noninvertible maps and applications: an introductory overview outline what is a noninvertible map...

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

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

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

noninvertible maps and applications: an introductory overview outline what is a noninvertible map...

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