2-dim discrete dynamical systems: iterated maps of the plane

1 / 119

# 2-dim discrete dynamical systems: iterated maps of the plane - PowerPoint PPT Presentation

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 )

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## 2-dim discrete dynamical systems: iterated maps of the plane

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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:

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. x0given ; y0 = 0

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

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

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.

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

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

UNSTABLE FOCUS

STABLE FOCUS

Iml

1

-1

1

Rel

-1

• CENTER

IMPROPER NODE

STAR NODE

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)

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

unstable node

center

if

Det= 1, -2<Tr<2

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:

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.

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
• 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
• There are different ways to exit the unit circle:

Flip bifurcation

(period doubling)

Neimark-Sacker bifurcation

Fold bifurcation

Bifurcattion lines and the creation of new invariant sets

Line of Neimark-Sacker

Line of flip

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,a) = a + x-x2

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

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

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

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)

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 + ….

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

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)

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.

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

a = 1.01

a = 1.02

a = 1.05

a = 1.1

a = 1.3

a = 1.4

a = 1.505

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

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

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 )

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

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

P

LC

Z2

Z0

G

O

R1

R2

LC-1

P

LC-1

A0

Z2

LC

G

Z0

A1

B0

O

h1

R1

R2

B1

C1

C7

G

C2

LC

C3

O

C6

C4

LC-1

C5

P

LC

O

LC-1

P

O

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

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

T:

T

F

F’= T(F)

LC

-1

LC

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)

LC

2

LC

5

LC

6

LC

1

LC

LC

-1

LC

3

4

LC

f:

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

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

1

q2

.

.

ES

ES

c2

E2

E2

.

.

c1

E1

E1

0

0

1

q1

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

y

y’

LC-1

Z2

Z0

U-1,1

U

U-1,2

LC

R1

R2

x

x’

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

CS-1

CS-1

U

R1

R2

R1

R2

T(U)

V

Z2

Z2

CS

CS

Z0

Z0

Q

LC-1

Z2

P

LC

Z0

contact

Z2

LC

Z0

LC-1

Z2

LC

Z0

Z2

LC

Z0

LC-1

Z2

LC

Z0

LC-1

Z2

LC

Z0

LC-1

Z2

LC

Z0

LC-1

Z2

LC

Z0

LC-1

Z2

LC

Z0

LC-1

Z2

LC

Z0

LC-1

Z2

LC

Z0

LC-1

Z2

LC

Z0

1

1

4

5

2

2

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):

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

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

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

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)

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

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

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”

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)

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

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

+

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)

m1 and m2 measure the intensity of the positive externality

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

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)

Dynamical system:

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

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

.

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

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

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

y’

y

Z0

Z2

Z4

x’

x

.

In the homogeneous case

and

has a cusp point in

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.

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 …

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

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

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)

The map

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

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

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

7

q2

Z0

Z2

Z4

-0.5

q1

-0.5

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).

13

13

E*

R1

R2

E2

E2

R2

E*

R1

O

E1

E1

13

13

O

13

Z0

13

Z0

H0

q2

q2

Z2

Z2

Z4

Z4

q1

O

13

O

13

q1