1, x2 , x3 
x
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Where f, g and h are complicate non-linear
functions of class C2 (twice continuously
differentiable) in all their arguments.
The parameters:
,i  0
i  , i 1,2...n
are real and positive.
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A stationary (equilibrium) point
of our system is any solution of :
 x
 x
 0
x
1
2
3
Assuming the existence of
such
a solution at some point
P*   x*, x*, x* 
2 3
 1
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THE JACOBIAN MATRIX

The local dynamical properties of system
from (2.1) to (2.3), at a hyperbolic
equilibrium point P*, can be described in
terms of the Jacobian matrix , for brevity.
*
J  P *  J
In fact, the nature of the eigenvalues of J*,
plays a key role.
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We consider the system as a one-parameter
family of differential equations dependent
of the parameter .
We fixe the other parameters .
We assume that in our model
exists a parameters set where the Jacobian
*
J  P *  J
has two eigenvalues complex conjugate :
 ( ),  
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EMERGENCE OF STABLE CYCLES
First we use rigorous arguments to show as an
equilibrium of the model could be destabilized
into a stable cycle in the dynamic of R3 under
the following two alternative assumptions:
a) A steady state has three stable roots.
b) A closed orbit has a two dimensional
manifolds in which it is asymptotically stable .
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APPLICATIONS

Next we apply these results to a general
non-linear fixed-price disequilibrium
IS-LM model as formulated by
Neri U. and Venturi B. (2007).
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APPLICATIONS

Neri U. and Venturi B. (2007) discuss the
effect
of a change of the adjustment
parameter in the money market, via the Hopf
bifurcation’s approach in a three dimensional
fixed-price disequilibrium IS-LM model.
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THE ECONOMIC MODEL
S = savings,

T= tax collections,

G=government expenditure,

B = interests payment on
perpetuities,

L = liquidity preference
function

m = is the real money
supply.

I = investment,

r =interest rate,

y = output (income)

w =wealth

α >0 and µ>0
are the adjustment parameters
in their respective markets

.
y   I r, y   S y D , w  G  T  y  B


 
.
r  Lr, y, w  m
.
m  G  B  T ( y)
(2)
(3)
(1)
Specifically, our “generalized model” is a non-linear system
in the independent state variables r, y and m.
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THE ECONOMIC MODEL
(1) describes the traditional disequilibrium of
dynamic adjustment in the product market;
(2) describes the corresponding disequilibrium in
the money market;
(3) represents the governmental budget
constraint.
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THE ECONOMIC MODEL

Next, we define the disposable income y
and the wealth w as follows:
y  y  B  T  y  B
D
B
wm
r
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THE ECONOMIC MODEL


We assume that the functions: I, S, T are of class
C2
Recall that a stationary (equilibrium) point of our
system is any solution of.
y  r m  0
Assuming the existence of such a solution at
some point P*(y*,r*,m*), we want to analyze its
local properties (e.g. stability, etc.) around P*.
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THE ECONOMIC MODEL
We shall rewrite our system in the equivalent
form:
r  Lr, y, w  m  f r, y, m
y    I r, y   S  y D ( y),w   G  T  y  B  g r, y, m




  G  B  T ( y) h y 
m
with h:
R R
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THE ECONOMIC MODEL
Remark1: Unlike Schinasi G.J. ,1982 , we allow the
functions L and S (liquidity and savings) to depend on
wealth w.
Since G and B are fixed a choice of the policies, implies:
'h  y   T
y
because all deficit must be financed either by creation
of money or by creation of new debt.
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THE ECONOMIC MODEL

The monotonicity, of the restrictions:
,
r  I r ,.,. r  Lr ,.,. w  L(.,., w) y S ., y,. w S .,., w
,
,
is assumed and these assumptions imply the
(economic) conditions:
.
Ir  0
Lr  0
Lw  0
Sy  0
Sw  0
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Dynamical analysis

Theorem 1. A hyperbolic stationary point of the
system (1), (2), (3) is locally asymptotically stable
if the following assumptions hold at:
   f r  g y  N 
1
with N given by (*) at P* :
N = the first integer such that N a2> a3 , at P*,
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Dynamical analysis
i) the marginal propensity to invest out
of income is greater than unity
0  Ty  1
I  S y 1  T   T g y  0

y
y 
y
ii) S w f r  (1  Lw ) g r 0  Lw  1
iii)
g 0
r
 f g  g L     1S T
 r y r y
w y


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Dynamical analysis

Corollary 1. Let the hypotheses of Theorem 1 hold at
a hyperbolic stationary point of the our system.
Then the steady state for each α >0 and all
 
   f  1g  P*   N     P*   0
r  y     
is locally asymptotically stable.
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Dynamical analysis

Corollary 2. Assume Corollary 1 and let
J*=J(P*) as before.
Then, there exists a value µ>0 for which
J* has a pair of purely imaginary eigenvalues.
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Theorem 2. Assume the hypotheses of Theorem1
1


except that    fr g y  N 
is now replaced by :
 g  P*
  y 
0  
2






*


 f  P 
r  
Then, there exists a continuous function µ(δ) with
µ(0)=μ’ and for all δ small enough , there exists a
continuous family of non-constant positive
periodic solutions
[r*(t ,), y*(t,), m*(t,)]
for the dynamical system (1), (2), (3) which collapse
to the stationary point P * .
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THE ECONOMIC MODEL
REMARK For a three-dimensional
non linear dynamical model the general version
of the
Hopf bifurcation theorem,
ensures the existence of a
small amplitude periodic solutions
bifurcating
from the steady state only
in the center manifold
(a two-dimensional subspace of 3).
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Dynamical analysis

From an economic point of view, sub-critical or
super-critical orbits are both reasonable.

Since the third real root of the Jacobian matrix is
negative the existence of a super-critical Hopf
bifurcations becomes very interesting in the
analysis of macroeconomic fluctuations for a ISLM model

A stable economy, by the increase or decrease of
its control parameters, could be destabilized into a
stable cycle in the dynamic of R3
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CONCLUSIONS

The sub-critical Hopf bifurcations may correspond
to the Keynesian corridor (Leijonhufvud, 1973):
the economy has stability inside the corridor
while it will loose the stability outside the corridor.

In such a case the dynamics are either converging
to an equilibrium point or the trajectories go
somewhere else, and it is also possible that another
attracting set exists, but often the alternative is
diverging trajectories.
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CONCLUSIONS

We have seen that fluctuations derive from the
mechanisms through which money markets reflect
and respond to the developments in the real
economy.

Our analysis provides an example of the classical
thesis concerning endogenous explanations to the
existence of fluctuations in some real world economic
variables
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The general version of the
Hopf bifurcation theorem
For a three-dimensional non linear dynamical
model the general version of the Hopf
bifurcation theorem, ensures the existence of a
small amplitude periodic solutions bifurcating
from the steady state only
in the center manifold
(a two-dimensional subspace of 3).
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Shil’nikov Theorem
Shil’nikov showed that if the real eigenvalues
has large magnitude than the real part of the
complex eigenvalues of the
Jacobian of system,
then there are horseshoes present in return
maps near the homoclinic
orbit of the model.
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The orbits, in the dynamics of the center
manifold, can generally be either
attracting or repelling.
In the case of an attracting orbit
(the so- called sub-critical case)
trajectory on the center manifold are locally
attracted by this orbit,
which becomes a limit set.
.
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In this situation the stationary point
is an unstable solution
meanness from an economic point of view
(unless the initial conditions happen
to coincide with the stationary value).
Conversely, if the cycle is unstable
(the so- called super-critical case)
the steady state is attracting.
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This case is particularly relevant from
an economic point of view
only
if the initial conditions imply that
the economy fluctuates right from the beginning.
The study of stability of emerging orbits
on the centre manifold can be performed
by calculating the sign up of-up-third order
derivative of the nonlinear part of the system,
when written in normal form.
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In fact, the Hopf bifurcation theorem proves the existence of
closed orbits but it gives no information on their number and
their stability.
Using the non linear parts of an equation system,
a stability coefficient
(as formulate for example by Guckenheimer J.- Holmes
P.,1983)
may be calculated in order to determine
the stability properties of the closed orbits
(see Foley , 1989, Feichtinger ,1992, Mattana - Venturi,1999,
Anedda C. -Venturi B. ,2003).
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Mattana P.- Venturi B. , 1999, analyzed a simplified
three-dimensional version of Lucas’s model,
a two sector endogenous growth model with externality.
Considering the externality as a bifurcation
parameter, they proved, the existence of small
amplitude periodic solutions, Hopf bifurcating from
the steady state in the center manifold.
Venturi B., 2002, have elaborated a numerical
simulation of this model.
.
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In Fig.1 is plotted the dynamics of one orbit
Hopf bifurcating from the steady state
of the reduced version of
the Lucas model
(see Venturi B., 2002)
The orbits is super-critical
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Figure 1
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THE ECONOMIC MODELS
 We
review a generalized two sector models of
endogenous growth, with externalities, as formulated
by Mulligan B.- Sala-I-Martin X.,1993.
 We
show that in this class of economic models,
considering the externality as bifurcation parameter, the
conditions of existence of periodic orbits and chaotic
solutions come true.
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The Mulligan B. - Sala-I-Martin X. model deal with the
maximization of a standard utility function:
 c1  1  t
max 
e dt
0 1
where
c = is per-capita consumption
= is a positive discount factor
= is the inverse of the intertemporal elasticity of
substitution.
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
The constraints to the growth process are
represented by the following equations





k  A(u (t ) u h(t ) h )(v(t ) v k (t ) k )hˆ(t ) hˆ kˆ(t )
  k (t )  c(t )
k





kˆ


h  B((1  u (t )) u h(t ) h )((1  v(t )) u k (t ) k )hˆ(t ) kˆ(t )
hˆ

kˆ

  h(t )
h
k (0)  k  0,
0
h ( 0)  h  0
0
k =is the physical capital,
h =is the human capital
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 k  h  1
k  h  1
 h  u ,
k  v
k, , h are the private share of physical and the
human capital in the output sector
 h  u ,
k  v
k , h are the private share of human
capital in the education sector
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u and v =
are the fraction of aggregate
human and physical capital
used in the final output sector at instant t
(1- u) and (1- v)
are the fractions used in the education sector,
A and B
are the level of the technology in each sector,
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 k̂
is a positive externality, parameter in the production of
physical capital
 ĥ
is a positive externality parameter in the production of
human capital
All the parameters:
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live inside the following set:
= (0, 1)(0, 1) (0, 1)  (0, 1) (0, 1)  (0,1)
(01) (0  1) (0  1)  4.
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The representative agent’s problem is solved by
defining the current value Hamiltonian:
c1  1
H

1
ˆ
ˆ






  A(u (t ) u h(t ) h )(v(t ) v k (t ) k )hˆ(t ) h kˆ(t ) k   k (t )  c(t ) 
1
k



ˆ ˆ






   B((1  u (t )) u h(t ) h )((1  v(t )) u k (t ) k )hˆ(t ) h kˆ(t ) k   h(t )
2
h 


i, i = 1, 2 =Lagrange multipliers
(co-state variables).
 = is a depreciation factor
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and obtaining the first-order necessary conditions for
an interior solutions.
Plus the usual two transversality conditions:


t
lim e
1(t ) k (t )  0
t 


t
lim e
2 (t )h(t )  0
t 
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The general model just presented collapses to Uzawa-Lucas’
when depreciation  is neglected and the following
restrictions are imposed:
v     0
k
 ˆ   ˆ  v   k  0
k
h
u  h  1  k
u  h  1
A  B 1
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According to the strategy used by
Mulligan B.C. - Sala-I-Martin X.,1993,
we express the two multipliers 1 and 2
in terms of their corresponding control variables
c and u
and obtain an autonomous system of
four differential equations in
the four variables k, h, c, u.
A solution of this autonomous system is called a Balanced
Growth Path (BGP) if it entails a set of functions of time
solving
the optimal control problem presented such that
k, h and c grow at a constant rate and u is a constant.
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We choose a standard combination of the original variables
that is stationary on BGP.
1   /   1



x1  h
k
x u
2
c
x3 
k
We get a first order autonomous system of
three differential equations
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B.G.P that does not include positive externalities admits only
the saddle-path stable solutions
(see Mulligan B.C. - Sala-I-Martin X.,1993).
We get a first order autonomous system of three
differential equations
 1 1 
x1  x1
x2
  (1  x 2 ) x1  x1 x3

  1
2
x 2  x2  
x2  x2 x 3

 1 1 
x 3  x  x1 x 2 x3  x3
2
3
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Where:



  ;











1





.
 1
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We put our model into normal form
(see Guckenheim - Holmes 1983, pp. 573).
The system becomes:
w 1  TrJ * w1  F w1 , w2 , w3 ,  
1
w 2  w2  BJ *w3  F w1 , w2 , w3 ,  
2
w 3  BJ *w2  w3  F w1 , w2 , w3 ,  
3
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In the equilibrium point is translated the origin P*(0,0,0) of
the model and the eigenvalues are
linearized in P*.
The real eigenvalue is
  TrJ
R
*
and the complex conjugate eigenvalues are
 j   i B
j  1,2
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We consider the dynamic of the system when the
parameter values are evaluated in a parameters
set where emerges a super-critical Hopf
bifurcation.
We choose the value 
(the exponent of the externality factor )
very close to the bifurcation value *.
Setting:  = 0.75;  = 0.055;  = 0.054;
 = 0.1 and  = 0.05,
We found that the system has a homoclinic orbit.
(See Fig. 3.)
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Figure 2
-15
-20
-25
-30
-35
20
0.25
0
0.2
0.15
-20
0.1
-40
0.05
0
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The loss of stability and the presence
of periodic solutions Hopf bifurcating from
the steady state
can lead to the occurrence of
an homoclinic orbit, Fig. 2,
and,
under Shil’nikov assumption,
of chaotic solutions
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The aim of this work is to point out some basic
ideas that may be useful to prove the transition to
bounded and complex behavior,
and to explain how
the presence of Hopf bifurcations in a general
class of economic-financial models can be
interesting from
an economic and dynamic point of view.
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