Nonlinear difference equations, bifurcations and chaos: An introduction

Research in Economics

Volumn 62, Issue 3, 2008, Pages 122-177

  Grandmont, Jean Michel ^a,b  

^a CA' FOSCARI UNIVERSITY OF VENICE   (Italy)

^b CREST   (France)

Author keywords

Bifurcations; Business cycles; Chaos; Nonlinear dynamics

Indexed keywords

EID: 50849138497     PISSN: 10909443     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.rie.2008.06.003     Document Type: Article

References (58)

1. Azariadis C. Self-fulfilling prophecies. J. Econom. Theory 25 (1981) 380-396
2. Banks J., Brooks J., Cairns G., Davis G., and Stacey P. On Devaney's definition of chaos. Trans. Amer. Math. Soc. 99 (1992) 332-334
3. Benedicks M. Non uniformly hyperbolic dynamics: Hénon maps and related dynamical systems. In: Li T.-t. (Ed). Proceedings of the International Congress of Mathematicians, Beijing 2002. vol. III (2003), Higher Education Press, Beijing 265-278
4. Benedicks M., and Carleson L. The dynamics of the Hénon map. Ann. of Math. 133 (1991) 73-169
5. Benedicks M., and Young L.-S. Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps. Ergodic Theory Dynam. Systems 12 (1992) 13-37
6. Benedicks M., and Young L.-S. Sinai-Bowen-Ruelle measures for certain Hénon maps. Invent. Math. 112 (1993) 541-576
7. Benhabib J., and Day R. A characterization of erratic dynamics in the overlapping generations model. J. Econom. Dynam. Control 4 (1982) 37-55
8. Bruin H., Keller G., Nowicki T., and van Strien S. Wild Cantor attractors exist. Ann. of Math. 143 (1996) 97-130
9. Carr J. Applications of Center Manifold Theory (1981), Springer Verlag, New York
10. Cass D., and Shell K. Do sunspots matter ?. J. Political Econom. 91 (1983) 193-227
11. 2 au Voisinage d'un Point Fixe Elliptique. In: Iooss et al. (1983)
12. Chenciner A. Bifurcations de points fixes elliptiques. I. Courbes invariantes. IHES-Publ. Math. 61 (1985) 67-127
13. Chenciner A. Bifurcations de points fixes elliptiques. II. Orbites pé riodiques et ensembles de cantor invariants. Invent. Math. 80 (1985) 81-106
14. Chenciner A. Bifurcations de points fixes elliptiques. III. Orbites périodiques de "petites" pé riodes et elimination résonnante des couples de courbes invariantes. IHES-Publ. Math. 66 (1988) 5-91
15. Collet P., and Eckmann J.P. Iterated Maps on the Interval as Dynamical Systems (1980), Birkhaüser, Boston
16. De Melo W., and van Strien S. One-dimensional dynamics: The Schwarzian derivative and beyond. Bull. Amer. Math. Soc. 18 (1988) 159-162
17. De Melo W., and van Strien S. One-Dimensional Dynamics (1993), Springer Verlag, New York
18. Devaney R.L. An Introduction to Chaotic Dynamical Systems. 2nd ed (2003), Westview Press, Boulder
19. {ring operator} 96, University of Amsterdam
20. Gaunersdorfer A., Hommes C.H., and Wagener F.O.O. Bifurcation routes to volatility clustering under evolutionary learning. J. Econom. Behavior Organization 67 (2008) 27-47
21. Grandmont J.-M. On endogenous competitive business cycles. Econometrica 53 (1985) 995-1045
22. Grandmont, J.-M., 1986. In: Hildenbrand, W., Mas-Colell, A., (Eds.), Periodic and Aperiodic Behaviour in Discrete One-Dimensional Dynamical Systems. In: Hildenbrand and Mas-Colell (1986)
23. Grandmont J.-M., Kirman A.P., and Neuefind W. A new approach to the uniqueness of equilibrium. Rev. Econom. Studies 41 (1974) 289-291
24. Grandmont J.-M., Pintus P., and de Vilder R. Capital-labor substitution and competitive nonlinear endogenous business cycles. J. Econom. Theory 80 (1998) 14-59
25. Guckenheimer J., and Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983), Springer Verlag, New York
26. Hartman P. Ordinary Differential Equations (1964), Wiley, New York
27. Hildenbrand W., and Mas-Colell A. Contributions to Mathematical Economics, In Honour of Gerard Debreu (1986), North-Holland, New York
28. Hirsch M.W., and Smale S. Differential Equations, Dynamical Systems, and Linear Algebra (1974), Academic Press, New York
29. Hommes, C., de Vilder, R., 1995. Sunspot equilibria in an implicitly defined overlapping generations model. Mimeo. University of Amsterdam
30. Iooss G. Bifurcations of Maps and Applications (1979), North-Holland, New York
31. In: Iooss G., Helleman R.H.G., and Stora R. (Eds). Chaotic Behaviour of Deterministic Systems (1983), North-Holland, New York
32. Jakobson M.V. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm. Math. Phys. 81 (1981) 39-88
33. Jakobson M.V., and Swiatek G. One-dimensional maps. In: Hasselblatt B., and Katok A. (Eds). Handbook of Dynamical Systems. vol. 1, Part 1 (2002), Elsevier, Amsterdam 599-664 (Chapter 10)
34. Johnson S. Singular measures without restrictive intervals. Commun. Math. Phys. 110 (1985) 185-190
35. Kennedy J.A., and Stockman D.R. Chaotic equilibria in models with backward dynamics. J. Econom. Dynam. Control 32 (2008) 939-955
36. Kozlovski, O.S., 1998. Structural stability in one-dimensional dynamics. Ph.D. Thesis. University of Amsterdam
37. Kozlovski O.S. Getting rid of the negative Schwarzian derivative condition. Ann. Math. 152 (2000) 743-762
38. Kuznetsov Y.A. Elements of Applied Bifurcation Theory. 2nd ed (1998), Springer Verlag, New York
39. Lanford, O.E. III, 1983. In: Iooss, G., Helleman, R.H.G., Stora, R. (Eds.), Introduction to the Mathematical Theory of Dynamical Systems. In: Iooss et al. (1983)
40. Li S.H. Dynamical properties of the shift maps on the inverse limit spaces. Ergodic Theory Dynam. Systems 12 (1992) 95-108
41. Li T., and Yorke J.A. Period three implies chaos. Amer. Math. Monthly 82 (1975) 985-992
42. Marsden J.E., and McCracken M. The Hopf Bifurcation and its Applications (1976), Springer Verlag, New York
43. May R.B. Simple mathematical models with very complicated dynamics. Nature 261 (1976) 459-467
44. Medio A., and Raines B. Backward dynamics in economics. The inverse limit approach. J. Econom. Dynam. Control 31 (2007) 1633-1671
45. Misiurewicz M. Absolutely continuous measures for certain maps of an interval. Publ. Math. l'IHES 53 (1981) 17-51
46. Misiurewicz, M., 1983. Iooss, G., Helleman, R.H.G., Stora, R., Maps of an Interval. In: Iooss et al. (1983)
47. Nusse H.E. Persistence of order and structure in chaos. Physica D 20 (1986) 374-386
48. Palis Jr. J., and De Melo W. Geometric Theory of Dynamical Systems (1982), Springer Verlag, New York
49. Palis Jr. J., and Takens F. Topological equivalence of normally hyperbolic dynamical systems. Topology 16 (1977) 335-345
50. Parthasarathy K.R. Probability Measures on Metric Spaces (1967), Academic Press, New York
51. Sarkovskii A.N. Coexistence of cycles of a continuous map of the line into itself. Ukr. Mat. Z. 16 (1964) 61-71
52. Shell, K., 1977. Monnaie et allocation intertemporelle. Mimeo. Séminaire d'Econométrie Roy-Malinvaud, CNRS, Paris. Title and abstract in French, text in English
53. Singer D. Stable orbits and bifurcations of maps of the interval. SIAM J. Appl. Math. 35 (1978) 260
54. Smale S. The Mathematics of Time (1980), Springer Verlag, New York
55. Thieullen Ph., Tresser C., and Young L.S. Positive Lyapounov exponent for generic one-parameter families of unimodal maps. J. Anal. Math. 64 (1994) 121-172
56. Whitley D. Discrete dynamical systems in dimensions one and two. Bull. London Math. Soc. 15 (1983) 177-217
57. Wilkinson, A., 2008. Smooth ergodic theory. Mimeo. Northwestern University. R. Meyers (Ed.), Encyclopedia of Complexity and Systems Science, Springer Verlag, New York (in press)
58. Woodford M. Stationary sunspot equilibria in a finance constrained economy. J. Econom. Theory 40 (1986) 128-137