1. 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

2. 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.. LC Several distinct inverses are defined in p’ : i.e. the inverse relation p = T -1 (p’) is multivalued T : R n  R n p’ = T (p)

3. ZkZk Z k+2 Z k : region of R n where k distinct inverses are defined LC (critical manifold) locus of points having two merging preimages R n can be divided into regions (or zones) according to the number of rank-1 preimages x’ = f(x) = ax (1-x) Z 0 - Z 2 map: if x’ < a/4 then where: critical point c = a/4 Example: 1-dimensional NIM Df(c -1 ) = 0 and c = f(c -1 )

4. Folding by T Unfolding by T -1 c -1

5. c -1 M c -1 m cMcM cmcm Piecewise differentiable noninvertible map

6. Z0Z0 Z4Z4 Z2Z2 c c1c1 f 2 f c c1c1

7. Logistic map x’ = f(x) = ax (1-x)

8. c1c1 c 4 = c 6 = p 1 * ; c 5 = c 7 = p 2 * c3c3 c2c2 c c4c4 c5c5 c7c7 c6c6 c1c1 c3c3 c2c2 c c4c4 a < a 2 a = a 2 c 2 = c 3 = p * a 2 < a < a 1 a = a 1 c1c1 c3c3 c2c2 c c1c1 c

9. c 1 =f(c) c 2 =f(c 1 ) c c 3 =f(c 2 )

10. 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 LC -1 LC = T(LC -1 ) T Z2Z2 Z0Z0 R2R2 R1R1 Riemann Foliation

11. Equivalently, a point has several distinct rank-1 preimages, i.e. several inverses are defined in it, which “unfold” the plane LC -1 LC Z2Z2 Z0Z0 R2R2 R1R1 SH 1 SH 2

12. Z 1 - Z 3 - Z 1 LC -1 (b) LC (b) Z3Z3 Z1Z1 SH 1 SH 2 LC (a) LC -1 (a) SH 3 Z1Z1 LC -1 Z3Z3 LC Z1Z1 Z1Z1 SH 1 SH 2 SH 3 Z 1 < Z 3

13. AB C A’ B’ C’ 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’ F A’ B’ C’ AB C T T is orientation reversing if det A < 0 F’ F

14. 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 )

15. Example: 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 x=0

16.  ’  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

17. F F’  F  T A plane figure across LC -1 is folded along LC LC -1 LC

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

20. LC 2 LC 1 LC LC 3 LC -1

21. LC 2 LC 1 LC -1 LC 5 LC 3 LC 6 LC LC 4

23. 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 ?

24. 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

25. 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(x 0 )>x 0, the trajectory is increasing, whereas if f(x 0 )<x 0 the trajectory is decreasing p*p* q*q* r*r* p*p* q*q* r*r*

26. f(x) = a arctan (x-1) a = 3 a = 1 a = 0.5 basin boundary fold bifurcation

27. a = 0.5 a = 0.2 Continuous and decreasing maps The only possible invariant sets are one fixed point and cycles of period 2, being f 2 =f°f increasing The 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) = – ax 3 + 1

28. a = 0.7

29. Z2Z2 Z0Z0 c c -1 p q p q r q -1 Nononvertible maps. Several preimages

30. Noninvertible map: f (x) = a x (1– x) = 1/2 c=a/4

31. Z3Z3 Z1Z1 c max p q c min Z1Z1 z r

32. Z3Z3 Z1Z1 c max p q c min Z1Z1 z r c -1 q -1 1 q -1 2

34. 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

36. 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

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

39. 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

40. 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

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

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

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

45. Z0Z0 Z2Z2 LC -1 LC

46. Z0Z0 Z2Z2

47. Z0Z0 Z2Z2 LC -1 LC

48. Z0Z0 Z2Z2 LC -1 LC

49. Z0Z0 Z2Z2 LC -1 LC

50. Z0Z0 Z2Z2 LC -1 LC

51. Z0Z0 Z2Z2 LC -1 LC

52. Z0Z0 Z2Z2 LC -1 LC

54. Z0Z0 Z2Z2 LC -1 LC

55. Z0Z0 Z2Z2 LC -1 LC

56. 1 6 2 5 3 41 2 3

57. Adaptive expectations Dynamical system: 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 Cournot Game

58. r2r2 r1r1 r2r2 r1r1           0 01 1 q1q1 q2q2 0 01 1 q1q1 q2q2 Non monotonic reaction functions may lead to several coexisting equilibria Logistic reaction functions 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?

59. 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 

60. 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

61. y x y’y’ x’x’ Z4Z4 Z2Z2 Z0Z0          -0.51.5          0.5 1.5 K 0.5 Critical curves

62. y x y’y’ x’ Z4Z4 Z2Z2 Z0Z0. In the homogeneous case has a cusp point in and

63. Proposition (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.

64. 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

65. 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

66. 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

68. 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)

69. S VIVI V II V III V0V0 PIPI P II P III P IV Q II Q III Local Stability Vertices V 0 and V II are always repelling; Interior FP S (if it exists) is a saddle point or a repelling node Q II and P II are created together (saddle-node) P III and Q III are created together (saddle-node) P II and P IV cannot coexist P III and P I cannot coexist

70. 1 1 Fig. 3 x1x1 x2x2 O F1F1 F2F2 V III Q III P III P II Q II VIVI B(P II ) B(V I ) B(P III ) B(V III ) S V II

71. 1 1 x1x1 x2x2 (a) Fig. 5 LC x2x2 1 x1x1 (b) LC (c) 1 1 x1x1 x2x2 P III Q III O VIVI V II V III 1 O P III Q III LC V II 0.965 0.35 B(P III ) B(V III ) B(V I ) P III Q III V III VIVI S V II S 1 1 0.965 0.35 x1x1 x2x2 (d) LC Z1Z1 Z3Z3 H1H1 H2H2 H3H3 P III Q III V II Z1Z1

72. H1H1 H2H2 H3H3 Fig. 6 0 0 1.1 x1x1 q III 1 p III 1 q III 1 p III 1 c1c1 q III 1 p III 1 c1c1 1 x1x1 1 0.3 0.6 x1x1

73. 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

74. E*E* q1q1 q2q2 8 120 0 v 1 = 0.24 v 2 = 0.48 c 1 = 3 c 2 = 5 a = 10 b = 0.5 O Each coordinate axis is trapping since q i (t) = 0 implies q i (t+1) = 0 The restriction of the map T to that axis is conjugate to the standard logistic map

75. E*E* q1q1 q2q2 7 0 0 v 1 = 0.24 v 2 = 0.55 c 1 = 3 c 2 = 5 a = 10 b = 0.5 Z2Z2 Z4Z4 Z0Z0 E*E* q1q1 q2q2 7 11 0 0 v 1 = 0.24 v 2 = 0.55 c 1 = 3 c 2 = 5 a = 10 b = 0.5 Z2Z2 Z4Z4 Z0Z0 contact

76. E*E* q1q1 q2q2 11 0 0 v 1 = 0.24 v 2 = 0.55 c 1 = 3 c 2 = 5 a = 10 b = 0.5 Z2Z2 Z4Z4 Z0Z0 E*E* q2q2 0 0 v 1 = 0.24 v 2 = 0.7 c 1 = 3 c 2 = 5 a = 10 b = 0.5 Z2Z2 Z4Z4 Z0Z0

77. q1q1 q2q2 6 11 0 0 v 1 = 0.24 v 2 = 1.0747 c 1 = 3 c 2 = 5 a = 10 b = 0.5

78. q1q1 q2q2 7 9.50 0 v 1 = 0.4065 v 2 = 0.535 c 1 = 3 c 2 = 5 a = 10 b = 0.5

79. q1q1 q2q2 7 9.5 v 1 = 0.4065 v 2 = 0.535 c 1 = 3 c 2 = 5 a = 10 b = 0.5 Z2Z2 Z4Z4 Z0Z0 -0.5