1 Dynamical Systems: A Tutorial -- 2 Introduction to Dynamic Optimization Theory -- 3 Periodic and Chaotic Programs of Optimal Intertemporal Allocation in an Aggregative Model with Wealth Effects -- 4 Optimal Chaos, Nonlinearity, and Feasibility Conditions -- 5 The Local Bifurcation of Ramsey Equilibrium -- 6 On the Structure of Ramsey Equilibirum: Cycles, Indeterminacy and Sunspots -- 7 Robust Ergodic Chaos in Discounted Dynamic Optimization Models -- 8 Non-Linear Dynamics and Chaos in Optimal Growth: A Constructive Exposition -- 9 Ergodic Chaos in Optimal Growth Models with Low Discount Rates -- 10 On the Minimum Rate of Impatience for Complicated Optimal Growth Paths -- 11 An Exact Discount Factor Restriction for Period-Three Cycles in Dynamic Optimization Models -- 12 On the Least Upper Bound of Discount Factors that are Compatible with Optimal Period-Three Cycles -- 13 On the Relationship between Discounting and Entropy of Dynamic Optimization Models -- References.

This collection of essays brings together some articles on dynamic optimization models that exhibit chaotic behavior. Chapters 3, 4, 5, 6, 7, and 9 appeared in a Symposium on Chaotic Dynamical Systems in Economic Theory (Volume 4, Number 5, 1994). Also, Chapters 10,11, and 12 appeared in the Journal of Economic The­ ory. We would like to thank the authors, and Academic Press for permission to reprint. We are grateful to Professor C.D. Aliprantis for suggesting the idea of a book structured around the Economic Theory Symposium, and without the support and patience of Dr. Mueller this project could not have been completed. We would like to thank Ms. Amy Gowan who cheerfully per­ formed the arduous task of typing the manuscript. Thanks are also due to Xiao Qing Yu, Tridip Ray and Malabika Majumdar for their help at various stages in the preparation of the manuscript. For a course on dynamic optimization addressed to students with a good background in economic theory and real analysis, one can assign Chapter 2 as a partial introduction to the basic tech­ niques. Chapters 3 and 4 can be assigned to provide examples of simple optmization models generating complicated behavior.