Transcript Diapositiva 1

Noninvertible maps and applications:
An introductory overview
Outline
•What is a noninvertible map
•The method of critical sets
•Some history
•Some recent applications
•The concept of absorbing area and related bifurcations
•Non connected and multiply connected basins
T : Rn  Rn
p’ = T (p)
Noninvertible map
means “Many-to-One”
.
p1
p2
T
. p’
.
p1
Equivalently, we say that
p’ has several rank-1 preimages
p2
T
.
.
T1-1
.p’
T2-1
Several distinct inverses are defined in p’ :
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
Zk+2
Zk: region of Rn where k distinct
inverses are defined
LC (critical manifold)
locus of points having two merging preimages
Example: 1-dimensional NIM
x’ = f(x) = ax (1-x)
Z0 - Z2 map:
if x’ < a/4 then
f 1( x' )  f11( x' )  f21( x' )  x1, x2
where:
aa  4 x'
1

2
2a
aa  4 x'
1
x 2  f 21 ( x' )  
2
2a
x1  f 1 1 ( x' ) 
critical point c = a/4
c1  f 21 (c)  f 21 (c) 
1
2
Df(c-1) = 0 and c = f(c-1)
Folding by T
c-1
Unfolding by T-1
Piecewise differentiable noninvertible map
cM
cm
c-1M
c-1m
Z0
c
c
f2
Z4
c1
Z2
f
c1
Logistic map
x’ = f(x) = ax (1-x)
a1
a2
a3 a
a < a2
c
a = a2
c4
c4
c6
c2
c2
c3
c5
c
c3
c7
c1
c1
c4= c6 = p1*; c5= c7 = p2*
a2 < a < a1
c
a = a1
c
c2
c3
c1
c1
c2= c3= p*
c
c2=f(c1)
c3=f(c2)
c1=f(c)
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
LC = T(LC-1)
R2
Z2
Z0
Riemann Foliation
Equivalently, a point has several distinct rank-1 preimages,
i.e. several inverses are defined in it, which
“unfold” the plane
SH1
SH2
LC-1
LC
Z2
R1
R2
Z0
Z1 - Z3 - Z1
LC-1(b) LC-1(a)
SH1
Z1
SH2
SH3
LC (a)
LC (b)
Z3
Z1
Z1 < Z3
SH1
SH3
SH2
Z1
Z3
LC-1
Z1
LC
 x'
a
Linear map T:     11
 y ' a 21
area (F’) = |det A |area (F)
|det A | < 1 (>1) contraction (expansion)
a12   x 
a 22   y 
Meaning of the sign of |det A |
T is orientation preserving
if det A > 0
C’
T
C
F
A
B’
F’
B
A’
T is orientation reversing
if det A < 0
T
C
F
A
A’
B
F’
B’
C’
For a continuous map the fold LC-1 is included in the set where det DT(x,y)
changes sign.
T is orientation preserving near points (x,y) such that det DT(x,y)>0
orientation reversing if det DT(x,y)<0
The critical set LC = T ( LC-1 )
If T is continuously differentiable LC-1 is included in the set
where det DT(x,y) = 0
Example: T :  x'  ax  y

2
1
1
T
 y'  x  b

 x   y 'b
:

 y  x' y 'b
T
1
2

 x  y 'b
:

 y  x' y 'b
Z2 = {(x,y) | y > b }
Z0 = {(x,y) | y < b }
LC = {(x,y) | y = b }
LC-1 = {(x,y) | x = 0 }
 a 1
DT  

2
x
0


det DT = -2x =0 for x=0
T({x=0}) = {y=b}
T11
R1
LC-1
R2
T
1
2
SH1
SH2
Z2
x=0
LC
Z0
y=b
Curves across LC-1 are mapped into curves tangent to LC
Simple across LC-1 may be mapped into mapped with a double point
g ’  Tg
T
g
LC-1
LC
A plane figure across LC-1 is folded along LC
T
F
F’  TF
LC-1
LC
 x'  ax  y
T :
2
 y'  x  b
LC = {(x,y) | y = b }
LC-1 = {(x,y) | x = 0 }
LC2
LC1
LC-1
LC3
LC
LC2
LC5
LC6
LC1
LC-1
LC4
LC
LC3
Basins of attraction of noninvertible iterated maps
* basins in 1- dimensional discrete dynamical systems
- generated by invertible maps
- generated by noninvertible maps
contact bifurcations and non connected basins
* 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)
Continuous and increasing maps
•The only invariant sets are the fixed points.
•When many fixed points exist they are alternatingly stable and unstable:
the unstable fixed points are the boundaries that separate the basins of
the stable ones.
• Starting from an initial condition where the graph is above the diagonal,
i.e. f(x0)>x0, the trajectory is increasing, whereas if f(x0)<x0 the trajectory
is decreasing
r*
r*
p*
q*
p*
q*
f(x) = a arctan (x-1)
basin
boundary
a = 0.5
a=3
fold bifurcation
a=1
Continuous and decreasing maps
The only possible invariant sets are one fixed point and cycles of period 2, being f2=f°f
increasing
The periodic points of the 2-cycles are located at opposite sides with respect to the
unique fixed point, the unstable ones being boundaries of the basins of the stable ones.
If the fixed point is stable and no cycles exist, then it is globally stable.
f(x) = – ax3 + 1
a = 0.2
a = 0.5
a = 0.7
Nononvertible maps. Several preimages
Z0
c
q-1
p
p
Z2
q
q
c-1
r
Noninvertible map: f (x) = a x (1– x)
y
Z0
c=a/4
Z2
0
c-1= 1/2
1
x
z
Z1
cmax
p
Z3
cmin
Z1
r
q
z
Z1
q-12
q-11
cmax
p
Z3
cmin
q
r
Z1
c-1
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
 x(t  1)  ax(t )  y(t )
T :
2
y
(
t

1)

x
(
t
)
b

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
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
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
Two kinds of complexity
(b)
x
1.5
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
SH2
SH1
T21
y
y’
T11
LC-1
U-1,1
U-1,2
R1
Z2
U
R2
x
Z0
LC
x’
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
Bischi, G.I. and M. Kopel "Equilibrium Selection in a Nonlinear Duopoly Game
with Adaptive Expectations" Journal of Economic Behavior and Organization, vol.
46 (2001) pp. 73-100
Cournot Game



Max 1 q1 , q2e (t  1) ; Max  2 q1e (t  1), q2
q1
Best Replies (or reaction functions)
From beliefs to realizations
q2
q1 (t )  r1 (q2e (t ))
q2 (t )  r2 (q1e (t ))
Adaptive expectations
q1e (t  1)  q1e (t )   q1 (t )  q1e (t )   q1 (t )  1   q1e (t )
q 2e (t  1)  q 2e


(t )   q (t )  q (t ) 
2
e
2
  q 2 (t )  1   q 2e (t )
Dynamical system: T : q1e (t ), q2e (t )  q1e (t  1), q2e (t  1)
e
e
e



q
(
t

1
)

1


q
(
t
)


r
(
q
 1
1
1
1 1
2 (t ))
T : e
e
e



q
(
t

1
)

1


q
(
t
)


r
(
q
2
2
2 2
1 (t ))
 2
t

Non monotonic reaction functions may lead to several coexisting equilibria
r1 q2   m1q2 1  q2 
Logistic reaction functions
1
q2
r1
m1  3 m2  3.5
r2 q1   m2 q1 1  q1 
m1  3.7 m2  3.5
1
q2
r2
r2
r1
0
0
q1
1
0
0
q1 1
Problem of equilibrium selection
•Which equilibrium is achieved through an evolutive (boundedly rational) process?
•Stability arguments are used to select among multiple equilibria
•What happens when several coexisting stable Nash equilibia exist?
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
0
O
O
(b )
Z4
0
(a)
x 2.3
Z4
0
(b)
(2)
O1
x 1.4
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
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
Proposition (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
LC
Z0
Z2
(1)
H2
E2
S
LC
E2
Z2
S
H (11)
Z4
(b)
1
LC
E1
H (12)
H0
( 3)
H2
0
0
0
( 2)
H2
LC ( b )
Z4
(b)
LC1
2 = 0.7
(a)
1
Z0
LC ( a )
LC ( b )
1 = 0.59
1.2
y
(a)
LC1
(a)
m1 = m2 = 3.6
x 1.1
0
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
A2
S
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., H. Dawid and M. Kopel "Gaining the Competitive Edge Using Internal
and External Spillovers: A Dynamic Analysis" Journal of Economic Dynamics
and Control vol. 27 (2003) pp. 2171-2193
Bischi, G.I., H. Dawid and M. Kopel"Spillover Effects and the Evolution of Firm
Clusters" Journal of Economic Behavior and Organization vol. 50, pp.47-75
(2003)
Local Stability
QIII
PIII
VIII
PIV
VII
PII
S
QII
V0
PI
Vertices V0 and VII are always
repelling;
Interior FP S (if it exists) is a saddle
point or a repelling node
QII and PII are created together
(saddle-node)
PIII and QIII are created together
(saddle-node)
VI
PII and PIV cannot coexist
PIII and PI cannot coexist
F1
VIII
QIII
1 B(V )
III
PIII
VII
S
x2
PII
B(PIII)
F2
B(PII)
QII
B(VI)
O
x1
Fig. 3
1
VI
VIII
1
x B(VIII)
VII
QIII PIII
VIII
VII
QIII PIII
1
S
2
S
x2
B(PIII)
LC
LC
B(VI)
O
x1
(a)
1
QIII PIII
VI
VII
1
O
QIII PIII
H1
H2
H3
VI
1
VII
1
x2
x2
x1
(c)
LC
LC
LC
Z3
Z1
LC
Z1
0.965
0.35
Fig. 5
0.965
(b)
x1
1
0.35
(d)
x1
1
0.6
1.1
pIII 1
pIII 1
c1
qIII 1
qIII 1
0
x1
0
0.3
1.1
x1 0.6
0.3
1
H3
H2
pIII 1
H1
c1 qIII 1
0.3
0.3
x1
Fig. 6
1
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)
i (q1, q2 )  qi a  b(q1  q2   ci qi .
Gradient dynamics
i
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