noninvertible maps and applications: an introductory overview outline what is a noninvertible map...
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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
Noninvertible map means “Many-to-One”
. T
p’
p1
p2 T..
Equivalently, we say thatp’ has several rank-1 preimages
. T1-1
p’
p1
p2 T2-1.
.
211
21
11 ,)()()'( pppTpTpT
Several distinct inverses are defined in p’ :
i.e. the inverse relation p = T-1(p’) is multivalued
T : Rn Rn p’ = T (p)
Zk
Zk+2Zk: region of Rn where k distinctinverses are defined
LC (critical manifold) locus of points having two merging preimages
Rn can be divided into regions (or zones) according to
the number of rank-1 preimages
x’ = f(x) = ax (1-x)
Z0 - Z2 map:if x’ < a/4 then
where:
a
xaaxfx
2
'4
2
1)'(1
11
211
21
11 ,)'()'()'( xxxfxfxf
a
xaaxfx
2
'4
2
1)'(1
22
critical point c = a/4
2
1)()( 1
21
21 cfcfc
Example: 1-dimensional NIM
Df(c-1) = 0 and c = f(c-1)
Folding by T
Unfolding by T-1
c-1
c-1M c-1
m
cM
cm
Piecewise differentiable noninvertible map
Z0
Z4
Z2
c
c1
f 2
f
c
c1
a1 a2 aa3
Logistic map x’ = f(x) = ax (1-x)
c1
c4= c6 = p1*; c5= c7 = p2
*
c3
c2
c
c4
c5
c7
c6
c1
c3
c2
cc4
a < a2a = a2
c2= c3= p*
a2 < a < a1 a = a1
c1
c3
c2
c
c1
c
c1=f(c)
c2=f(c1)
c
c3=f(c2)
A noninvertible map of the plane“folds and pleats”' the plane
so that distinct points are mapped into the same point.
LC-1
LC = T(LC-1)
T
Z2 Z0
R2R1
Riemann Foliation
Equivalently, a point has several distinct rank-1 preimages, i.e. several inverses are defined in it, which “unfold” the plane
LC-1
LC
Z2 Z0R2R1
SH1
SH2
Z1 - Z3 - Z1
LC-1(b)
LC (b)
Z3 Z1
SH1
SH2
LC (a)
LC-1(a)
SH3
Z1
LC-1
Z3LC
Z1
Z1
SH1
SH2SH3
Z1 < Z3
A B
C
A’
B’
C’T
Linear map T:
y
x
aa
aa
y
x
2221
1211
'
'
T is orientation preservingif det A > 0
area (F’) = |det A |area (F) |det A | < 1 (>1) contraction (expansion)
Meaning of the sign of |det A |
F’F
A’B’
C’
A B
C TT is orientation reversingif det A < 0
F’
F
T is orientation preserving near points (x,y) such that det DT(x,y)>0 orientation reversing if det DT(x,y)<0
For a continuous map the fold LC-1 is included in the set where det DT(x,y) changes sign.
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 )
Example:
bxy
yaxxT 2'
':
byxy
byxT
byxy
byxT
''
':
''
': 1
21
1
Z2 = {(x,y) | y > b }
Z0 = {(x,y) | y < b }
LC = {(x,y) | y = b }
LC-1 = {(x,y) | x = 0 }
02
1
x
aDT
det DT = -2x =0 for x=0
T({x=0}) = {y=b} Z0
Z2
R1 R2LC-1
LC
SH1
SH2
11T
12T
y=bx=0
’T
Curves across LC-1 are mapped into curves tangent to LC
Simple across LC-1 may be mapped into mapped with a double point
LC-1
LC
F F’F
T
A plane figure across LC-1 is folded along LC
LC-1
LC
bxy
yaxxT 2'
':
LC = {(x,y) | y = b }
LC-1 = {(x,y) | x = 0 }
LC2
LC1
LC
LC3
LC-1
LC2
LC1
LC-1
LC5
LC3
LC6
LC
LC4
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 ?
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)
“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.
Homines amplius oculis quam auribus credunt, deinde quia longum iter est per praecepta, breve et efficax per exempla.Seneca, Epistula VI
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
p*
q*
r*
p*
q*
r*
f(x) = a arctan (x-1)
a = 3
a = 1
a = 0.5
basinboundary
fold bifurcation
a = 0.5
a = 0.2
Continuous and decreasing mapsThe only possible invariant sets are one fixed point and cycles of period 2, being f2=f°f increasingThe 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 = 0.7
Z2
Z0
c
c-1
p
q
p
q
r
q-1
Nononvertible maps. Several preimages
x
y
Z0
Z2
0 1c-1
Noninvertible map: f (x) = a x (1– x)
= 1/2
c=a/4
Z3
Z1
cmax
p
q
cmin
Z1
z
r
Z3
Z1
cmax
p
qcmin
Z1
z
r
c-1
q-11
q-12
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):
where T-n(x) 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
0
( ( ))n
n
B A T U A
2
( 1) ( ) ( ):
( 1) ( )
x t ax t y tT
y t x t b
q1
q2
00
1
1
ES
c1
c2
c1
c2
ES
.
.
.
.
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
ES
E1
E2
q1
q2
00
1
1
ES
E1
E2
c1
c2
.
.
.
.
Two kinds of complexity
k = 1; v1 = v2 = 0.851 ; 1= 2 =0.6 ; c1 = c2 = 3
y
x
1.5
1.500
E*
(a)
k = 1; v1 = v2 = 0.852 ; 1= 2 =0.6 ; c1 = c2 = 3
y
x
1.5
1.500
E*
(b)
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
x
y
T
T
2
( 1) ( ) ( ):
( 1) ( )
x t ax t y tT
y t x t b
2 inverses
T
T
2 fixed points
2
'
'
x ax y
y x b
map
2
2
(1 )
( 1) 0
x ax y
y x b
y a x
x a x b
byxy
byxT
byxy
byxT
''
':
''
': 1
21
1
Z2 = {(x,y) | y > b }
Z0 = {(x,y) | y < b }
LC = {(x,y) | y = b }
LC-1 = {(x,y) | x = 0 }
02
1
x
aDT
det DT = -2x =0 for x=0
T({x=0}) = {y=b}Z0
Z2
R1
R2
LC-1LC
SH1SH2
11T
12T
R1 R2
Z0
Z2
CS
CS-1
U
T(U)R1 R2
Z0
Z2
CS
CS-1
V
11 ( )T V
12 ( )T V
LC
LC-1
SH2
SH1
R1 R2
Z2Z0
11T
12T
UU-1,2
U-1,1
x’
y’
y
x
Z0
Z2
LC-1
LC
P
Q
contact
Z0
Z2
LC
Z0
Z2
LC-1
LC
Z0
Z2 LC
Z0
Z2
LC-1
LC
Z0
Z2
LC-1
LC
Z0
Z2
LC-1
LC
Z0
Z2
LC-1
LC
Z0
Z2
LC-1
LC
Z0
Z2
LC-1
LC
Z0
Z2
LC-1
LC
Z0
Z2
LC-1
LC
1
6
2 5
3
41
2
3
)(1)()()()()1(
)(1)()()()()1(
222222
111111
tqtqtqtqtqtq
tqtqtqtqtqtqeeee
eeee
ttqrtq
tqrtqe
e
))(()(
))(()(
122
211
))(()(1)1(
))(()(1)1(:
122222
211111
tqrtqtq
tqrtqtqT
eee
eee
Adaptive expectations
Dynamical system: )1(),1()(),(: 2121 tqtqtqtqT eeee
Best Replies (or reaction functions)From beliefs to realizations
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
1 2
1 1 2 2 1 2, ( 1) ; ( 1),e e
q qMax q q t Max q t q Cournot Game
r2
r1
r2
r1
00 1
1
q1
q2
00 1
1
q1
q2
Non monotonic reaction functions may lead to several coexisting equilibria
Logistic reaction functions r q q q r q q q1 2 1 2 2 2 1 2 1 11 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 equilibriain the case of homogeneous expectations
223
2126
p
h
22 32
1
1
1
00 1 2 3
s sS sEi
transcritical O =
S
pitchfork E1 =
E1 =
S
4 5
1 3
1 5
61
sEi,C2
Z4
Z2
Z0
E2
E1
S
O 11( )
O
LC b( )
LC a( )
0
0
2.3
2.3
y
x
1 = 2 = 3.4 1 = 2 = 0.2 < 1/(+1)
(a) Z4
Z2
E2
E1
O 11( )
O
LC b( )
LC a( )
0
0
1.4
1.4
y
x
1 = 2 = 3.4 1 = 2 = 0.5 > 1/(+1)
O 1
3( )
O 12( )
Z0
(b)
K
y
x
y’
x’
Z4
Z2
Z0
LC a 1( )
LC b 1( )
LC b( )
LC a( )
-0.5 1.5
1.5
0.50.5 1.5
1.5
K
0.5
Critical curves 21 ( , ) | det ( , ) 0LC x y DT x y 1( ).LC T LC
1 1 1
2 2 2
1 1 2( , )
1 2 1
yDT x y
x
1 21
1 2 1 2
1 11 1:
2 2 4LC x y
y
x
y’
x’
Z4
Z2
Z0LC a
1( )
LC b 1( ) LC b( )
LC a( )
( )1 1 1 1 1
1 1, with
2bLC K k k k
z
( ) ( )1( )b bLC T LC
1 1 3(1 ), ,
4K k k where k
.In the homogeneous case
has a cusp point in
and
Proposition (Homogeneous behavior)
If , 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 then the two basins are simply connected sets;if then the two basins are non connected sets, formed by infinitely many simply connected components.
0
0
1.2
1.1
y
x
1 = 2 = 3.6 1 = 0.55 2 = 0.7
Z4
Z2
Z0
LC a 1( )
LC b 1( )
LC b( )
LC a( )
E2
E1
S
0
0
1.2
1.1
y
x
1 = 2 = 3.6 1 = 0.59 2 = 0.7
Z4
Z2
Z0LC a
1( )
LC b 1( )
LC b( )
LC a( )
E2
E1
S)1(
1H
)2(1H
H 21( )
H 22( )
H 24( )
H 23( )
H0
Case of heterogenous players
0
0
1.1
1.1
y
x
1 = 2 = 3.9 1 = 0.7 2 = 0.8
S
A2
A1
E1
0
0
1.1
1.1
y
x
1 = 2 = 3.95 1 = 0.7 2 = 0.8
S
A2
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
1 1 1 1 1 2 2 3 3
2 2 2 2 2 3 3 1 1
3 3 3 3 3 1 1 2 2
1 1 1
: 1 1 1
1 1 1
q q q q q q
T q q q q q q
q q q q q q
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)
S
VI
VIIVIII
V0 PI
PII
PIII
PIV
QII
QIII
Local StabilityVertices V0 and VII are always repelling;
Interior FP S (if it exists) is a saddle point or a repelling nodeQII and PII are created together (saddle-node)PIII and QIII are created together (saddle-node) PII and PIV cannot coexistPIII and PI cannot coexist
1
1
Fig. 3
x1
x2
O
F1
F2
VIII QIII PIII
PII
QII
VI
B(PII)
B(VI)
B(PIII)
B(VIII)S
VII
1
1x1
x2
(a)
Fig. 5
LC
x2
1
x1
(b)
LC
LC
(c)
1
1x1
x2
PIIIQIII
OVI
VIIVIII
1O
PIIIQIII
LC
VII
0.9650.35
B(PIII)
B(VIII)
B(VI)
PIIIQIIIVIII
VI
S
VII
S
1
10.965
0.35 x1
x2
(d)
LCLC
Z1
Z3
H1 H2 H3PIIIQIII VII
Z1
H1
H2
H3
Fig. 6
0
0
1.1
1.1
x1
qIII 1
pIII 1
qIII 1
pIII 1
c1
qIII 1
pIII 1
c1
1x1
1
0.30.3
0.30.3
0.6
0.6
x1
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
1 2 1 2( , ) ( .i i i iq q q a b q q c q
1 2( 1) ( ) ( ) ( ( ), ( )) ; 1, 2ii i i i
i
q t q t v q t q t q t iq
profit function (linear cost and demand)
Gradient dynamics
' 21 1 1 1 1 1 1 1 2
' 22 2 2 2 2 2 2 1 2
(1 ( )) 2
:
(1 ( )) 2
q v a c q bv q bv q q
T
q v a c q bv q bv q q
The map
E*
q1
q2
8
120
0
v1 = 0.24 v2 = 0.48 c1 = 3 c2 = 5 a = 10 b = 0.5
O
)2(1O
)1(1O
)3(1O
11
12
12
' 2(1 ( )) 2 .j j j j j jq v a c q bv q j i
Each coordinate axis is trapping since qi(t) = 0 implies qi(t+1) = 0
The restriction of the map T to that axis is
conjugate to the standard logistic map
E*
q1
q2
7
0
0
v1 = 0.24 v2 = 0.55 c1 = 3 c2 = 5 a = 10 b = 0.5
)(1bLC
)(1aLC
)(bLC
)(aLC
Z2
Z4
Z0
E*
q1
q2
7
11
0
0
v1 = 0.24 v2 = 0.55 c1 = 3 c2 = 5 a = 10 b = 0.5
)(1bLC
)(1aLC)(bLC
)(aLC
Z2
Z4
Z0
)1(1h
)2(1h
)1(2h
)2(2h
h
contact
E*
q1
q2
110
0
v1 = 0.24 v2 = 0.55 c1 = 3 c2 = 5 a = 10 b = 0.5
)(1bLC
)(1aLC
)(bLC
)(aLC
Z2
Z4
Z0
)2(1O
E*
q2
0
0
v1 = 0.24 v2 = 0.7 c1 = 3 c2 = 5 a = 10 b = 0.5
Z2
Z4
Z0
)2(1O
q1
q2
6
110
0
v1 = 0.24 v2 = 1.0747 c1 = 3 c2 = 5 a = 10 b = 0.5
)(1aLC)(aLC
q1
q2
7
9.50
0
v1 = 0.4065 v2 = 0.535 c1 = 3 c2 = 5 a = 10 b = 0.5
q1
q2
7
9.5
v1 = 0.4065 v2 = 0.535 c1 = 3 c2 = 5 a = 10 b = 0.5
)(1bLC
)(1aLC
)(bLC
Z2
Z4
Z0
)1(1h
)2(1h
)1(2h
)2(2h
h
)(aLC
-0.5
-0.5
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