Transcript Nessun titolo diapositiva

2-dim discrete dynamical
systems: iterated maps of the plane
y
T : 2  2
 xt 1  f ( xt , yt )

 yt 1  g ( xt , yt )
Example
 xt 1  axt  yt

2
y

x
t b
 t 1
T
T
T
T
x
Cournot Duopoly
Two firms produce a homogeneous good and interact in a competitive market,
choosing the quantities: q1 (t) e q2 (t)
Inverse demand function: p = f (q1+q2) = a – b (q1 + q2)
Production costs: ci (q1, q2) = ci qi , i = 1,2
Each period profit: Pi qi f (q1+ q2) – ci (q1, q2)
At each stage, they simultaneously decide, solving the problems
maxq1 P1 q1, q2 
maxq2 P2 q1, q2 
On the basis of the previous assumptions, we obtain
a  c1
1 ( e)

(e)
q
(
t

1)

r
(
q
(
t
))


q
(
t
)

1 2
 1
2 2
2b

q (t  1)  r (q (e) (t ))   1 q ( e) (t )  a  c2
2 1
 2
2 1
2b
And assuming naive expectations: q j (t 1)  qi (t ) i  1,2
a  c1
1

q
(
t

1)


q
(
t
)

2
 1
2
2b

q (t  1)   1 q (t )  a  c2
1
 2
2
2b
e
Higher order difference equations
Example: xt+1 + bxt + cxt-1= 0
With two initial conditions: x0, x1
Let yt = xt-1
Then the difference equation in transformed into:
xt+1 = bxtcyt
yt+1 = xt
Nonautonomous difference equation.
Example. xt+1 = f(xt,t)
with i.c. x0 given
Let yt = t
Then
xt+1 = f(xt,yt)
yt+1 = yt + 1
with i.c. x0 given ; y0 = 0
xt 2  2xt 1  xk  3ut  0
xt 1  yt ; zt  t
 xt 1  yt

 yt 1  xk  2 yk  3u ( zt )
z  z 1
t
 t 1
Linear 2-dim. map
 xt 1   a11 a12   xt 
 y   a a   y 
 t 1   21 22   t 
Remembering the case of 1-dim linear maps let’s consider the trial solution:
 xt 
t


y 
 t
 v1 
v 
 2
And substitute it in the law of evolution:
 t 1v1  a11 t v1  a12 t v2
 t 1v2  a21 t v1  a22 t v2
And after dividing by t we get (A I v = v
i.e. the proposed trial is a particular solution provoded that
L is an eigenvalue and v is a corresponding eigenvector for the matrix A
Characteristic equation det (AI) = 0 becomes:
P() = 2 Tr∙ + Det = 0
where Tr = a11+a22 ;Det = a11a22 a12a21
(I) D=Tr24Det >0 then 1 and 2 real and distinct eigenvalues exist
with correnponding linearly independent eigenvectors v1, v2, that give
rise to two independent soutions  t v and 2t v2
1 1
(II) D=0 coincident eigenvalues , with eigenvector v give two
independend solutions tv and ttv
(III) D <0,
1,2=  1 Tr  i D  ei    cos  i sin   with   Det ; cos   Tr
2
2
Two independent complex conjugate solutions
2
Any linear combination of solutions is a solution, hence the generic
solution of the linear homogeneous system is;
(I) Real and distinct eigenvalues of A, 1 and 2. Denoting by v1 e v2 two
eigenvectors respectively associated with them, we obtain
xt  C1v11t  K1v 22t
(II) Real and equal eigenvalues of A:
xt  c11t  c 2t 1t
where c1 and c2 are two suitable vectors dependent on two arbitrary chosen
constants
(III) Complex conjugated eigenvalues, the real part and the imaginary part of the
two independengt complex solutins are solutions,being:



1
1
Re z  z  z ; Im z 
zz
2
2i

xt   t   c1h1  c1h 2  sin( t )   c1h1  c2h 2  cos( t ) 
where h = h1 + ih2 is an eigenvector associated with .
Stability of the unique equilibrium:
Im
||<1
i.e. all eigenvalues
inside the unoi circle
of the complex plane
1
-1
1
Re
-1
The origin is an asymptotically stable equilibrium point iff all the
eigenvalues are smaller than 1 in modulus. Local stability and global are
equivalent
The origin is stable, but not asymptotically, iff the modulus of the eigenvalues
is not larger than 1 and all the eigenvalues with unit modulus are regular
Otherwise the origin is unstable
STABLE NODE
Im
UNSTABLE NODE
Im
1
1
-1
1 Re
-1
1
-1
Re
-1
Im
SADDLE
Im
SADDLE
1
1
-1
-1
1
1
Re
Re
-1
-1
Im
1
-1
1
1
-1
UNSTABLE FOCUS
Im
STABLE FOCUS
Re
-1
1
-1
Re
Im
CENTER
1
-1
1 Re
-1
IMPROPER NODE
STAR NODE
• Second order
– real and distinct eigenvalues:
xt  C1v11t  K1v 2 2t
• if |1| < 1and |2| < 1 , the origin is globally
asymptotically stable (stable node)
• if |1| > 1and |2| > 1 , the origin is unstable (unstable node)
• if |1| < 1and |2| > 1 , the origin is unstable (saddle)
– equal eigenvalues :
xt  c11t  c 2t 1t
• if || < 1, the origin is gloablly asymptotically stable (stable node)
• if || < 1, the origin is unstable (unstable node)
• if the matrix A is diagonal: the origin è stable if || < 1, unstable if
|| > 1 (star node)
i
    e
– complex conjugated eigenvalues
xt   t   c2h1  c1h 2  sin  t   c1h1  c2h 2  cos t 
• if  < 1, the origin is globally asymptotically stable (stable focus)
• if  > 1 , the origin is unstable (unstable focus)
• if  = 1, the origin is stable (center)
Stability triangle
unstable node
1+Tr+Det=0
(Flip curve)
D = Tr24Det=0
1Tr+Det=0
(Fold curve)
unstable focus
center
if
detA = 1, -2<trA<2
Det= 1 (N-S curve)
stable focus
saddle
stable
node
saddle
unstable node
center
if
Det = 1, -2<Tr<2
a) 1  trA  det A  0
b) 1  trA  det A  0
c) 1  det A  0
Cournot Duopoly
The model we considered is described by the system of two I^ order linear
difference equations
a  c1
1

q
(
t

1)


q
(
t
)

2
 1
2
2b

q (t  1)   1 q (t )  a  c2
1
 2
2
2b
The matrix of the system is
 0
1 
2
A
 1
0 
 2

with distinct real eigenvalues: 1,2   1 2
 1
1
1
1

,
and the eigenvectors associated with are
2   2
1
1



2
Solution:
 1
 1  a  c2  2c1
xn  h    k   
3b
 2
2
n
n
 1
 1  a  2c2  c1
y n  h    k   
3b
 2
2
n
1.5
1
0.5
n
0
-0.5
2
4
6
8
10
• Easily extended to dim >2
– The origin is an asymptotically stable equilibrium point
iff all the eigenvalues are smaller than 1 in modulus.
Global stability in IRn
– The origin is stable, but not asymptotically, iff the
modulus of the eigenvalues is not larger than 1 and all
the eigenvalues with unit modulus are regular
– Otherwise the origin is unstable.
Nonlinear maps of the plane: local stability of a fixed point
f ( x, y)  x  0
Let (x*,y*) be a solution of :
g ( x, y)  y  0
Linear approximation around (x*,y*)
f
f

x

f
(
x
*,
y
*)

|
x

x
*

|( x*, y*)  y  y *  h.o.t.


( x*, y *)
 t 1
x
y


g
 y  g ( x*, y*)  g |
x

x
*

|( x*, y*)  y  y *  h.o.t.


t 1
( x*, y *)

x
y

f
f

 xt 1  x*  x |( x*, y*)  xt  x *  y |( x*, y*)  yt  y *  h.o.t.


g
 y  y*  g |
x

x
*

|( x*, y*)  yt  y *  h.o.t.


t 1
( x*, y *)
t

x
y

Linear homogeneous system in X = xx* ; Y = yy*
 X t 1  a11 X t  a12Yt

Yt 1  a21 X t  a22Yt
 a11
With A  
 a21
a12 
a22 
Jacobian matrix
Stability of the equilibrium points
• An equilibrium point x* is locally stable if for any
neighborhood U of x* there esists a neighborhood VU
such that any solution starting in V belongs to U for any t.
• If V can be chosen such that
x  t   x* ,
t  
x* is said locally asymptotically stable
• An equilibrium point is unstable if it is not stable
• If x* is an asymptotically stable equilibrium point, the set
of the initial condition giving rise to the trajectories
converging to x* is the basin of attraction of x*
• If the basin of attraction of x* coincides with the whole
state space W then x* is globally asymptotically stable.
Local bifurcations in a discrete
dynamical system
xt 1  f  xt , 
with f  0,0  0
• There are different ways to exit the unit circle:
1  0  1
1  0  1
1,2  0   e i0
Fold bifurcation
Flip bifurcation
(period doubling)
Neimark-Sacker
bifurcation
Bifurcattion lines and the creation of new invariant sets
Line of Neimark-Sacker
detA
trA
(fold) 1  trA  det A  0
(flip) 1  trA  det A  0
N-S) det A  1
Where A is the Jacobian matrix
computed at the equilibrium
considered
An eigenvalue equals to 1
Saddle-Node bifurcation: two fixed points appear, one stable and one unstable
Normal form: f(x,) =  + x  x2
x
x
x
An eigenvalue equals to 1: Pitchfork bifurcation:a fixed point becomes unstable
(stable) and two further fixed points appear, both stable (unstable)
Normal form:f(x,) =  x + x  x3
x
supercritical
x
x
subcritical
An eigenvalue equals to -1: Flip bifurcation (period doubling bifurcation):
the fixed point becomes unstable and a stable period 2 cycle appears, surrounding
it. It corresponds to a pitchfork bifurcation of the second iterated of the map.
Normal form:f(x,) = 1x + x3
supercritical
Neimark-Sacker bifurcation:
The eigenvalues of the Jacobian matrix DT(P*) evaluated at the fixed
point P* are complex and cross the unit circle for   0.
1,2  0   e
 i0
d | 1 ( ) |
d
0
d
  0
(transversality condition)
eik0  1, k=1,2,3,4
(non resonance conditions)
1
2
two alternative situations
P* becomes unstable and an attracting closed curve S appears around it
(supercritical)
S
P* becomes unstable merging with a repelling closed curve U,existing
when it is stable
(subcritical)
U
Neimark-Sacker bifurcation:The eigenvalues of the Jacobian matrix
DT(P*) evaluated at the fixed point P* are complex and cross the unit
circle.
1,2  0   ei0
d | 1 ( ) |
d
0
d
  0
(transversality condition)
ik0
e
 1, k=1,2,3,4
(non resonance conditions)
After rescaling  P* = 0, o = 0
complex variable:
change of variable:
polar coordinates:
linear terms nonlinear terms
z=x1+ix2
z=w+h(w)
w=reib
z  1() z + g(z,z,)
w  m1() w + c1w2w + ….
r  r (1+ d +
ar2
+ ….)
b  b + 0  e + br2 + ….
 d
r
a
b  Im( e iθ c1 (0))
a  0 supercritical
a  Re(e c1 (0))  0 
a  0 subcritical
 iθ
As the bifurcation parameter moves away from the N-S bifurcation value:
The circle slightly deforms, but:
 remains an invariant curve
 maintains its “stability”
 approches a circle for 0
 Amplitude  |    0 |
On the invariant curve:
 dense quasiperiodic orbits or
 a finite number of periodic orbits,
saddles and nodes, appearing and
disappearing via Saddle-Node
Arnold tongues
schematic
m1()
SN bifurcations
θ

inside the Arnold
tongues the rotation
number is rational
2p
q
Infinitely many tongues, of thickness
 d (q-2)/2
(d is the distance from the unit circle)
bifurcation point
(o0
S5
N5
S4
S
N4
U
S3
N6
N3
S6
N1
S2
S1
N2
Inside an Arnol’d tongue 1/6
for a stable closed invariant curve
(supercritical Neimark-Sacker)
N1
S1
Inside an Arnol’d tongue 1/6
for an unstable closed invariant curve
(subcritical Neimark-Sacker)
Frequency locking:
Two cycles appear via Saddle-Node bifurcation
The invariant closed curve is given by a saddle-node connection
The cycles disappear via Saddle-Node bifurcation.
x'  y


Example: Iterated map T T : 
2

 y '  y  x  x
fixed points: O = (0,0)
P = (,)
 0
DT  
2 x  
1
1

Supercritical Neimark-Sacker bifurcation of O occurs at  = 1
O stable focus for  <1
unstable focus for  >1
  1.01
  1.02
  1.05
  1.1
  1.3
  1.4
  1.505
T : Rn  Rn
p’ = T (p)
p1
Noninvertible map
means “Many-to-One”
p2
Equivalently, we say that
p’ has several rank-1 preimages
.
.
.
.
p1
p2
T
. p’
T
T1-1
.
p’
T2-1
Several distinct inverses are defined: T 1 ( p' )  T11 ( p)  T21 ( p)  p1, p2 
i.e. the inverse relation p = T-1(p’) is multivalued
Rn can be divided into regions (or zones) according
to the number of rank-1 preimages
Zk
LC
Zk+2
Zk: region where k distinct inverses are defined
LC (critical manifold): locus of points having two merging preimages
Linear map T : (x,y)→(x’,y’)
 x '   a11 a12   x   b1 
 y '    a a   y   b 
   21 22     2 
area (F’) = |det A |area (F), i.e. |det A | < 1 (>1) contraction (expansion)
Meaning of the sign of |det A|
T is orientation preserving if det A > 0
a11=2 a12= -1 a21=1 a22=1 b1= b2= 0 ; Det = 3
C
y
y
T
C’
F
A
a11=1 a12=1.5 a21=1 a22 =1
y
F’
B
T is orientation reversing if det A < 0
C
B’
B’
y
T
B’
F’
F
A
C’
B
A’
A’
x
b1= b2= 0; Det = - 0.5
x
x
x
For a continuous map the fold LC-1 is included in the set
where det DT(x,y) changes sign
in fact,
T is orientation preserving near points (x,y) such that det DT(x,y)>0
orientation reversing if det DT(x,y) < 0
If T is continuously differentiable
LC-1 is included in the set where det DT(x,y) = 0
The critical set LC = T ( LC-1 )
A noninvertible map of the plane “folds and pleats”' the plane
so that distinct points are mapped into the same point.
T
LC-1
R1
A point has several distinct preimages,
i.e. several inverses are defined in it,
which “unfold” the plane
A region Zk is seen as
the superposition of k LC-1
sheets, each
associated with a
different inverse,
connected by folds
along LC
LC = T(LC-1
R2
Z2
Z0
SH1
SH2
LC
R1
R2
Z2
Riemann Foliation
Z0
Example:

x '  y
T :
2
y
'

y


x

x





1  x 
T1 : 
2
y  x'

2
4
 y ' x '
fixed points: O = (0,0)
P = (,)



x 

2
T21 : 
y  x'


2
4
 y ' x '
Z2 = {(x,y) | y > x– 2/4}
LC = {(x,y) | y = x –2/4}
Z0 = {(x,y) | y < x– 2/4 y < b }
LC-1 = {(x,y) | x = /2 }
 0
DT  
2 x  
1

1
det DT =  2x = 0 for x = /2
T({x = /2 }) = {y = x –2/4}
Supercritical Neimark-Sacker bif. at  = 1
P
LC
Z2
Z0

O
R1 R2
LC-1
P
LC-1
A0

Z2
h01 h2
0
B0
O
Z0
A
1
h1
B1
LC
R1 R2

C1
C
7
C2
C3
LC
O
C
6
C4
C
5
LC-1
P
LC
O
LC-1
P
O
 x(t  1)  ax(t )  y(t )
T :
2
 y(t  1)  x(t )  b
a = 1 b = -2
Mappa non invertibile
y
y
.
3
T 2
P1
.
.
P = T(P1) = T(P2)
1
3
T11 2
1
1
P
T
.
.
T21
1
-3
-2
1
-1
2
3
x
-3
-2
T
1
-1
-1
-1
-2
-2
2 inverse
1
1
 x   y 'b
:
 y  x' y 'b
.
P21
P2
x
P
T
1
2
 x  y 'b
:
 y  x' y 'b
2
3
 x'  ax  y
T :
2
y
'

x
b


 x   y 'b
T11 : 

 y  x' y 'b
 a 1
DT  
det DT = -2x =0 for x=0

 2 x 0

 x  y 'b
T21 : 

 y  x' y 'b
T({x=0}) = {y=b}
LC = {(x,y) | y = b }
LC-1 = {(x,y) | x = 0 }
Z2 = {(x,y) | y > b }
Z0 = {(x,y) | y < b }
T11
R1
LC-1
R2
T
1
2
SH1
SH2
Z2
x=0
LC
y=b
Z0
 x'  ax  y
T :
2
y
'

x
b

T:
x1 (t  1)  ax1 (t )  x2 (t )
x2 (t  1)  b  x12 (t )
T
F
F’= T(F)
LC -1
LC
LC-1
y
D
LC-1
C
y
C
B
A
B
A
O
B’
A’
LC
C’
A’
D’
B’
LC
C’
O’
x
x
LC-1
y
LC-1
y
B
B
A
C
C
A
C’
LC
B’
A’
(a)
A’
C’
B’
LC
x
(b)
x
f:
x1 (t  1)  ax1 (t )  x2 (t )
x2 (t  1)  b  x12 (t )
LC2
LC1
LC-1
LC3
LC
LC2
LC5
LC6
LC1
L
C
LC4
LC
-1
LC3
x'  ax  y
T :
2
 y'  x  b
Two kinds of complexity
k = 1; v1 = v2 = 0.852 ; b1= b2 =0.6 ; c1 = c2 = 3
k = 1; v1 = v2 = 0.851 ; b1= b2 =0.6 ; c1 = c2 = 3
1.5
1.5
y
y
E*
E*
0
0
0
(a)
x
1.5
0
(b)
x
1.5
G.I. Bischi and M. Kopel
“Multistability and path dependence in a dynamic brand competition model”
Chaos, Solitons and Fractals, vol. 18 (2003) pp.561-576
G.I. Bischi, C. Chiarella and M. Kopel “The Long Run Outcomes and Global
Dynamics of a Duopoly Game with Misspecified Demand Functions”
International Game Theory Review, Vol. 6, No. 3 (2004) pp. 343-380
1
q2
ES
.c
.c
ES
2
.c
1
.c
2
1
0
0
q1
1
1
q2
ES
.c
ES
2
E2
.c
1
.
E2
.
E1
E1
0
0
q1
1
Basins in 2- dimensional discrete dynamical systems
- noninvertible maps, contact bifurcations, non connected basins
- some examples from economic dynamics
- some general qualitative situations
- particular structures of basins and bifurcations related to 0/0
What about dimension > 2 ?
Homines amplius oculis quam auribus credunt, deinde quia longum
iter est per praecepta, breve et efficax per exempla.
Seneca, Epistula VI
“the systematic organization, or exposition, of a mathematical theory is always
secondary in importance to its discovery ... some of the current mathematical
theories being no more that relatively obvious elaborations of concrete examples”
Birkhoff, Bull. Am. Math. Soc., May 1946, 52(5),1, 357-391.
Attempts to provide a truly coherent approach to bifurcation theory have been
singularly unsuccessful. In contrast to the singularity theory for smooth maps,
viewing the problem as one of describing a stratification of a space of dynamical
system quickly leads to technical considerations that draw primary attention from
the geometric phenomena which need description. This is not to say that the
theory is incoherent but that it is a labyrinth which can be better organized in
terms of examples and techniques than in terms of a formal mathematical
structure. Throughout its history, examples suggested by applications have been a
motivating force for bifurcation theory.
J. Guckenheimer (1980) “Bifurcations of dynamical systems”, in Dynamical
Systems, C. Marchioro (Ed.), C.I.M.E. (Liguori Editore)
Some results presented in this book were essentially obtained via a numerical way,
guided by fundamental considerations based on critical curves properties.
Certain abstractly inclined readers might find occasions to feel irritated by such a
“modus operandi”. Unfortunately, taking into account the complexity of the
matter and its particular nature, even in the simplest situations, it seems unlikely
to carry out the study with success from another process.
Moreover, without using the critical curve tool and the basic considerations
mentioned above, simple numerical investigations do not permit to advance in this
field.
Mira, Gardini, Barugola and Cathala “Chaotic dynamicd in two-dimensional
noninvertible maps”, World Scientific, 1996
"... Both the formulation and the proof of this lemma are geometric rather than
analytic, as is often the case in nonlinear dynamics. We emphasize though that
this is a formal lemma, which is not based upon (but very much inspired by)
computer simulations..."
Brock and Hommes, "A rational route to randomness", Econometrica 65 (1997)
SH2
SH1
T21
y
y’
T11
LC-1
U-1,1
U-1,2
R1
Z2
U
R2
x
Z0
LC
x’
y
 x(t  1)  ax(t )  y(t )
T :
2
y
(
t

1)

x
(
t
)
b

T
map
2 fixed points
 x  ax  y

2
y

x
b

 y  (1  a ) x
 2
 x  (a  1) x  b  0
 x '  ax  y

2
y
'

x
b

2 inverses
 x   y 'b
T :
 y  x' y 'b
T
T
 x  y 'b
T :
 y  x' y 'b
1
1
T
1
2
x
Z2 = {(x,y) | y > b }
LC = {(x,y) | y = b }
Z0 = {(x,y) | y < b }
LC-1 = {(x,y) | x = 0 }
T11
 a 1
DT  

2
x
0


det DT = -2x =0 for x=0 R1
T({x=0}) = {y=b}
T
R2
LC-1
1
2
SH2
LC
SH1
Z2
Z0
CS-1
CS-1
T21 (V )
U
R1
T11 (V )
R2
R1 R2
T(U)
Z2
Z0
V
Z2
CS
Z0
CS
Q
LC-1
Z2
P
Z0
LC
contact
Z2
Z0
LC
LC-1
Z2
Z0
LC
Z2
Z0
LC
LC-1
Z2
Z0
LC
LC-1
Z2
Z0
LC
LC-1
Z2
Z0
LC
LC-1
Z2
Z0
LC
LC-1
Z2
Z0
LC
LC-1
Z2
Z0
LC
LC-1
Z2
Z0 LC
LC-1
Z2
Z0 LC
1
1
2
2
4
5
6
3
3
After “exempla” some “precepta”
The basin of an attractor A is the set of all points that generate
trajectories converging to it:
B(A)= {x| Tt(x)  A as t +}
Let U(A) be a neighborhood of A whose points converge to it. Then
U(A)  B(A), and also the points that are mapped into U after a finite
number of iterations belong to B(A):

B  A 
T
n
(U ( A))
n 0
T-n(x)
where
represents the set of the rank-n preimages of x.
From the definition it follows that points of B are mapped into B both
under forward and backward iteration of T
T(B)  B, T-1(B) = B
;
T(B) B, T-1(B)= B
This implies that if an unstable fixed point or cycle belongs to B then
B must also contain all of its preimages of any rank.
If a saddle-point, or a saddle-cycle, belongs to B, then B must also
contain the whole stable set
Augustine Cournot (1838)
Récherches sur les principes matématiques de la théorie de la richesse
2 firms producing at time t homogeneous goods
q1 (t) and q2 (t) outputs
p = f (q1+q2) inverse demand function
ci (qi )
cost functions,
The profits of the two quantity-settimg firms are:
Pi= qi f (q1+ q2) – ci (qi) i=1,2
At time period t each firm decides (t+1)-output by solving a
profit-maximization problem

maxq1 P1 q1, q2e


maxq2 P2 q1e , q2

Each firm considers the output of its competitor as given

 

P q (t  1), q   r q (t  1) 
q1 (t  1)  arg maxq1 P1 q1 , q2e (t  1)  r1 q2e (t  1)
q2 (t  1)  arg maxq2
2
e
2
2
2
e
1
q2
Expectation of agent i about the rival’s choice
qej (t  1)
Rational expectations (perfect foresight):
q1 = r1(q2)
.
Cournot-Nash Equilibrium
q2 = r2(q1)
qej (t 1)  q j (t 1) i  1,2
q1
 q1 (t )  r1 (q 2 (t ))

q 2 (t )  r2 (q1 (t ))
t
 q1*  r1 (q2* )  r1 (r2 (q1* )
One-shot (static) game  *
*
*
q2  r2 (q1 )  r2 (r1 (q2 )
The game directly goes to the intersection(s) of the reaction curves (Cournot-Nash
equilibrium) in one shot
Cournot (Naive) expectations:
 q1 (t  1)  r1 (q2 (t ))
T :
q2 (t  1)  r2 (q1 (t ))
qej (t 1)  q j (t ) i  1,2
t
Two-dimensional dynamical system:
given (q1(0),q2(0)) the repeated application of the map T:(q1,q2) (r1(q2), r2(q1))
gives the time evolution of the duopoly game.
This repeated game may converge to a Cournot-Nash equilibrium in the long run,
i.e. boundedly rational players may achieve the same equilibrium
as fully rational players provided that the “myopic” game is played several times
Evolutionary interpretation of Nash equilibrium (Nash’s concern)
Linear demand p = a – b (q1 + q2) ;
Linear cost Ci = ci qi
i = 1,2
Quadratic Profit: Pi  a – b (q1 + q2))qi – ci qi =
F.O.C.
S.O.C.
P1
 a  2bq1  bq2  c1  0
q1

1
a  c1
q1 (t  1)  r1 (q2 (t ))   q2 (t ) 
2
2b
P 2
 a  2bq2  bq1  c2  0 
q2
1
a  c2
q2 (t  1)  r2 (q1 (t ))   q1 (t ) 
2
2b
 2P1  2P 2

 2b  0
q12
q22
 q1  r1 ( q2 )
q2  r2 ( q1 )
Equilibrium: 
A  2c1  c2
q 
3B
*
1
q2* 
q2
Cournot-Nash Equilibrium
A  2c2  c1
3B
q1
Linear/linear Cournot game and best reply dynamics with naive expectations
1
a  c1
q1 (t  1)  r1 (q2 (t ))   q2 (t ) 
2
2b
1
a  c2
q2 (t  1)  r2 (q1 (t ))   q1 (t ) 
2
2b
r1
q2
r2
Reaction function of firm 1
q1  r1  q2 
r1
Reaction function of firm 2
q*2
q2  r2  q1 
q*1
r2
q1
Developments and complexities
The firms in the Cournot (1838) model (mineral water producers) decide quantities,
then the price at each time period is obtained from the inverse demand finction.
Bertrand (1883) criticized this approach and preferred to assume that firms compete
by deciding prices, and assumed differentiated products, each with its price.
The problem is mathematically equivalent.
Edgeworth (1925) considered the case of homogeneous products and stated that
oligopoly markets, in contrasts with the cases of monopoly and perfect
competition, may be indeterminate, i.e. uniqueness of equilibrim is not ensured.
Moreover, assuming quadratic costs, prices may never reach an equilibrium
position and continue to oscillate ciclycally forever.
Teocharis (1960) proves that the linear/linear discrete time Cournot model is only
stable in the case of duopoly.
McManus & Quandt (1961), Hahn (1962), Okuguchi (1964) show that this statement
depends on the kind of adjustment considered and the kind of expectations formation.
However, Fisher (1961) stresses that in general “the tendency to instability does rise
with the number of sellers for most of the processes considered”
Linear demand: p = a – b (q1 + q2)
Quadratic cost: Ci = ci qi + ei qi2
i = 1,2
Quadratic Profit: Pi  a – b (q1 + q2))qi – (ci qi + ei qi2 )
Linear reaction functions:
P1
b
a  c1
 a  2(b  e1 )q1  bq2  c1  0  q1 (t  1)  r1 (q2 (t ))  
q2 (t ) 
q1
2(b  e1 )
2(b  e1 )
P 2
b
a  c2
 a  2(b  e1 )q2  bq1  c2  0  q2 (t  1)  r2 (q1 (t ))  
q1 (t ) 
q2
2(b  e2 )
2(b  e2 )
eigenvalues: z1, 2  
b
2
1
b  1 b   2 
b2 < 4(b+e1)(b+e2)
(Stable)
stability if
b2 < 4(b+e1)(b+e2)
b2 > 4(b+e1)(b+e2)
(Unstable)
Linear demand, quadratic costs, case b2 > 4(b+e1)(b+e2)
E unstable, E1 , E2 , stable
L2
basin of E1
x2
basin of E2
E2
R2
R1
0
c2
c1
0
basin of 2-cycle (c1,c2)
E
R2
R1
E1
x1
L1
Non monotonic reaction curves
Rand, D., 1978. Exotic Phenomena in games and duopoly models.
Journal of Mathematical Economics, 5, 173-184.
A Cournot tâtonnement is considered with unimodal (one-hump)
reaction functions, and he proves that chaotic dynamics arise, i.e.
bounded oscillations with sensitive dependence on initial conditions etc..
Postom and Stewart (1978 ) "Catastrophe Theory and its Applications",
Book seller example:
“...If you start producing books, when no one else is, you will not sell
many.There will be no book habit among people, no distribution
industry…
On the other hand if other producers exist producing books in huge
numbers, you will be invisible…and again you will sell rather few.
Your sales will be best when your competitors’ output will be
intermediate…”
New mathematics
“… Adequate mathematics for planning in the presence of such
phenomena is a still far distant goal…”
Tonu Puu (1991) “Chaos in Duopoly pricing” Chaos, Solitons & Fractals
Shows how an hill-shaped reaction function is quite simply obtained by
using linear costs and replacing the linear demand function by the
economists’ “second-favourite” demand curve, the constant elasticity
demand
1
1
qi
p 
P

 ci qi
i
Q q1  q2
q1  q2
qj
qj
P i


c

0
for
q


q

i
i
j
qi q1  q2 2
ci

q1 (t  1)  r1 (q2 (t ))  q2 (t ) 


 q (t  1)  r (q (t ))  q (t ) 
2
1
1
 2
q2
c1
q1
c1
–
+
Van Witteloostuijn, A., Van Lier, A. (1990) Chaotic patterns in Cournot competition.
Metroeconomica.
Van Huyck, J., Cook, J., & Battalio, R. (1984). Selection dynamics, asymptotic stability,
and adaptive behavior. Journal of Political Economy, 102, 975–1005.
Dana, R.A., & Montrucchio, L. (1986). Dynamic complexity in duopoly games.
Journal of Economic Theory, 40, 40–56.
Everything goes !
Kopel, M. (1996) “Simple and complex adjustment dynamics in Cournot Duopoly
Models”. Chaos, Solitons, and Fractals.
Linear demand function, cost function Ci = Ci(q1,q2) with positive cost externalities
(spillover effect which gives some advantages due to the presence of the competitor)
r1 q2   m1q2 1  q2  r2 q1   m2 q1 1  q1 
m1 and m2 measure the intensity of the positive externality
Adaptive adjustment (inertia, or anchoring )
q1 (t  1)  1  1  q1 (t )  1 r1 (q2 (t ))
T :
q2 (t  1)  1  2  q2 (t )  2 r2 (q1 (t ))
Bischi, G.I., C. Mammana and L. Gardini (2000) «Multistability and cyclic attractors
in duopoly games», Chaos, Solitons and Fractals.
 q1 (t  1)  r1 (q2 (t ))
Cournot with naive expectations (Best reply dynamics): T : 
q2 (t  1)  r2 (q1 (t ))
And reaction functions r1 q2   m1q2 1  q2  ; r2 q1   m2 q1 1  q1 
m1 = m2 = 3.4
1 = 2 = 1
1
y
E2
C2( 2 )
ES
C2(1)
E1
0
0
x
1
Bischi, G.I. and M. Kopel (2001) «Equilibrium Selection in a Nonlinear Duopoly
Game with Adaptive Expectations» Journal of Economic Behavior and Organization
Problem of equilibrium selection:
•Which equilibrium is achieved through an evolutive
(boundedly rational) process?
•What happens when several coexisting stable Nash
equilibia exist?
Cournot Game (from beliefs to realizations)
q1 (t )  r1 (q2e (t ))
q2 (t )  r2 (q1e (t ))
Adaptive expectations


q (t  1)  q (t )   q (t )  q (t ) 
q1e (t  1)  q1e (t )   q1 (t )  q1e (t )   q1 (t )  1   q1e (t )
e
2
e
2

2
e
2
 
  q 2 (t )  1   q 2e (t )
Dynamical system: T : q1e (t ), q2e (t )  q1e (t  1), q2e (t  1)
q1e (t  1)  1   1 q1e (t )   1 r1 (q 2e (t ))
T : e
q 2 (t  1)  1   2 q 2e (t )   2 r2 (q1e (t ))

Existence and local stability of the equilibria
in the case of homogeneous expectations 1  2  
3
1
1 6
1
WsEi,C2

 p 
1
m 1
transcritical O = S
WsS
WsO
0
0
pitchfork E1 = E1 = S

1
2
  h 
6  12 m m  2
3  2m  m 2
2
m 2  2m  3
WsEi
3 1 5
4
5
m
m1 = m2 = 3.4
1 = 2 = 0.2 < 1/(m+1)
1 = 2 = 0.5 > 1/(m+1)
O 1.4
y
2.3
y
Z0
LC
m1 = m2 = 3.4
(1)
1
D
(1)
O1
Z0
LC ( a )
D
( 3)
O1
(a)
E2
E2
Z2
K
S
LC ( b )
E1
0
LC
E1
Z2
Z4
0
O
O
(b )
Z4
0
(a)
x 2.3
0
(b)
(2)
O1
x 1.4
Noninvertible (“Many-to-One”) map
.
.
T
p1
Distinct points are mapped into the same point
p2
.
p’
T
Folding action of T
p1
Equivalently, we say that p’ has several rank-1 preimages
p2
.
.
T1-1
T2-1
SH2
y
LC-1
U-1,2
SH1
T21
1
1
T
U-1,1
y’
Z2
U
R1 R2
x
Unfolding action of T-1
Z0
LC
x’
.
p’
Critical curves


LC  T ( LC1 ).
LC1  ( x, y ) 2 | det DT ( x, y )  0
 1  1
1m1 1  2 y  
DT ( x, y )  


m
1

2
x
1




2
 2 2

LC1 :
1 
1  1  1  1   2 

x   y 
2 
2
41 2 m1m2

m1  m2  3
1  2  0.5
1.5
m1  m2  3
1  2  0.5
1.5
(a)
1
y
LC
y’
LC ( a )
Z0
Z2
(b)
LC1
K
Z4
0.5
-0.5
x
0.5
1.5 0.5
LC (b )
x’
1.5
Critical curves separate regions Zk , Zk+2 characterized by different numbers of
preimages. Each region Zk can be seen as the superposition of k sheets om which the k
distinct “inverses” are defined, so the critical lines LC represent foldings, and the
inverses “unfold” sheets along LC.

LC1  ( x, y ) 
2

| det DT ( x, y )  0
LC  T ( LC1 ).
 1  1
1m1 1  2 y  
DT ( x, y )  


m
1

2
x
1




2
 2 2

LC1 :
1 
1  1  1  1   2 

x

y


 

2 
2
41 2 m1m2

In the homogeneous case
LC( b1)  D  K 1   k 1 , k 1 
  m  1  1
2m
LC-1
with k1 
LC (b)  T ( LC(b1) )
K   k , k  , where
has a cusp point in
LC
  m  1  1 m  3(1   ) 

k
4m
y’
y
LC ( a )
(a)
LC1
Z2
Z4
(b)
1
LC
.
In the homogeneous case
and
z
LC ( b )
x’
x
LC(b1)  D  K1   k1 , k1 
LC (b)  T ( LC(b1) )
has a cusp point in
Z0
K  k, k  ,
where
with k1 
  m  1  1
2m
  m  1 1 m  3(1   ) 

k
4m
Theorem (Homogeneous behavior)
If m1  m2 m, 12, and the bounded trajectories converge to one
of the stable Nash equilibria E1 or E2, then the common boundary 
B(E1)  B(E2) which separates the basin B(E1)from the basin B(E2) is
given by the stable set WS(S) of the saddle point S.
If m11 then the two basins are simply connected sets;
if m11 then the two basins are non connected sets, formed by
infinitely many simply connected components.
Case of heterogenous players
m1 = m2 = 3.6
1 = 0.55
1.2
y
2 = 0.7
Z0
Z2
LC ( b )
. .
(b)
LC1
Z0
LC ( a )
(1)
H2
E2
S
Z4
Z2
.
H (11)
. .
(a)
LC1
.
Z4
LC
E1
H
0
x 1.1
0
( 2)
H2
E2
(b)
1
0
Theorem …
2 = 0.7
S
LC ( b )
( 2)
1
( 3)
H2
0
1 = 0.59
1.2
y
(a)
LC1
LC ( a )
m1 = m2 = 3.6
H0
E1
(4)
H2
x 1.1
m1 = m2 = 3.9
1 = 0.7
2 = 0.8
m1 = m2 = 3.95
1.1
y
1 = 0.7
2 = 0.8
1.1
y
A2
.
S
A2
A1
.
S
.
E1
0
0
0
x 1.1
0
x 1.1
Agiza, H.N., Bischi, G.I. and M. Kopel «Multistability in a Dynamic Cournot
Game with Three Oligopolists», Mathematics and Computers in Simulation, 51
(1999) pp.63-90
q1  1  1  q1  1m1  q2 1  q2   q3 1  q3  

 
T : q2  1  2  q2  2 m2  q3 1  q3   q1 1  q1  
 


q

1


q


m
q
1

q

q
1

q







3
3
3
3
3
1
1
2
2



Bischi, G.I. and A. Naimzada, "Global Analysis of a Duopoly Game with
Bounded Rationality", Advances in Dynamic Games and Applications, vol.5,
Birkhauser (1999) pp. 361-385
profit function (linear cost and demand)
Pi (q1, q2 )  qi a  b(q1  q2   ci qi .
Gradient dynamics
Pi
qi (t  1)  qi (t )  vi qi (t )
(q1 (t ), q2 (t ))
qi
;
The map
 q1'  (1  v1 (a  c1 ))q1  2bv1q12  bv1q1q2

T :
q '  (1  v (a  c ))q  2bv q 2  bv q q
2
2
2
2 2
2 1 2
 2
i  1, 2
Each coordinate axis is trapping since qi(t) = 0 implies qi(t+1) = 0
The restriction of the map T to that axis is
q'j  (1 v j (a  c j ))q j  2bv j q2j
j i .
conjugate to the standard logistic map
v1 = 0.24
8
v2 = 0.48 c1 = 3
c2 = 5 a = 10 b = 0.5
O(12)
q2
11
E*
2
O(31)
 21
1
0
O(11)
O
0
q1
12
v1 = 0.24
7
q2
v2 = 0.55 c1 = 3
c2 = 5 a = 10 b = 0.5
LC (b )
LC(b1)
Z0
Z2
LC (a)
0
E*
Z4
0
v1 = 0.24
v2 = 0.55 c1 = 3
c2 = 5 a = 10 b = 0.5
11
7
contact
LC( 1a )
q2
h(12)
q1
h
h(11)
Z0
h(12 )
( 2)
2
h
Z2
LC (a)
Z4
0
0
E*
(b )
1
LC
LC
(b )
LC( 1a )
q1
v1 = 0.24 v2 = 0.55 c1 = 3 c2 = 5 a = 10 b = 0.5
( 2)
1
Z0
O
q2
LC
O(12)
Z0
q2
LC( 1a )
(b )
1
v1 = 0.24 v2 = 0.7 c1 = 3 c2 = 5 a = 10 b = 0.5
E*
Z2
LC
0
E*
(a )
Z4
0
Z2
LC (b )
q1
Z4
0
11
0
v1 = 0.24
v2 = 1.0747 c1 = 3
c2 = 5 a = 10 b = 0.5
6
q2
LC
0
0
(a )
LC( 1a )
q1
11
7
v1 = 0.4065
v2 = 0.535 c1 = 3
c2 = 5 a = 10 b = 0.5
q2
0
0
q1
9.5
v1 = 0.4065
7
v2 = 0.535 c1 = 3
c2 = 5 a = 10 b = 0.5
q2
LC (b )
h(12)
Z0
h
h(11)
Z2
(b )
1
LC
LC( 1a )
( 2)
1
h
LC (a)
h( 22)
Z4
-0.5
-0.5
q1
9.5
Bischi, G.I. and F. Lamantia «Nonlinear Duopoly Games with Positive Cost
Externalities due to Spillover Effects» Chaos, Solitons & Fractals, vol. 13 (2002).
Dynamic adjustment process:
i.e
13
13
 11
O(12)
O(13)
O(12)
 11
2
2
R1
E2
O(13)
E*
R2
E2
R2
E*
 21
O
1
E1
O(11)
R1
13
O
1
 21
E1
O(11)
13
13
q2
13
Z0
q2
LC(b)
Z2
Z0
(b)
H0
LC
Z2
(a)
LC1
(b )
LC1
(b )
LC1
H (11)
H (12)
LC(a )
Z4
O
Z4
q1
13
O
q1
13