1. x y T T T T 2-dim discrete dynamical systems: iterated maps of the plane Example

2. Higher order difference equations Example: x t+1 + bx t + cx t-1 = 0 With two initial conditions: x 0, x 1 Let y t = x t-1 Then the difference equation in transformed into: x t+1 =  bx t  cy t y t+1 = x t

3. Nonautonomous difference equation. Example. x t+1 = f(x t,t) with i.c. x 0 given Let y t = t Then x t+1 = f(x t,y t ) y t+1 = y t + 1 with i.c. x 0 given ; y 0 = 0

5. 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 t we get (A  v = v 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

6. Characteristic equation det (A  I) = 0 becomes: P( ) = 2  Tr∙ + Det = 0 where Tr = a 11 +a 22 ;Det = a 11 a 22  a 12 a 21 (I)  =Tr 2  4Det >0 then 1 and 2 real and distinct eigenvalues exist with correnponding linearly independent eigenvectors v 1, v 2, that give rise to two independent soutions and (II)  =0 coincident eigenvalues  with eigenvector v give two independend solutions t v and t t v (III)  <0, 1,2 = Two independent complex conjugate solutions

7. (I) Real and distinct eigenvalues of A, 1 and 2. Denoting by v 1 e v 2 two eigenvectors respectively associated with them, we obtain (II) Real and equal eigenvalues of A: where c 1 and c 2 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 solutions are solutions as well, being: where h = h 1 + i h 2 is an eigenvector associated with. eneric solution of the linear homogeneous system is; Any linear combination of solutions is a solution, hence the generic solution of the linear homogeneous system is;

8. Re Im 1 1 Re Im 1 1 Re Im 1 1 Re Im 1 1 Re Im 1 1 Re Im 1 1 STABLE NODE UNSTABLE NODE SADDLE STABLE FOCUS UNSTABLE FOCUS

9. Re Im 1 1 CENTER IMPROPER NODE STAR NODE

10. saddle stable node unstable node saddle unstable node Stability triangle  = Tr 2  4Det=0 1  Tr+Det=0 (Fold curve) 1+Tr+Det=0 (Flip curve) stable node stable focus unstable focus center if Det = 1, -2<Tr<2 Det= 1 (N-S curve) P( ) = 2  Tr∙ + Det = 0 unstable node

11. Stability of the unique equilibrium: | |<1 i.e. all eigenvalues inside the unoi circle of the complex plane Re Im 1 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

12. Second order – real and distinct eigenvalues: if | 1 | < 1and | 2 | < 1, the origin is globally asymptotically stable (stable node) if | 1 | > 1and | 2 | > 1, the origin is unstable (unstable node) if | 1 | 1, the origin is unstable (saddle) –equal eigenvalues : if | |  < 1, the origin is gloablly asymptotically stable (stable node) if | |  < 1, the origin is unstable (unstable node) if the matrix A is diagonal: the origin è stable if | |  < 1, unstable if | |  > 1 (star node) –complex conjugated eigenvalues if  < 1, the origin is globally asymptotically stable (stable focus) if  > 1, the origin is unstable (unstable focus) if  = 1, the origin is stable (center)

13. 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 IR n –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.

14. Nonlinear maps of the plane: local stability of a fixed point Let (x*,y*) be a solution of : Linear approximation around (x*,y*) With Linear homogeneous system in X = x  x* ; Y = y  y* Jacobian matrix

15. 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  then x * is globally asymptotically stable.

16. Local bifurcations in a discrete dynamical system There are different ways to exit the unit circle: Fold bifurcation Flip bifurcation (period doubling) Neimark-Sacker bifurcation

17. Bifurcattion lines and the creation of new invariant sets Line of saddle-node Line of flip Line of Neimark-Sacker Where A is the Jacobian matrix computed at the equilibrium considered

18. Normal form: f(x,  ) =  + x  x 2 An eigenvalue equals to 1 Saddle-Node bifurcation: two fixed points appear, one stable and one unstable

19. subcritical supercritical Normal form:f(x,  ) =  x + x  x 3 An eigenvalue equals to 1: An eigenvalue equals to 1: Pitchfork bifurcation:a fixed point becomes unstable (stable) and two further fixed points appear, both stable (unstable)

20. supercritical Normal form:f(x,  ) =  x + x 3 An eigenvalue equals to -1: 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.

21.   two alternative situations  k=1,2,3,4 (non resonance conditions) 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  . (transversality condition)

22. P * becomes unstable merging with a repelling closed curve  U,existing when it is stable (subcritical) P * becomes unstable and an attracting closed curve  S appears around it (supercritical)

23. z  1 (  ) z + g(z,z,  ) complex variable: nonlinear terms z=x 1 +ix 2 After rescaling  After rescaling  P* = 0,  o = 0 linear terms change of variable: w  1 (  ) w + c 1 w 2 w + …. z=w+h(w) polar coordinates: w=re i  r  r  (1+ d  + ar 2 + ….)   +    e  + br 2 + ….  k=1,2,3,4 (non resonance conditions) Neimark-Sacker bifurcation:The eigenvalues of the Jacobian matrix DT(P*) evaluated at the fixed point P * are complex and cross the unit circle. (transversality condition)

24. The circle slightly deforms, but: remains an invariant curve remains an invariant curve maintains its “stability” maintains its “stability” approches a circle for approches a circle for   Amplitude On the invariant curve: dense quasiperiodic orbits or dense quasiperiodic orbits or a finite number of periodic orbits, saddles and nodes, appearing and disappearing via Saddle-Node a finite number of periodic orbits, saddles and nodes, appearing and disappearing via Saddle-Node As the bifurcation parameter moves away from the N-S bifurcation value:

25. Arnold tongues inside the Arnold tongues the rotation number is rational bifurcation point bifurcation point (    1()1() SN bifurcations schematic Infinitely many tongues, of thickness Infinitely many tongues, of thickness  d (q-2)/2 (is the distance from the unit circle) (d is the distance from the unit circle)

26. 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.

27. fixed points: O = (0,0) P = (  ) Supercritical Neimark-Sacker bifurcation of O occurs at  = 1 O stable focus for  <1 unstable focus for  >1 Example: Iterated map T

28.       

29. Problema del monopolista: Più produco e più guadagno? q = quantità prodotta p = prezzo unitario di vendita c = costo unitario di produzione Profitto = Ricavo – Costo = p q – c q = (p – c) q Teorema. Se p > c allora il profitto cresce ogniqualvolta cresce la produzione Ma ci sono sempre dei consumatori disposti a comprare ciò che si produce al prezzo imposto dal monopolista ?

30. q (quantità venduta) e p (prezzo di vendita) non sono indipendenti Il prezzo decresce al crescere della quantità ovvero la quantità acquistata è funzione decrescente del prezzo Esempio: Funzione di domanda lineare p q p = a – b q funzione inversa di domanda A=p max per merce rara p→0 pur di vendere Tutta la produzione

31.  = f (q) = – b q 2 + (a – c) q Profitto quantità prodotta è una parabola! profitto del monopolista = p q – c q = (a – b q) q – cq

32. Problema del duopolio A. Cournot, Récherches sur les principes matématiques de la théorie de la richesse, 1838. Due produttori, 1 e 2, vendono lo stesso prodotto Il produttore 1 produce e immette nel mercato q 1 con costi c 1 q 1 Il produttore 2 produce e immette nel mercato q 2 con costi c 2 q 2 prezzo: p = A – B Q TOT = A – B ( q 1 + q 2 ) Profitto produttore 1:  1 = pq 1 – c 1 q 1 = [ A – B ( q 1 + q 2 )]q 1 – c 1 q 1 Profitto produttore 2:  2 = pq 2 – c 2 q 2 = [ A – B ( q 1 + q 2 )]q 2 – c 2 q 2

33.  1 = [ A – B ( q 1 + q 2 )]q 1 – c 1 q 1 = – Bq 1 2 + (A – c 1 –Bq 2 )q 1 Max per  2 = [ A – B ( q 1 + q 2 )]q 2 – c 2 q 2 = – Bq 2 2 + (A – c 2 Bq 1 )q 2 Max per q2q2 q1q1 q 2 = r 2 (q 1 ) q 1 = r 1 (q 2 ) Equilibrio: Equilibrio di Cournot-Nash

34. Cournot Duopoly Two firms produce q 1 (t) e q 2 (t) Inverse demand function: p = f (q 1 +q 2 ) = a – b (q 1 + q 2 ) Production costs: c i (q 1, q 2 ) = c i q i, i = 1,2 Each period profit:    q i f (q 1 + q 2 ) – c i (q 1, q 2 ) At each stage, they simultaneously solve the problems and get Assuming naive expectations:

35. 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:

36. q1q1 q2q2 q 2 = r 2 (q 1 ) q 1 = r 1 (q 2 ) Cournot-Nash Equilibrium. Perfect foresight: One-shot (static) game The game goes to the intersection(s) of the reaction curves (Cournot-Nash equilibrium) in one shot Expectation of agent i about the rival’s choice

37. Two-dimensional dynamical system: given (q 1 (0),q 2 (0)) the repeated application of the map T:(q 1,q 2 )  (r 1 (q 2 ), r 2 (q 1 )) 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) Cournot (Naive) expectations:

38. Linear demand p = a – b (q 1 + q 2 ) Linear cost C i = c i q i i = 1,2 Quadratic Profit:    a – b (q 1 + q 2 ))q i – c i q i R1R1 R2R2 R1R1 R2R2 F.O.C. S.O.C.

39. 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 considered and 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”

40. Linear demand: p = a – b (q 1 + q 2 ) Quadratic cost: C i = c i q i + e i q i 2 i = 1,2 Quadratic Profit:  i  a – b (q 1 + q 2 ))q i – (c i q i + e i q i 2 ) Linear reaction functions: eigenvalues: stability if (Stable) (Unstable) b 2 < 4(b+e 1 )(b+e 2 ) b 2 > 4(b+e 1 )(b+e 2 )

41. R1R1 R1R1 R2R2 R2R2 E2E2 E1E1 x1x1 x2x2 L2L2 L1L1 0 0 E c1c1 c2c2 Linear demand, quadratic costs, case b 2 > 4(b+e 1 )(b+e 2 ) E unstable, E 1, E 2, stable basin of E 1 basin of E 2 basin of 2-cycle (c 1,c 2 )

42. 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..

43. Postom and Stewart "Catastrophe Theory and its Applications", Pitman 1978 “… not treat such phenomena as program bugs, and not to steer their patameters or their model away from realism to eliminate them, such explicit models “drawn from life” will become common…” 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…”

44. 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…”

45. 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 – +

46. 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 !

47. Kopel, M. (1996) Simple and complex adjustment dynamics in Cournot Duopoly Models. Chaos, Solitons, and Fractals. Linear demand function, cost function C i = C i (q 1,q 2 ) with positive cost externalities (spillover effect which gives some advantages due to the presence of the competitor)  1 and  2 measure the intensity of the positive externality

48. Adaptive adjustment (inertia, or anchoring )

49. 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

52. Cournot Game (from beliefs to realizations) Adaptive expectations Dynamical system: 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?

53. Existence and local stability of the equilibria in the case of homogeneous expectations        1 0 012 3  s sSsSsEisEi transcritical O = S pitchfork E 1 = E 1 = S 45 1 3  s  E i,C 2 

54. Z4Z4 Z2Z2 Z0Z0 E2E2 E1E1 S 0 0 2.3 y x  1 =  2 = 3.4  1 =  2 = 0.2 < 1/(  +1)  (a) Z4Z4 Z2Z2 E2E2 E1E1 0 0 1.4 y x  1 =  2 = 3.4  1 =  2 = 0.5 > 1/(  +1)  Z0Z0 (b) K

55. Modello bidimensionale (lineare) dell'interazione fra fondamentalisti e chartisti nel mercato di un titolo Sia p(t) il logaritmo del prezzo del titolo considerato nel periodo t, p* il logaritmo del prezzo considerato di equilibrio dai fondamentalisti. Sia D f (t) = a (p*  p(t)) a > 0 l'eccesso di domanda da parte dei fondamentalisti D c (t)= b (p(t)  p(t  1)) b>0 l'eccesso di domanda dei chartisti, che prevedono un mantenimento dell'attuale trend dei prezzi, e quindi comprano (vendono) se il trend osservato è crescente (decrescente). Il prezzo si aggiusta nel modo usuale: p(t+1)= p(t) +  D(t)= p(t) +  [D f (t) + D c (t)]  > 0

56. Mettendo un po' in ordine il modello diventa: p(t+1) = [1 +  ]p(t)  p(t  1)+  p* un'equazione alle differenze del secondo ordine: occorre assegnare p(0) e p(1) per poter ottenere p(2) ecc. Può essere riscritto in una forma equivalente di due equazioni alle differenze del primo ordine introducendo la variabile "di servizio" z(t) = p(t  1) da cui otteniamo il modello bidimensionale nel piano (z,p): z(t+1) = p(t) p(t+1) =  z(t) + [1 +  ]p(t) +  p* Modello lineare a tempo discreto: bidimensionale: da z(0), p(0) si ottengono per iterazione le traiettorie nel piano (z(t), p(t)).

57. Imponendo le condizioni di stazionarietà: z(t+1)=z(t)=z p(t+1)=p(t) = p si ottiene come unico equilibrio z = p = p* ovvero se il sistema converge a un equilibrio non può che essere il prezzo ipotizzato dai fondamentalisti. Ma sarà sempre stabile? Matrice Jacobiana (matrice dei coefficienti essendo il modelli lineare) Tr = 1+  ; Det =  Condizioni di stabilità: 1  Tr+Det>0; 1+Tr+Det>0; Det<1, diventano:

58.  >0 sempre verificata;  >0, ovvero  , ovvero  riportando queste condizioni sul piano dei parametri a,b otteniamo la regione di stabilità

59. In ogni caso il sistema esplode quando l'equilibrio perde stabilità in quanto il modello è lineare. Si può utilizzare una domanda dei chartisti non lineare di forma sigmoidale invece di lineare: ad esempio 2/  arctan(  *(p(t) - p(t-1))) oppure tanh(  *(p(t) - p(t-1))) che significa una certa prudenza, almeno in presenza di variazioni molto ampie dei prezzi. Allora abbiamo la biforcazione Neimark- Sacker quando  supera una certa soglia, con conseguente creazione di oscillazioni persistenti (autosostenute) quasi-periodiche e anche oscillazioni caotiche. Analogamente quando  esce attraverso la curva di biforcazione flip si crea un ciclo di periodo 2 stabile e poi la “cascata” di raddoppi periodo che porta alle dinamiche caotiche.

60. Noninvertible map means “Many-to-One”. T p’ p1p1 p2p2 T.. Equivalently, we say that p’ has several rank-1 preimages. T 1 -1 p’ p1p1 p2p2 T 2 -1.. ZkZk Z k+2 Z k : region where k distinct inverses are defined LC (critical manifold): locus of points having two merging preimages LC Several distinct inverses are defined: i.e. the inverse relation p = T -1 (p’) is multivalued R n can be divided into regions (or zones) according to the number of rank-1 preimages T : R n  R n p’ = T (p)

61. Linear map T : (x,y)→(x’,y’) T is orientation preserving if det A > 0 area (F’) = |det A |area (F), i.e. |det A | 1) contraction (expansion) Meaning of the sign of |det A| T is orientation reversing if det A < 0 y x x y a 11 =2 a 12 = -1 a 21 =1 a 22 =1 b 1 = b 2 = 0 ; Det = 3 A B C A’ B’ C’ F F’ T a 11 =1 a 12 =1.5 a 21 =1 a 22 =1 b 1 = b 2 = 0; Det = - 0.5 x x y A B C A’ B’ C’ F’ T F B’ y

62. 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 in fact, 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 )

63. LC -1 LC Z2Z2 Z0Z0 R2R2 R1R1 SH 1 SH 2 Riemann Foliation LC -1 LC = T(LC -1 ) T Z2Z2 Z0Z0 R2R2 R1R1 A noninvertible map of the plane “folds and pleats”' the plane so that distinct points are mapped into the same point. A region Z k is seen as the superposition of k sheets, each associated with a different inverse, connected by folds along LC A point has several distinct preimages, i.e. several inverses are defined in it, which “unfold” the plane

64. Z 2 = {(x,y) | y > x– 2 /4} Z 0 = {(x,y) | y < x– 2 /4 y < b } LC = {(x,y) | y = x – 2 /4} LC -1 = {(x,y) | x = /2 } det DT =  2x = 0 for x = /2 T({x =  /2 }) = {y = x –  2 /4} Example: fixed points: O = (0,0) P = (  ) Supercritical Neimark-Sacker bif. at  = 1

65. LC -1 O P LC  Z0Z0 Z2Z2 R2R2 R1R1

66. LC -1 O P LC  Z0Z0 Z2Z2 R2R2 R1R1 A0A0 B0B0 A1A1 B1B1 h1h1

67. LC -1 O LC  C1C1 C2C2 C3C3 C4C4 C5C5 C6C6 C7C7

68. LC -1 O LC P

69. P O

70. x y T T T T Mappe iterate del piano

71. Point mapping f : R n  R n x’ = f ( x ) x  R n Takes a point in R n and moves it to a new position If S is a set of points then f(S) = { f(x) | x  S} Lineland : f : R  R x’ = f(x) f f A B x x’ A’ B’ f A B A’ B’ f A B A’B’ Flatland f : R 2  R 2 f S f (x,y) S’ (x’,y’)

72. 2-dimensional linear maps: contractions, expansions, rotations etc. 2dim-linear x’ = ax + by +c y’ = dx + ey +f rotation

73. A B C A’ B’ C’ F’ T Linear map T: T is orientation preserving if det A > 0 area (F’) = |det A | area (F) |det A | 1) contraction (expansion) Meaning of the sign of |det A | F A’ B’ C’ T is orientation reversing if det A < 0 F’ F A B C T

75. Henon map: nonlinear, invertible transforms a line y=k into a parabola is a linear contraction in x direction for |b|<1 is a reflection through the diagonal S T(S) T 2 (S) T 3 (S) det DT = b

76. LC = T(LC - 1 ) Equivalently, we say that p’ has several rank-1 preimages LCLC LC - 1 SH 2 SH 1 R1R1 R2R2 Z2Z2 Z0Z0 U U - 1,2 U - 1,1 x’x’ y’y’ y x T p’ p1p1 p2p2 T... Noninvertible (Many-to-One) map: Distinct points are mapped into the same point Folding action of T LC -1. T 1 -1 p’ p1p1 p2p2 T 2 -1.. Unfolding action of T - 1

77. -2 x.. P = T(P 1 ) = T(P 2 ) x y. 1 2 3 1 2 3 -2-3.. P y. 1 2 3 1 2 3 -2-3 -2 2 inverses Noninvertible maps: many to one

78. 2D example T: det DT = -2x =0 for x=0 T({x=0})  {y=b} LC = {(x,y) | y = b } LC -1 = {(x,y) | x = 0 }

79. F T LC LC F’= T(F)

80. LC LC -1 A B B’ A’ A B O’ C’ x x y y C D D’ O C A’B’ LC LC -1 A B B’ A’ A B C’ xx y y C C A’ B’ Itera…

81. miraquad

83. LC -1 LC 1 LC LC 2 LC 3 LC 4 LC 5

84. T: miraquad

92. http://paulbourke.net/fractals/ Fractals, Chaos http://paulbourke.net/fractals/clifford/ Clifford Attractors Definition x n+1 = sin(a y n ) + c cos(a x n ) y n+1 = sin(b x n ) + d cos(b y n ) where a, b, c, d Are parameters that define each attractor. a = -1.4, b = 1.6, c = 1.0, d = 0.7

93. a = 1.1, b = -1.0, c = 1.0, d = 1.5

94. a = 1.6, b = -0.6, c = -1.2, d = 1.6

95. a = 1.7, b = 1.7, c = 0.06, d = 1.2 a = 1.3, b = 1.7, c = 0.5, d = 1.4

96. a = 1.5, b = -1.8, c = 1.6, d = 0.9 Lovely renderingsLovely renderings by Thomas Burt.

97. Let us choose a polynomial of the form: f(x,y) = a x 2 + b y 2 + c x y + d x + e y + f where a, b, c, d, e, f are constants discussed later and x, y are the usual coordinates in 2-space. The simplest way to turn this polynomial into a map is known as coordinate rotation : y new = f(x old, y old ) x new = y old OK, say you, what about those 6 constants? A willy-nilly choice of those will not give you a strange attractor. There may be (probably is) some way to telling if a particular set of constants produces an attractor but, I am not a mathematician, which means I don't really care -- I basically write a program to pick them at random and see what comes out. Most sets diverge, some converge to a point, some converge to a boring loop, and only few produce good-looking pictures. These things are called attractors, not because they're attractive, but because they attract reasonable points Here is a short C/C++ program I wrote. Here What is a strange attractor? I, of course, do not know the formal, mathematical, definition of Chaotic Attractors, but I will do my best to correctly guess it. Strange Attractor is a collection of points such that each point is a function of another point What kind of function? Everything from polynomials to transcendentals. Like a randomly-appearing mosaic - instead of individual features appearing one after the other, dots light up and eventually compose distinct shapes.

98. Coupled Twisted Logistic: four parameter algebraic map of 4 th degree Chaos and symmetry

100. Symmetric case: Coutl simmetrica

103. Conjugate to the map by the linear homeomorphism from which:

110. http://people.mbi.ohio-state.edu/mgolubitsky/ Chaos and Symmetry. Mike Field, Martin Golubitsky,

111. Symmetric Icons Program DEFDBL I, P-Q, X-Z ON ERROR GOTO errortrap DEF fnxpix (x) = nstartx + scalex * (x + scale) DEF fnypix (y) = npixely - scaley * (y + scale) GOSUB initialize GOSUB menu loops: GOSUB iterate x = xnew: y = ynew PSET (fnxpix(x), fnypix(y)) a$ = INKEY$ IF a$ = "c" THEN iterates = 1: CLS : GOSUB parameters IF a$ = "i" THEN GOSUB parameters restart: IF a$ = "m" THEN GOSUB menu iterates = iterates + 1 GOTO loops iterate: zzbar = x * x + y * y p = alpha * zzbar + lambda zreal = x: zimag = y FOR i = 1 TO n - 2 za = zreal * x - zimag * y zb = zimag * x + zreal * y zreal = za: zimag = zb NEXT i zn = x * zreal - y * zimag p = p + beta * zn xnew = p * x + gamma * zreal - omega * y ynew = p * y - gamma * zimag + omega * x RETURN menu: GOSUB parameters PRINT USING "(X,y) = ##.#### ##.####"; x; y PRINT "Scale =", scale PRINT "ESC to exit program" PRINT "R for RETURN" CLS GOTO menu initialize: CLS scale = 1! nscreen = 12: npixelx = 640: npixely = 480 nstartx = 160 SCREEN nscreen GOSUB setscreen x =.01: y =.003: n = 4: iterates = 1 lambda = -1.8: alpha = 2: beta = 0: gamma = 1!: omega = 0 RETURN initialpoint: CLS PRINT "Enter r to reset coordinates automatically" PRINT "Enter x to INPUT coordinates" setscreen: CLS scaley = npixely / (2 * scale) scalex = (npixelx - nstartx) / (2 * scale) RETURN

115. Two kinds of complexity k = 1; v 1 = v 2 = 0.851 ;  1 =  2 =0.6 ; c 1 = c 2 = 3 y x 1.5 0 0 E*E* (a) k = 1; v 1 = v 2 = 0.852 ;  1 =  2 =0.6 ; c 1 = c 2 = 3 y x 1.5 0 0 E*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

116. q1q1 q2q2 0 0 1 1 ESES c1c1 c2c2 c1c1 c2c2 ESES.... 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

117. ESES E1E1 E2E2 q1q1 q2q2 0 0 1 1 ESES E1E1 E2E2 c1c1 c2c2....

119. 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 ?

120. 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

121. 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)

123. LC LC -1 SH 2 SH 1 R1R1 R2R2 Z2Z2 Z0Z0 U U -1,2 U -1,1 x’ y’ y x

124. x y T T 2 inverses T T 2 fixed pointsmap Z 2 = {(x,y) | y > b } Z 0 = {(x,y) | y < b } LC = {(x,y) | y = b } LC -1 = {(x,y) | x = 0 } det DT = -2x =0 for x=0 T({x=0}) = {y=b} Z0Z0 Z2Z2 R1R1 R2R2 LC -1 LC SH 1 SH 2 y=b

125. R1R1 R2R2 Z0Z0 Z2Z2 CS CS -1 U T(U) R1R1 R2R2 Z0Z0 Z2Z2 CS CS -1 V

126. Z0Z0 Z2Z2 LC -1 LC P Q contact Z0Z0 Z2Z2 LC

127. Z0Z0 Z2Z2 LC -1 LC

128. Z0Z0 Z2Z2

129. Z0Z0 Z2Z2 LC -1 LC

130. Z0Z0 Z2Z2 LC -1 LC

131. Z0Z0 Z2Z2 LC -1 LC

132. Z0Z0 Z2Z2 LC -1 LC

133. Z0Z0 Z2Z2 LC -1 LC

134. Z0Z0 Z2Z2 LC -1 LC

136. Z0Z0 Z2Z2 LC -1 LC

137. Z0Z0 Z2Z2 LC -1 LC

139. 1 6 2 5 3 41 2 3

140. 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| T t (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

141. Noninvertible (“Many-to-One”) map. T p’ p1p1 p2p2 T.. Equivalently, we say that p’ has several rank-1 preimages. T 1 -1 p’ p1p1 p2p2 T 2 -1.. Distinct points are mapped into the same point LC LC -1 SH 2 SH 1 R1R1 R2R2 Z2Z2 Z0Z0 U U -1,2 U -1,1 x’ y’ y x Folding action of T Unfolding action of T -1

142. Critical curves separate regions Z k, Z k+2 characterized by different numbers of preimages. Each region Z k 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. LC -1 LC In the homogeneous case has a cusp point in

143. Theorem (Homogeneous behavior) If          , and the bounded trajectories converge to one of the stable Nash equilibria E 1 or E 2, then the common boundary  B(E 1 )   B(E 2 ) which separates the basin B(E 1 )from the basin B(E 2 ) is given by the stable set W S (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.

144. 0 0 1.2 1.1 y x  1 =  2 = 3.6  1 = 0.55  2 = 0.7 Z4Z4 Z2Z2 Z0Z0 E2E2 E1E1 S 0 0 1.2 1.1 y x  1 =  2 = 3.6  1 = 0.59  2 = 0.7 Z4Z4 Z2Z2 Z0Z0 E2E2 E1E1 S Case of heterogenous players Theorem …

145. 0 0 1.1 y x  1 =  2 = 3.9  1 = 0.7  2 = 0.8 S A2A2 A1A1 E1E1 0 0 1.1 y x  1 =  2 = 3.95  1 = 0.7  2 = 0.8 S A2A2

146. Agiza, H.N., Bischi, G.I. and M. Kopel (1999) «Multistability in a Dynamic Cournot Game with Three Oligopolists», Mathematics and Computers in Simulation. Bischi, G.I., L. Mroz and H. Hauser (2001) «Studying basin bifurcations in nonlinear triopoly games by using 3D visualization» Nonlinear Analysis, Theory, Methods and Applications.