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2-dim discrete dynamical systems: iterated maps of the plane. y. T. T. T. T. x. Example. Cournot Duopoly. Two firms produce a homogeneous good and interact in a competitive market, choosing the quantities: q 1 ( t ) e q 2 ( t )

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cournot duopoly
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

On the basis of the previous assumptions, we obtain

And assuming naive expectations:

slide3

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

slide4

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. x0given ; y0 = 0

slide6

Linear 2-dim. map

Remembering the case of 1-dim linear maps let’s consider the trial solution:

And substitute it in the law of evolution:

And after dividing by lt we get (A lI) v = lv

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

slide7

Characteristic equation det (AlI) = 0 becomes:

P(l) = l2Tr∙l + Det = 0

where Tr = a11+a22 ;Det = a11a22 a12a21

(I) D=Tr24Det >0 then l1 and l2 real and distinct eigenvalues exist with correnponding linearly independent eigenvectors v1, v2, that give rise to two independent soutions and

(II) D=0 coincident eigenvalues l, with eigenvector v give two independend solutions ltv and tltv

(III) D <0,

l1,2=

Two independent complex conjugate solutions

slide8

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, l1 and l2. Denoting by v1 e v2 two eigenvectors respectively associated with them, we obtain

(II) Real and equal eigenvalues of A:

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:

where h = h1 + ih2 is an eigenvector associated with l.

slide9

Stability of the unique equilibrium:

|l|<1

i.e. all eigenvalues

inside the unoi circle

of the complex plane

Iml

1

-1

1

Rel

-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
slide10

Iml

1

-1

1

Rel

Iml

Iml

Iml

Iml

Iml

1

1

1

1

1

-1

-1

-1

-1

-1

-1

1

1

1

1

1

Rel

Rel

Rel

Rel

Rel

-1

-1

-1

-1

-1

STABLE NODE

UNSTABLE NODE

SADDLE

SADDLE

UNSTABLE FOCUS

STABLE FOCUS

slide11

Iml

1

-1

1

Rel

-1

  • CENTER

IMPROPER NODE

STAR NODE

slide12
Second order
    • real and distinct eigenvalues:
      • if |l1| < 1and |l2| < 1 , the origin is globally

asymptotically stable (stable node)

      • if |l1| > 1and |l2| > 1 , the origin is unstable (unstable node)
      • if |l1| < 1and |l2| > 1 , the origin is unstable (saddle)
    • equal eigenvalues :
      • if |l|< 1, the origin is gloablly asymptotically stable (stable node)
      • if |l|< 1, the origin is unstable (unstable node)
      • if the matrix A is diagonal: the origin è stable if |l|< 1, unstable if

|l|> 1 (star node)

    • complex conjugated eigenvalues
      • if r < 1, the origin is globally asymptotically stable (stable focus)
      • if r > 1 , the origin is unstable (unstable focus)
      • if r = 1, the origin is stable (center)
slide13

Stability triangle

unstable node

D = Tr24Det=0

1+Tr+Det=0

(Flip curve)

1Tr+Det=0

(Fold curve)

unstable focus

center

if

detA = 1, -2<trA<2

Det= 1 (N-S curve)

stable focus

stable node

stable node

saddle

saddle

unstable node

center

if

Det= 1, -2<Tr<2

slide14

Cournot Duopoly

The model we considered is described by the system of two I^ order linear difference equations

The matrix of the system is

with distinct real eigenvalues:

and the eigenvectors associated with are

Solution:

slide15

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.
slide16

Nonlinear maps of the plane: local stability of a fixed point

Let (x*,y*) be a solution of :

Linear approximation around (x*,y*)

Linear homogeneous system in X = xx* ; Y = yy*

Jacobian matrix

With

stability of the equilibrium points
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* 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
Local bifurcations in a discrete dynamical system
  • There are different ways to exit the unit circle:

Flip bifurcation

(period doubling)

Neimark-Sacker bifurcation

Fold bifurcation

slide19

Bifurcattion lines and the creation of new invariant sets

Line of Neimark-Sacker

Line of saddle-node

Line of flip

Where A is the Jacobian matrix computed at the equilibrium considered

slide20

An eigenvalue equals to 1

Saddle-Node bifurcation: two fixed points appear, one stable and one unstable

Normal form: f(x,a) = a + x-x2

slide21

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,a) = a x + x-x3

supercritical

subcritical

slide22

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,a) = -(1+a)x + x3

supercritical

slide23

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 a = a0.

l1

,k=1,2,3,4

(non resonance conditions)

l2

(transversality condition)

two alternative situations

slide24

P* becomes unstable and an attracting closed curve GS appears around it

(supercritical)

  • P* becomes unstable merging with a repelling closed curve GU,existing
  • when it is stable (subcritical)
slide25

Neimark-Sacker bifurcation:The eigenvalues of the Jacobian matrix DT(P*)evaluated at the fixed point P* are complex and cross the unit circle.

,k=1,2,3,4

(non resonance conditions)

(transversality condition)

After rescaling  P* = 0,ao = 0

nonlinear terms

linear terms

complex variable:

z® l1(a) z + g(z,z,a)

z=x1+ix2

change of variable:

z=w+h(w)

w® m1(a) w + c1w2w + ….

r® r(1+ da + ar2 + ….)

polar coordinates:

w=reib

b® b+ q0 + ea + br2 + ….

slide26

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 aa0
  • Amplitude

On the invariant curve:

  • dense quasiperiodic orbits or
  • a finite number of periodic orbits, saddles and nodes, appearing and disappearing via Saddle-Node
slide27

Arnold tongues

inside the Arnold tongues the rotation number is rational

m1(a)

schematic

SN bifurcations

bifurcation point (a=ao=0)

  • Infinitely many tongues, of thickness  d (q-2)/2

(d is the distance from the unit circle)

slide28

Inside an Arnol’d tongue 1/6

for a stable closed invariant curve

(supercritical Neimark-Sacker)

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.

slide29

Example: Iterated map T

fixed points: O = (0,0)

P = (a,a)

Supercritical Neimark-Sacker bifurcation of O occurs at a = 1

O stable focus for a <1

unstable focus for a >1

slide30

a = 1.01

a = 1.02

a = 1.05

a = 1.1

a = 1.3

a = 1.4

a = 1.505

slide31

T : Rn Rn p’ = T(p)

.

p1

T

.

Noninvertible map

means “Many-to-One”

.

p’

p2

T

.

p1

.

T1-1

Equivalently, we say that

p’ has several rank-1 preimages

.

p’

p2

T2-1

Several distinct inverses are defined:

i.e. the inverse relation p = T-1(p’) is multivalued

Zk

LC

Rn can be divided into regions (or zones) according

to the number of rank-1 preimages

Zk+2

Zk: region where k distinct inverses are defined

LC (critical manifold): locus of points having two merging preimages

slide32

Linear map T : (x,y)→(x’,y’)

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

T is orientation reversing if det A < 0

a11=1 a12=1.5 a21=1 a22 =1 b1= b2= 0; Det = - 0.5

a11=2 a12= -1 a21=1 a22=1 b1= b2= 0 ; Det = 3

y

y

C

C’

C’

C

y

T

T

y

B’

B’

F

F

B’

F’

F’

A

B

B

A

A’

A’

x

x

x

x

slide33

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 )

slide34

A noninvertible map of the plane “folds and pleats”' the plane

so that distinct points are mapped into the same point.

T

LC-1

LC = T(LC-1)

R2

R1

Z2

Z0

SH1

SH2

LC-1

LC

Z2

Z0

R2

R1

Riemann Foliation

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 sheets, each

associated with a different inverse, connected by folds along LC

slide35

fixed points: O = (0,0)

P = (l,l)

Example:

LC = {(x,y) | y = x –l2/4}

LC-1 = {(x,y) | x = l/2 }

Z2 = {(x,y) | y > x– l2/4}

Z0 = {(x,y) | y < x– l2/4 y < b }

det DT = a 2x = 0 for x = l/2

T({x = a/2 }) = {y = x –a2/4}

Supercritical Neimark-Sacker bif. at a = 1

slide36

P

LC

Z2

Z0

G

O

R1

R2

LC-1

slide37

P

LC-1

A0

Z2

LC

G

Z0

A1

B0

O

h1

R1

R2

B1

slide38

C1

C7

G

C2

LC

C3

O

C6

C4

LC-1

C5

slide39

P

LC

O

LC-1

slide40

P

O

slide41

Mappa non invertibile

y

a = 1 b = -2

y

.

.

.

.

3

3

P = T(P1) = T(P2)

2

2

P

.

.

1

1

x

x

-3

-2

1

3

2

-1

-3

-2

1

3

2

-1

-1

-1

-2

-2

2 inverse

slide42

det DT = -2x =0 for x=0 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 }

R1

SH1

R2

SH2

LC-1

Z2

y=b

LC

x=0

Z0

slide44

T:

T

F

F’= T(F)

LC

-1

LC

slide45

LC-1

LC-1

y

y

C

D

C

B

A

B

A

O

B’

C’

A’

B’

A’

D’

LC

LC

C’

O’

x

x

LC-1

LC-1

y

B

y

B

A

C

C

A

C’

B’

C’

B’

A’

A’

LC

LC

x

x

(b)

(a)

slide46

LC

2

LC

5

LC

6

LC

1

LC

LC

-1

LC

3

4

LC

f:

slide53

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

(b)

0

x

1.5

(a)

0

x

1.5

Two kinds of complexity

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

slide54

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

ES

c2

c2

.

.

c1

c1

0

0

1

q1

slide55

1

q2

.

.

ES

ES

c2

E2

E2

.

.

c1

E1

E1

0

0

1

q1

slide57

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 ?

slide58

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)

slide59

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)

slide61

SH2

SH1

y

y’

LC-1

Z2

Z0

U-1,1

U

U-1,2

LC

R1

R2

x

x’

slide62

map

2 fixed points

y

T

2 inverses

T

T

T

x

Z2 = {(x,y) | y > b }

Z0 = {(x,y) | y < b }

LC = {(x,y) | y = b }

LC-1 = {(x,y) | x = 0 }

SH1

det DT = -2x =0 for x=0

T({x=0}) = {y=b}

R1

SH2

R2

Z2

y=b

LC

LC-1

Z0

slide63

CS-1

CS-1

U

R1

R2

R1

R2

T(U)

V

Z2

Z2

CS

CS

Z0

Z0

slide64

Q

LC-1

Z2

P

LC

Z0

contact

Z2

LC

Z0

slide65

LC-1

Z2

LC

Z0

slide66

Z2

LC

Z0

slide67

LC-1

Z2

LC

Z0

slide68

LC-1

Z2

LC

Z0

slide69

LC-1

Z2

LC

Z0

slide70

LC-1

Z2

LC

Z0

slide71

LC-1

Z2

LC

Z0

slide72

LC-1

Z2

LC

Z0

slide74

LC-1

Z2

LC

Z0

slide75

LC-1

Z2

LC

Z0

slide77

1

1

4

5

2

2

6

3

3

slide78

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

slide79

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

Each firm considers the output of its competitor as given

slide80

q2

.

q1 = r1(q2)

Cournot-Nash Equilibrium

q2 = r2(q1)

q1

Expectation of agent i about the rival’s choice

Rational expectations (perfect foresight):

One-shot (static) game

The game directly goes to the intersection(s) of the reaction curves (Cournot-Nash equilibrium) in one shot

slide81

Cournot (Naive) expectations:

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)

slide82

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.

q2

S.O.C.

q2 = r2(q1)

Equilibrium:

Cournot-Nash Equilibrium

q1 = r1(q2)

q1

slide83

r1

r1

r2

r2

Linear/linear Cournot game and best reply dynamics with naive expectations

q2

Reaction function of firm 1

Reaction function of firm 2

q*2

q1

q*1

slide84

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 consideredand 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”

slide85

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:

b2 < 4(b+e1)(b+e2)

stability if

eigenvalues:

b2 > 4(b+e1)(b+e2)

b2 < 4(b+e1)(b+e2)

(Stable)

(Unstable)

slide86

Linear demand, quadratic costs, case b2 > 4(b+e1)(b+e2)

E unstable, E1 , E2, stable

L2

basin of E1

x2

basin of E2

c2

E2

basin of 2-cycle (c1,c2)

R2

R1

E

R1

R2

c1

0

x1

L1

0

E1

slide87

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..

slide88

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…”

slide89

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

+

slide90

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 !

slide91

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)

m1 and m2 measure the intensity of the positive externality

slide93

Bischi, G.I., C. Mammana and L. Gardini (2000) «Multistability and cyclic attractors in duopoly games», Chaos, Solitons and Fractals.

Cournot with naive expectations (Best reply dynamics):

And reaction functions

slide96

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)

Adaptive expectations

Dynamical system:

slide97

Existence and local stability of the equilibria

in the case of homogeneous expectations a1 = a2 = a

3

1

1

Ws(Ei,C2)

a

pitchfork E1 = E1 = S

transcritical O = S

Ws(O)

Ws(S)

Ws(Ei)

0

3

0

1

2

4

5

m

slide98

m1 = m2 = 3.4 a1 = a2 = 0.2 < 1/(m+1)

m1 = m2 = 3.4 a1 = a2 = 0.5 > 1/(m+1)

2.3

1.4

y

y

Z0

Z0

D

D

E2

E2

K

E1

Z2

Z2

S

E1

Z4

0

0

Z4

0

(a)

x

2.3

0

(b)

x

1.4

slide99

.

Noninvertible (“Many-to-One”) map

T

p1

.

p’

p2

T

.

p1

.

SH2

T1-1

SH1

.

y

y’

p’

LC-1

Z2

p2

T2-1

Z0

U-1,1

U

U-1,2

LC

R1

R2

x

x’

.

Distinct points are mapped into the same point

Folding action of T

Equivalently, we say that p’ has several rank-1 preimages

Unfolding action of T-1

slide100

m1 = m2 = 3 a1 = a2 = 0.5

m1 = m2 = 3 a1 = a2 = 0.5

1.5

1.5

y

y’

Z0

Z2

K

Z4

0.5

0.5

x

x’

-0.5

1.5

0.5

1.5

Critical curves

slide101

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.

In the homogeneous case

LC-1

LC

has a cusp point in

slide102

y’

y

Z0

Z2

Z4

x’

x

.

In the homogeneous case

and

has a cusp point in

slide103

Theorem (Homogeneous behavior)

If m1 = m2 =m,a1=a2=a, 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.

Ifa(m+1)<1 then the two basins are simply connected sets;

if a(m+1)>1then the two basins are non connected sets, formed by infinitely many simply connected components.

slide104

Case of heterogenous players

m1 = m2 = 3.6 a1 = 0.55 a2 = 0.7

m1 = m2 = 3.6 a1 = 0.59 a2 = 0.7

1.2

1.2

y

y

.

.

Z0

Z0

.

.

E2

Z2

Z2

E2

S

S

.

.

Z4

Z4

E1

E1

0

0

0

x

1.1

0

x

1.1

Theorem …

slide106

m1 = m2 = 3.9 a1 = 0.7 a2 = 0.8

m1 = m2 = 3.95 a1 = 0.7 a2 = 0.8

1.1

1.1

y

y

A2

.

.

A2

S

A1

S

.

E1

0

0

0

x

1.1

0

x

1.1

slide107

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 

slide108

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)

Gradient dynamics

The map

slide109

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

v1 = 0.24 v2 = 0.48 c1 = 3 c2 = 5 a = 10 b = 0.5

8

q2

E*

0

O

q1

0

12

slide110

v1 = 0.24 v2 = 0.55 c1 = 3 c2 = 5 a = 10 b = 0.5

7

q2

Z0

Z2

E*

Z4

0

0

v1 = 0.24 v2 = 0.55 c1 = 3 c2 = 5 a = 10 b = 0.5

11

contact

7

q2

q1

Z0

Z2

E*

Z4

0

q1

0

slide111

v1 = 0.24 v2 = 0.7 c1 = 3 c2 = 5 a = 10 b = 0.5

v1 = 0.24 v2 = 0.55 c1 = 3 c2 = 5 a = 10 b = 0.5

Z0

Z0

q2

q2

E*

Z2

Z2

E*

Z4

Z4

0

0

0

11

0

q1

slide115

Bischi, G.I. and F. Lamantia «Nonlinear Duopoly Games with Positive Cost Externalities due to Spillover Effects» Chaos, Solitons & Fractals, vol. 13 (2002).

slide117

13

13

E*

R1

R2

E2

E2

R2

E*

R1

O

E1

E1

13

13

O

slide119

13

Z0

13

Z0

H0

q2

q2

Z2

Z2

Z4

Z4

q1

O

13

O

13

q1