Bibliography. [1] Abraham, R. & J. Marsden. Foundations of Mechanics. Benjamin & Cummings, Reading, MA, 1978.

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1 Bibliography [1] Abraham, R. & J. Marsden. Foundations of Mechanics. Benjamin & Cummings, Reading, MA, [2] Allais, M. Economie et Interet. Imprimerie Nationale, Paris, [3] Allen, R.G.D. Mathematical Economics. (2nd edition), MacMillan, London, [4] Allen, R.G.D. Macro Economic Theory. MacMillan, London, [5] Anton, H. Elementary Linear Algebra. John Wiley & Sons, New York, [6] Anton, H. & C. Rorres. Elementary Linear Algebra with Applications. J. Wiley & Sons, New York, [7] Arnold, V. Bifurcations in Versal Families. Russian Mathematical Surveys, 27: ,1972. [8J Arnold, V.I. Mathematical Methods of Classical Mechanics. Springer-Verlag: Heidelberg, [9J Arrow, K.J. & L. Hurwicz. On the Stability of the Competitive Equilibrium, Part I - II. Econometrica 26: , October, : , January, [10] Arrow, K.J. & M. Kurz. Public Investment, The Rate of Return and Optimal Fiscal Policy. John Hopkins Press, Baltimore, Md, [11] Arrow, K.J. & M. McManus. A Note on Dynamic Stability. Econometrica: , July, [12] Arrow, K.J. & M. McManus. A Note on Dynamic Stability. Econometrica 26: , [13] Arrow, K.J., H. Block & L. Hurwicz. The Stability of Competitive Equilibrium II. Econometrica 27 (I): , January [14] Arrowsmith, D.K. & C.M. Place. An Introduction to Dynamical Stytems. Cambridge University Presss, Cambridge, [15] Athans, M. & P.L. Falb. Optimal Control. McGraw-Hill, New York, 1966.

2 296 [16] Bailey, N.T.J. The Mathematical Theory of Epidemics. Hafner, New York, [17] Balasko, Y. & R. Boyer. Une Analyse de l'effet du Progres Technique sur l'emploi. CEPREMAP, January [18] Barnett, S. & R.G. Cameron. Introduction to Mathematical Control Theory, 2nd edition. Clarendon, Oxford, [19] Barnett, W.A., Geweke, J. & Shell, K. Economic Complexity. Cambridge University Press, [20] Basset, L., H. Habibagachi & J. Quirk. Qualitative Economics and Morishima Matrices. Econometrica 35: , April, [21] Beckmann, M. & H. Ryder. Simultaneous Price and Quantity Adjustment in a Single Market. Econometrica 37: , July, [22] Bellman, R. Introduction to Matrix Analysis. McGraw-Hill, New York, [23] Ben-Porath, Y. The Production of Human Capital and the Life Cycle of Earnings. Journal of Political Economy 75: , [24] Benhabib, J. & R.H. Day. Rational Choice and Erratic Behaviour. Review of Economic Studies 48: ,1981. [25] Benhabib, J. & R.H. Day. Characterization of Erratic Dynamics in the Overlapping Generation Model. Journal of Economic Dynamics and Control 4: 37-55, [26] Benhabib, J. & T. Miyao. Some New Results on the Dynamics of the Generalised Tobin Model. International Economic Review 22(3): , October, [27] Benhabib, J. & K. Nishimura. The Hopf Bifurcation and the Existence of Closed Orbits in Multisection Models of Optimal Economic Growth. Journal of Economic Theory 21: , December, [28] Benhabib, J. & K, Nishimura. Stability of Equilibrium in Dynamic Models of Capital Theory. International Economic Review 22: , June, [29] Benhabib, J. & K. Nishimura. Competitive Equilibrium Cycles. Journal of Economic Theory 35: , [30] Block, W.A. & J.A. Scheinkman. "Global Asymptotic Stability of Optimal Control with Applications to Dynamic Economic Theory" in Pitchford J.D. & S.J. Turnovsky (eds): Applications of Control Theory to Economic Analysis. North Holland Co., Amsterdam, [31] Boldrin, M. & L. Montrucchio. On the Indeterminary of Capital Accumulation Paths. Journal of Economic Theory 40: 26-39,1986.

3 297 [32] Brock, W.A. "The Global Asymptotic Stability of Optimal Control: A Survey of Recent Results" in Intrilligator, M.D. (ed): Frontiers of Quantitative Economics. VoI3A, North Holland Co., Amsterdam, [33] Brock, W.A. & J.A. Scheinkman. Global Asymptotic Stability of Optimal Control Systems with Applications to the Theory of Economic Growth. Journal of Economic Theory 12: , February, [34] Brocker, Th. & L. Lander. Differentiable Germs and Catastrophes. Cambridge University Press, [35] Burmeister, E. & A.R. Dobell. Mathematical Theories of Economic Growth. MacMillan, London, [36] Carr, T. Applications of Centre Manifold Theory. Springer-Verlag, Heidelberg, [37] Cass, D. Optimal Growth in an Aggregate Model of Capital Accumulation: A Turnpike Theorem. Econometrica 34: October, [38] Cass, D. & K. Shell (eds). The Hamiltonian Approach to Dynamic Economics. Academic Press, New York, [39] Chang, W.W. & D.J. Smyth. The existence and Persistence of Cycles in a Nonlinear Model: Kaldor's 1940 Model Re-examined. Review of Economic Studies 38: 37-44, January, [40] Chiarella, C. Elements of a Nonlinear Theory of Economic Dynamics. Lecture Notes in Economics and Mathematical Systems No Springer-Verlag, Heidelberg, [41] Chipman, J.S. The Theory of Intersectoral Money Flows and Income Formation. John Hopkins Press, Baltimore, [42] Chow, S.N. & J.K. Hale. Methods of Bifurcation Theory. Springer-Verlag, Heidelberg, [43] Clark, C. Profit Maximization and the Extinction of Animal Species. Journal of Political Economy 81: , July-August, [44] Clark, Colin W. Mathematical Bioeconomics. John Wiley & Sons, New York, [45] Clark, C.W., F.H. Clarke & G.R. Munro. The Optimal Exploitation of Renewable Resource Stocks: Problems of Irreversible Investment. Econometrica 47(1): 25-27, January, [46] Clarke, F.H., M.N. Darrough & J.M. Heineke. Optimal Pricing Policy in the Presence of Experience Effects. Journal of Business 55: , October, 1982.

4 298 [47] Coddington, E.A. & N. Levinson. Theory of Ordinary Differential Equations. McGraw-Hill, New York, [48] Colonius, F. "Poincare-Bendixson Theory for Control Problems with Continuous Optimal Controls", in Feichtinger, G. (ed): Optimal Control Theory and Economic Analysis 3. Elsevier Science, North Holland, [49] Conlisk, J. Quick Stability Checks and Matrix Norms. Economica (NS) XL: , November, [50] Conrad, J.M. & C.W. Clark. Natural Resource Economics. Cambridge University Press, [51] Crandall, M. & P. Rabinovitz. Bifurcation from Simple Eigenvalues. Journal of Functional Analysis 8: , [52] Cropper, M.L., D.R. Lee & S.S. Pannu. The Optimal Extinction of a Renewable Natural Resource. Journal of Environmental Economics and Management 6: , [53] Day, R.H. Irregular Growth Cycles. American Economic Review 72: , June, [54] Day, R.H. The Emergence of Chaos from Classical Economic Growth. Quarterly Journal of Economics 98: , May, [55] Day, R.H. & W.J. Shafer. Keynesian Chaos. Working Paper, Department of Economics, University of Southern California, Los Angeles, [56] DeBach, P. Biological Control of Insect Pests and Weeds. Chapman & Hall, London, [57] Debreu, G. Theory of Value. Yale University Press, New Haven, [58] Deneckere, R. & S. Pelican. Competitive Chaos. Journal of Economic Theory 40: 13-25, October, [59] Denison, E.G. The Sources of Economic Growth in the U.S. and the Alternatives Before Us. Suppl. Paper No 13, C.E.D., New York, [60] Diamond, P.A. National Debt in a Neo Classical Growth Model. American Economic Review 55 (5): , December, [61] Diamond, P. Chaotic Behaviour of Systems of Difference Equations. International Journal of Systems Science 7: , [62] Dockner, E. "Local Stability-Analysis in Optimal Control Problems with Two State Variables" in Feichtinger, G. (ed): Optimal Control and Economic Analysis 2. Elsevier Science, North Holland, Amsterdam, [63] Dockner, E.J. & G. Feichtinger. Cyclical Consumption Patterns and Rational Addiction. Vienna, undated.

5 299 [64] Dockner, E.J. & G. Feichtinger. On the Optimality of Limit Cycles in Dynamic Economic Systems. Journal of Economics 53(.): 31-50, [65] Dockner, E.J., G. Feichtinger & A. Novak. Cyclical Production and Marketing Decisions: Application of Hopf Bifurcation Theory. International Journal of Systems Science, Vol 00, No 00, , [66] Domar, E.D. The Burden of Debt and the National Income. American Economic Review 34: , December, [67] Domar, E.D. Capital Expansion, Rate of Growth and Employment. Econometrica 14: , April, [68] Domar, E.D. Essays in the Theory of Growth. Oxford University Press, London, [69] Dorfman, R. An Economic Interpretation of Optimal Control Theory. American Economic Review 59: , December, [70] Dorfman, R., P.A. Samuelson & R.M. Solow. Linear Programming and Economic Analysis. McGraw-Hill, New York, [71] Duesenberry, J. S. Selected Problems in Economic Theory: Discussion. American Economic Review /S 49: , May, [72] Euler, L. Methodus inveniendi lineas curvas... in Opera Omnia I, V 24: , Fissli, Zurich, [73] Feichtinger, G. Limit Cycles in Dynamic Economic Systems. Annals of Operations Research. Vol 37: , August, [74] Feichtinger, G. & E.J. Dockner. Capital Accumulation, Endogenous Population Growth and Easterlin Cycles. Journal of Population Economics 3: 73-87, [75] Feichtinger, G. & G. Sorger. Optimal Oscillations in Control Models: How can Constant Demand lead to Cyclical Production. Operations Research Letters 5: , [76] Feigenbaum, M.J. Quantitative Universality for a Class of Nonlinear Transformation. Journal of Statistical Physics 19: 25-52, [77] Ferrar, W.L. Finite Matrices. Oxford: Clarendon Press, [78] Fiedler, M. & V. Ptak. On Matrices with Nonpositive Off- Diagonal Elements and Positive Principal Minors. Czechoslov. Mathematics Journal 12: , [79J Flaschel, P. Some Stability Properties of Goodwin's Growth Cycle: A Critical Elaboration. Journal of Economics 44(1): 63-69, [80] Freedman, H.I. Deterministic Mathematical Models in Population Ecology. Marcel Dekker, New York, 1980.

6 300 [81] Gabisch, G. & H.W. Lorenz. Business Cycle Theory. Lecture Notes in Economics and Mathematical Systems No 283, Springer-Verlag, [82] Gandolfo, G. Economic Dynamics: Methods of Models, second edition. North Holland, Amsterdam, [83] Gantmacher, F.R. The Theory of Matrices. Vol 1, 2, InterScience Publishers, Chelsea, New York, [84] Gardini, L., R. Lupini, C. Mammana, & M.G. Messia. Bifurcations and transitions to Chaos in the three-dimensional Lotka- Volterra Map. SIAM Journal Appl. Math. Vol. 47(3): , June, [85] Gause, G.F. The Struggle for Existence. Williams & Wilkins, Baltimore, [86] George, D. Equilibrium and Catastrophes in Economics. Scottish Journal of Political Economy, 28: 43-62, [87] Goel, N.S., S.C. Maitra & E.W. Montroll. On the Volterra and other nonlinear models of interacting populations. Review of Modern Physics, 43: , [88] Goh, B.S. "Robust Stability Concepts of Ecosystem Models" in Halfon, E. (ed): Theoretical Systems Ecology. Academic Press, [89] Goh, B.S. Management and Analysis of Biological Populations. Elsevier Scientific Pub. G., Amsterdam, [90] Goh, B.S., G. Leitmann & T.L. Vincent. Optimal Control of a Prey-Predator System. Mathematical Biosciences Vol 19: , [91] Gohberg, I., P. Lancaster & L. Rodman. Matrix Polynomials. Academic Press, New York, [92] Goodwin, R.M. Dynamic Coupling with Especial Reference to Markets Having Production Lags. Econometrica 15: , July, [93] Goodwin, R.M. The Nonlinear Accelerntor and the Persistence of Business Cycles. Econometrica 19: 1-17, January, [94] Goodwin, R.M. "A growth cycle" in Feinstein, C.H. (ed): Socialism, Capitalism and Economic Growth. Cambridge University Press, Cambridge, [95] Goodwin R.M. & L.F. Punzo. The Dynamics of a Capitalist Economy. Westview Press Boulder, Colorado, [96] Goodwin, R.M. et al. (eds). Nonlinear Models of Fluctuating Growth. Lecture Notes in Economics and Mathematical Systems No 228, Springer-Verlag, Heidelberg, [97] Grandmont, J.M. On Endogenous Competitive Business Cycles. Econometrica 53: , September, 1985.

7 301 [98] Grandmont, J.M. (ed). Nonlinear Economic Dynamics. Academic Press, Orlando, [99] Grimshaw, R. Nonlinear Ordinary Differential Equations. Blackwell, Oxford, [100] Guckenheimer, J. & P. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. 1983, 2nd edition 1986, Springer-Verlag, Heidelberg, [101] Haberler, G. Prosperity and Depression. Harvard University Press, Cambridge, Mass., 4th edition, [102] Hadley, G. Linear Algebra. Addison-Wesley, Reading, MA, [103] Hahn, F. Gross Substitutes and the Dynamic Stability of General Equilibrium. Econometrica 26: , January, [104] Hahn, F. "Stability" in Arrow, K.J. & M.D. Intrilligator (eds): Handbook of Mathematical Economics. Vol II, North Holland Publishing Co, Amsterdam, [105] Hahn, F.H. On the Stability of a Pure Exchange Equilibrium. International Economic Review 3: , May, [106] Hahn, F.H. & R.C.O. Matthews. The Theory of Economic Growth: A SUnJey. Economic Journal 74: , December, [107] Halfon, E. Theoretical Systems Ecology. Academic Press, New York, [108] Harrod, R.F. An Essay in Dynamic Theory. Economic Journal 49: 14-33, March, [109] Harrod, R.F. Towards a Dynamic Economics. MacMillan, London, [110] Hartman, P. Ordinary Differential Equations. Wiley, New York, [111] Hassard, B.D., N.D. Kazarinoff & U Y.H. Wan. Theory and Applications of Hopf Bifurcation. Cambridge University Press, [112] Hawkins, D. & H.A. Simon. Note: Some Conditions of Macroeconomic Stability. Econometrica 17: , July-October, [113] Hicks, J.R. A Contribution to the Theory of the Trade Cycle. Oxford University Press, [114] Hilton, P.J. (ed). Structural Stability, The Theory of Catastrophes and Applications in the Sciences. Springer, Berlin, [115] Hirsch, M.W. & S. Smale. Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, New York, 1974.

8 302 [116] Hussey, N.W. & L. Bravenboer, "Control of Pests in Glass-House Culture by the Introduction of Natural Enemies" in Huffaker (ed): Biological Control. Plenum Grass, New York, [117] Inman, D.J. Vibration with Control, Measurement and Stability. Prentice Hall, New Jersey, [118] Ichimura, S. Towards a General Nonlinear Macro Dynamic Theory of Economic Fluctuations. In Kurihara, K.K. (ed.): Post Keynesian Economics: , Rutgers Univ. Press, New Brunswick, [119] Isnard, C.A. & E.C. Zeeman. Some Models from Catastrophe Theory in the Social Sciences. In Zeeman, E.C.: Catastrophe Theory: Selected Papers , Addison-Wesley, Reading, MA, [120] Johnston, J. Econometric Methods. McGraw-Hill, New York, [121] Jorgenson. On a Dual Stability Theorem. Econometrica 28: , October, [122] Jorgenson, D.W. On stability in the sense of Harrod. Economica NS 27: , August, [123] Jorgenson, D.W. The Structure of Multi-sector Dynamic Models. International Economic Review 2(3): , September, [124] Kaldor, N. A Model of the Trade Cycle. Economic Journal 50: 78-92, March, [125] Kaplan, W. Ordinary differential Equations. Addison- Wesley, Reading, Mass, [126] Kauffman. Antichaos and Adaptation. Scientific American: 78-84, August, [127] Kauffman, S.A~ Origins of Order: Self Organization and Selection in Evolution. Oxford University Press, Oxford, [128] Kelly, A. The Stable, Centre-Stable, Centre, Centre-Unstable and Unstable Manifolds. Journal of Differential Equations 3: , [129] Kolmogorov, A. Sulla Teoria di Volterra della lotta per l'esistenzia. Gi. Inst. Ital. Attuari 7: 74-80, [130] Kolmogorov, A.N. & S.V. Fomin. Elements of the Theory of Functions and Functional Analysis. Graylock Press, New York, [131] Kurz, M. The General Instability of a Class of Competitive Growth Processes. Review of Economic Studies XXXV(2): , [132] Lancaster, P. Lambda Matrices and Vibrating Systems. Pergamon Press, Elmsford, New York, 1966.

9 303 [133] Lancaster, P. Theory of Matrices. Academic Press, New York, [134] Lancaster, P. Quadratic Eigenvalue Problems. Linear Algebra and its Applications 150: , [135] Leijonhufvud, A. Effective Demand Failures. Swedish Journal of Economics 75: 27-48, March, [136] Levinson, N. & O.K. Smith. A General Equation for Relaxation Oscillations. Duke Mathematical Journal 9: , [137] Lewin, R. Complexity: Science on the Edge of Chaos. MacMillan, [138] Li, T. & J. Yorke. Period Three Implies Chaos. American Mathematical Monthly 8: , [139] Lorenz, H.W. Goodwin's Nonlinear Accelerator and Chaotic Motion. Journal of Economics 47: , [140] Lorenz, H.W. "Optimal Economic Control and Chaotic Dynamics" in Feichtinger (ed): Optimal Control Theory and Economic Analysis 9. Elsevier Science Pub., Amsterdam, [141] Lorenz, E.N. Deterministic nonperiodic flow. Journal of Atmospheric Sciences 20: , [142] Lorenz, H.W. Complexity in Deterministic, Nonlinear Business-Cycle Models: Foundations, Empirical Evidence & Predictability. In Gori, F., Geronazzo, L. & M. Galeotti (etc.) Nonlinear Dynamics in Economics and Social Sciences, Lecture Notes in Economics and Mathematical Systems, No. 399, Springer Verlag, [143] Lorenz, Hans-Walter. Nonlinear Dynamical Economics and Chaotic Motion, 2nd edition. Springer-Verlag, Heidelberg, [144] Lotka, A.J. Elements of Physical Biology. Williams & Wilkins, Baltimore, [145] Lu, Y.C. Singularity Theory and an Introduction to Catastrophe Theory. Springer, Berlin, [146] Lucas, R.E. On the Mechanics of Economic Development. Journal of Monetary Economics 22: 3-42, July, [147] Majthay, A. Foundations of Catastrophe Theory. Pitman, London, [148] Marotto, F.R. Snap-Back Repellers Imply Chaos in R:'. Journal of Mathematical Analysis and Applications 63: , [149] Marsden, J.E. Qualitative Methods in Bifurcation Theory. Bulletin of the American Mathematical Society 84 No 6: , November, 1978.

10 304 [150] Marsden, J.E. & M. McCracken. The Hopf Bifurcation and its Applications. Springer-Verlag, Heidelberg, [151] Marshall, A. Principles of Economics. MacMillan, London, [152] May, R.M. Limit Cycles in Predator-Prey Communities. Science 177: , [153] May, R.M. Stability and Complexity in Model Ecosystems. Princeton University Press, [154] May, R.M. Biological Populations with Non Overlapping Generations: Stable points, Stable Cycles and Chaos. Science 186: , [155] May, R.M. Simple Mathematical Models with Very Complicated Dynamics. Nature 261: , [156] McKenzie. Stability of Equilibrium and the Value of Positive Excess Demand. Econometrica 28: , July, [157] McKenzie, L.W. The Matrix with Dominant Diagonal and Economic Theory. Proceedings of a Symposium on Mathematical Methods in the Social Sciences, Stanford University Press, Palo Alto, [158] Meade, J.E. External Economies and Diseconomies in a Competitive Situation. Economic Journal G2: 54-67, March, [159] Medio, A. Oscillations in Optimal Growth Models. Journal of Economic Behavior and Organisation 8: , [160] Medio, A. Oscillations in Optimal Growth Models. Journal of Economic Dynamics and Control 11: , June, [161] Medio, A. Discrete and Continuous Models of Chaotic Dynamics. Structural Change and Economic Dynamics, Vol 2: , [162] Medio, A. Chaotic Dynamics: Theory and Applications to Economics, Cambridge University Press, Cambridge, [163] Mensch, G., Kaasch, K., Kleinknecht, A. & R. Schnopp. Innovation Trends and Switching Between Full and Under-Employment Equilibrium International Institute of Management Discussion Papers 80-5, January, [164] Metzler, L.A. The Nature and Stability of Inventory Cycles. Review of Economic Studies 23: , August, [165] Metzler, L. Stability of Multiple Markets: The Hicks Conditions. Econometrica 13: , October, [166] Montrucchio, L. Optimal Decisions Over Time and Strange Attractors: An Analysis by the Bellman Principle. Mathematical Modelling 7: , 1986.

11 305 [167] Moreira, H.N. On Lienard's equation and the uniqueness of Limit Cycles in predator-prey systems. Journal of Mathematical Biology 28: , [168] Morishima, M. Prices, Interest and Profits in a dynamic Leontief System. Econometrica 26: , July, [169] Morishima, M. On the three Hicksian laws of Comparative Statics. Review of Economic Studies 27: , June, [170] Morishima, M. A Reconsideration of the Walras-Cassel-Leontief Model of General Equilibrium. In Arrow, K.J., S. Karlin & P. Suppes (eds.): Mathematical Methods in the Social Science, 1959 Proceedings of the First Stanford Symposium, Stanford, [171] Negishi, T. The Stability of the Competitive Equilibrium: A Survey Article. Econometrica 30: ,1962. [172] Negishi, T. "Tatonnement and Recontracting" in The New Palgrave: General Equilibrium. W.W. Norton, New York, [173] Newhouse, S., D. Ruelle & F. Takens. Occurrence of Strange Axiom Attractors near Quasi-Periodic Flows on Tm (m > 3). Communications in Mathematical Physics 64: 35-40, [174] Newman, P.K. Some Notes on Stability Conditions. Review of Economic Studies 27: 1-9, September, [175] Nikaido, H. Convex Structure and Economic Theory, Academic Press, New York, [176] Nirenberg, L. Topics in nonlinear Functional Analysis. Courant Institute of Mathematical Sciences, New York University, [177] Nitecki, Z. Differentiable Dynamics. MIT Press, Cambridge, MA, [178] Ott, Edward. Chaos in Dynamical Systems. Cambridge University Press, Cambridge, [179] Parker, F.D. "Management of Pest Populations by Manipulating Densities of both Hosts and Parasites through Periodic Releases" in Huffaker, C.B. (ed): Biological Control. Plenum Press, New York, [180] Peixoto, M.M. Structural Stability on Two-Dimensional Manifolds. Topology 1: , [181] Perko, L. Differential Equations and Dynamical Systems. Springer-Verlag, Heidelberg, [182] Perlis, S. Theory of Matrices, Vol. 1. Chelsea, New York, [183] Phillips, A.W. Stabilisation Policy in a Closed Economy. Economic Journal 64: , June, 1954.

12 306 [184] Pielou, E.C. Mathematical Ecology. Wiley-Interscience, New York, [185] Pitchford, J.D. "Two state variable problems" in Pitchford J.D. & S.J. Turnovsky (eds): Applications of Control Theory to Economic Analysis. North Holland Co, Amsterdam, [186] Poincare, H. Sur l'equilibre d'une masse fluide animee d'un mouvement de rotation. Acta Mathematica (16 September, 1885) 7: [187] Poincare, H. Oeuvres de Henri Poincare. Tome VIII, pp Published by the Academie des Sciences Gauthier-Villars, Paris, [188] Pomeau, Y. & P. Manneville. Intermittent Transition to Turbulence in Dissipative Dynamical Systems. Communication in Mathematical Physics 74: , [189] Pontryagin, L.S. Ordinary Differential Equation. Addison- Wesley, Reading Mass, [190] Pontryagin, L.S., V.G. Boltyanskii, R.V. Gamkrelidze & E.R. Mishchenko. The Mathematical Theory of Optimal Processes. Interscience Publishers, New York,1962. [191] Poston, T. & LN. Stewart. Taylor Expansions and Catastrophes. Pitman, London, [192] Poston, T. & LN. Stewart. Catastrophe Theory and its Applications. Pitman, London, [193] Putzer, E.J. Avoiding the Jordan Canonical Form in the Discussion of Linear Systems with Constant Coefficients. American Mathematical Monthly, Vol. 73: 2-7, Januar~ [194] Puu, T6nu. Nonlinear Economic Dynamics. Lecture Notes in Economics and Mathematical Systems No 336, Springer-Verlag, Heidelberg, [195] Quirk, J. & R. Rupert. Qualitative Economics and the Stability of Equilibrium. Review of Economic Studies 32: , October, [196] Ragozin, D.L. & G. Brown. Harvest Policies and Nonmarket Valuation in a Predator-Prey System. Journal of Environmental Economics and Management 12: , June, [197] Ramsey, F.P. A Mathematical Theory of Saving. Economic Journal 38: , December, [198] Rebelo, S. Long Run Policy Analysis and Long Run Growth. Working Paper, University of Rochester, Rochester, New York, [199] Rescigno, A. The Struggle for Life I: Two Species. Bull. Math. Biophysics 29: , 1967.

13 307 [200] Rescigno, A. & I.W. Richardson. On the Competitive Exclusion Principle. Bull. Math. Biophysics 27: 85-89, [201] Rescigno, A. & I.W. Richardson. The Struggle for Life I: Two Species. Bull. Math. Biophysics 29: , [202] Romer, P.M. Increasing Returns and Long Run Growth. Journal of Political Economy 94(5): , October, [203] Romer, P.M. Endogenous technical change. Journal of Political Economy 98(5)II: S71-S102, October, [204] Rose, H. The Possibility of Warranted Growth. Economic Journal 69: , June, [205] Rosen, R. Dynamical System Theory in Biology. Wiley- Interscience, New York, [206] Rosen, R.R. (ed). Foundations of Mathematical Biology. Vol III, Academic Press, New York, [207] Rubinow, S.I. Introduction to Mathematical Biology. Wiley-Interscience, New York, [208] Ruelle, D. Strange Attractors. Mathematical Intelligencer 2: , [209] Ruelle D. & F. Takens. On the Nature of Turbulence. Communications in Math. Physics 20: , Also: Notes concerning our paper "On the nature of turbulence," Commun. Math. Physics 23: , [210] Samuelson, P.A. Interaction between the Multiplier Analysis and Principle of Acceleration. Review of Economic Statistics 21: 75-78, May, [211] Samuelson, P.A. The Stability of Equilibrium: Comparative Statics and dynamics. Econometric 9: , April, [212] Samuelson, P.A. Foundations of Economic Analysis. Harvard University Press, Cambridge, MA, [213] Samuelson, P.A. Foundations of Economic Analysis. Harvard University Press, [214] Samuelson, P.A. An Exact Consumption-Loan Model on Interest with or without the Social Contrivance of Money. Journal of Political Economy 66(6): , December, [215] Samuelson, P.A. A Universal Cycle. Operations Research 3: , [216] Saunders, P.T. An Introduction to Catastrophe Theory. Cambridge University Press, Cambridge, Mass, 1980.

14 308 [217] Schinasi, G.J. Fluctuations in a Dynamic, Intermediate Run IS-LM Model: Applications of the Poincare-Bendixson Theorem. Journal of Economic Theory 28: , [218] Schultz, T.W. Investment in Human Capital. American Economic Review 51: 1-17, March, [219] Seierstad A. & K. Sydsaeter. Sufficient Conditions in Optimal Control Theory. International Economics Review 18(2): , June, [220] Shell, K. Toward a Theory of Inventive Activity and Capital Accumulation. American Economic Review P /P 56(2): 62-68, May, [221] Silkinov, L.P. A Contribution to the Problem of the Structure of an Extended Neighbourhood of a Saddle-Focus. Math. USSR Sb., 10(1): ,1970. [222] Singer, D. Stable Orbits and Bifurcation of Maps of the Interval. SIAM Journal of Applied Mathematics 35: , [223] Smale, S. Diffeomorphisms with Many Periodic Points. In Differential and Combinatorial Topology, S.S. Cairns (ed.): 63-80, Princeton University Press, Princeton, [224] Smale, S. Differentiable Dynamical Systems. Bulletin of the American Mathematical Society 73: , [225] Smith, J.M. Models in Ecology. Cambridge University Press, Cambridge, [226] Smithies, A. Economic Fluctuations and Growth. Econometrica 25: 1-52, January, [227] Smithin, J.N. & Tu, P.N.V. Disequilibrium Adjustment in a Classical Macroeconomic Model: A Note. Journal of Economics 47(2): , [228] Solow, R.M. A Contribution to the Theory of Economic Growth. Quarterly Journal of Economics 70: 65-94, February, [229] Solow, R.M. Competitive Valuation in a Dynamic Input-Output System. Econometrica 27: 30-53, January, [230] Solow, R.M. Note on Uzawa's Two-Sector Model of Economic Growth. Review of Economic Studies 29: 48-50, October, [231] Sonnenschein, H.F. (ed). Models of Economic Dynamics. Lecture Notes in Economics and Mathematical systems No 264, Springer-Verlag, Heidelberg, [232] Stutzer, M. Chaotic Dynamics and Bifurcation in a Macro- Model. Journal of Economic Dynamics and Control 2: , [233] Strang. Linear Algebra and its Applications. Academic Press, New York, 1976.

15 309 [234] Struble, R.A. Nonlinear Differential Equations. McGraw- Hill, New York, [235] Swan, T.W. Economic Growth and Capital Accumulation. Economic Record 32: , November, [236] Takeuchi, Y. & N. Adachi. "Oscillations in Prey-Predator Volterra Models" in Levin, S. (ed): Lecture Notes in Biomathematics. Vol 52, Springer-Verlag, [237] Tan, N.x. & P.N.V. Tu. Some Hopf Bifurcation Theorems at Simple Eigenvalues and Economic Applications. Unpublished paper, 1992, now retitled as Some New Hopf Bifurcation Theorems at Simple Eigenvalues, to appear in the forthcoming issue of Applicable Analysis, an International Journal, Vol 53/ 3-4, June [238] Taylor, A.E. Introduction to Functional Analysis. John Wiley, New York, [239] Thorn, R. Stabilite Structurelle et Morphogenese. Reading, Mass: Benjamin, [240] Thompson, J.M.T. & G.W. Hunt. A General Theory of Elastic Stability. Wiley, London, [241] Thompson, J.M.T. & G.W. Hunt. A Bifurcation of Theoryfor the Instabilities of Optimization and Design. Synthese, Vol. 36: , [242] Thompson, J.M.T. & H.B. Stewart. Nonlinear Dynamics and Chaos. Wiley & Sons, New York, [243] Tobin, J. Money and Economic Growth. Econometrica 33: , October, [244] Torre, V. Existence of Limit Cycles and Control in Complete Keynesian Systems by Theory of Bifurcations. Econometrica 45: , [245] Trotman, D.J.A. & E.C. Zeeman. "The Classification of Elementary Catastrophes of Co dimension ~ 5," in Hilton, P.J. (ed): Structural Stability, the Theory of catastrophes and Applications in the Science. Springer, Berlin, [246] Tu, P.N.V. The Economics of Educational Planning. Ph.D. Dissertation, A.N.U., Canberra, [247] Tu, P.N.V. Optimal Educational Investment Program in an Economic Planning Model. Canadian Journal of Economics 2: 52-64, February, [248] Tu, P.N.V. A Multisectoral Model of Educational and Economic Planning. Metroeconomica 22: ,1970. [249] Tu, P.N.V. Comparative Statics and Catastrophe Theory in Economics. Discussion Papers Series No 74, University of Calgary, July, 1982.

16 310 [250] Tu, P.N.V. Introductory Optimization Dynamics. Springer- Verlag, Heidelberg, [251] Tu, P.N.V. A Dynamic Macroeconomic Model. Discussion Paper Series No 102, University of Calgary, February, [252] Tu, P.N.V. Towards an Optimal Wildlife Management. Discussion Papers Series No 111, University of Calgary, June, [253] Tu, P.N.V. Introductory Optimization Dynamics. Second revised and enlarged edition, Springer-Verlag, Heidelberg, [254] Tu, P.N.V. Double Axisymmetry and Perturbed Hamiltonian Dynamical Systems in Optimal Economic Growth Models. Discussion Paper #135, Sept. 1992, Department of Economics, University of Calgary. [255] Turnovsky, S. Macroeconomic Analysis and Stabilization Policy. Cambridge Univ. Press, Cambridge, 1977, reprinted [256] Uzawa, H. On a Two-Sector Model of Economic Growth. Review of Economic Studies 28: 40-47, October, [257] Uzawa, H. Optimal Technical Change in an Aggregative Model of Economic Growth. International Economic Review 6: 18-31, January, [258] Uzawa, H Market Allocation and Optimal Growth. Australian Economic Papers 7: 17-27, June, [259] Van der Ploeg, F. Economic Growth and Conflict over the Distribution of Income. Journal of Economic Dynamics and Control 6: , [260] Van der Ploeg, F. Classical Growth Cycles. Metroeconomica 37(2): , [261] Varian, H.R. Catastrophe Theory and the Business Cycle. Economic Inquiry 17: 14-28, January, [262] Velupillai, K. Some Stability Properties of Goodwin's Growth Cycles. Journal of Economics 39: , [263] Verhulst, P.F. Notice sur la loi que la population suit dans son accroissement. Correspondance Mathematique et Physique 10: , [264] Volterra, V. Lel;ons sur la theorie mathematique de la lutte pour la vze. Gauthier-Villars, Paris, [265] Volterra, V. Principes de biologie mathematique. Acta Biother. 3: 1-36, [266] Waldrop, M.M. Complexity: the Emerging Science at the Edge of Chaos. Simon & Schuster, 1992.

17 311 [267] Waltman, P.E. The Equation of Growth. Bulletin Math. Biophys. 26: 39-43, [268] Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, Heidelberg, [269] Wilen, J. & G. Brown, Jr. Optimal Recovery Paths for Perturbations of Trophic Level Bioeconomic Systems. Journal of Environmental Economics and Management 13(3): , September, [270] Wirl, F. Routes to Cyclical Strategies in Two Dimensional Optimal Control Models: Necessary Conditions and Existence. Technical University of Vienna, [271] Wirl, F. The Ramsey Model Revisited: The Optimality of Cyclical Consumption Paths. Technical University of Vienna, [272] Wirl, F. The Ramsey Model Revisited: The Optimality of Cyclical Consumption Paths. Institute of Energy Economics, University of Vienna, January, [273] Wolfstetters. Fiscal Policy and the Classical Growth Cycle. Journal of Economics 42(4): , [274] Zarnowitz, V. Recent Work on Business Cycles in Historical Perspective: A Review of Theories and Evidence. Journal of Economic Literature 23: , June, [275] Zeeman, E.C. "Differential Equations for the Heartbeat and Nerve Impulse" in Waddington, C.H. (ed): Towards a Theoretical Biology. Aldine-Atherton, Chicago, [276] Zeeman, E.C. "Differential Equations for the Hearth and Nerve Impulse," in Peixoto (ed): Dynamical Systems. Academic Press, New York, [277] Zeeman, E.C. Catastrophe Theory. Scientific American 234 (PL 4): [278] Zeeman, E.C. Catastrophe Theory. Reading, Mass, Addison- Wesley, 1977.

18 INDEX a-limit set 150 Diffeomorphism 3 Attracting set 13 Difference Equations Attractors 13 Differential Equations 5-38 Asymptotic stability 100 Differential Operator D 19 Diagonalization 76 Bendixson-Poincare 151 Discrete Systems 115 Bernouilli equation 11 Bifurcation Eigenspace of flow Eigenvalues 69 - of map complex 69 - Flip real 69 - Fold 209 Eigenvectors 69 - Hopf Elementary Catastrophes Pitchford 198 Equations - saddle node Bernouilli 11 - supercritical characteristic 70 - sub critical difference 39 - transcritical differential 5 Biological control Lienard 155 Biology logistic 212 Blue Sky catastrophe Van de Pol 154 Equilibrium point 8 - Exchange of Stability 195, Feigenbaum number 214 Catastrophe Theory fold Floquet Theory cusp 233 Centre Manifold Theorem Chaos First return map in map Flow 2 - in flow 216 Focus 102 Characteristic exponent 185, 186 Fundamental Matrix- 98 Characteristic equation 70 Characteristic polynomial 70 Gradient systems 163 Codimension 229 Growth models Competing species 288 in Economics 178,258 Complex eigenvalues 93, 120 Conservative Hamiltonian Systems 171 Cusp 233 Cycle 149 Hamiltonian flow 171 Hamiltonian function 170 Hamiltonian system Hartman-Grobman Theorem 135 Homeomorphism 160 Degenerate 195,266 Homoclinic bifurcation 219 Determinants Homoclinic tangle 218

19 314 Hopf bifurcation Horseshoe map Hyperbolic fixed point Implicit Function Theorem Improper Node Idempotent matrices Intermittency IS-LM economic models Jordan canonical form Lagrangian Dynamic System Laplace transformation Li-York Theorem Liapunov - characteristic exponent - function - Second method - stability Liapunov-Smith reduction Lienard-Smith reduction Lienard system Limit Cycles Linearization theorem Manifold Map Melkinov theory Maximum Principle Morse set ,218 34, , Multiplier-accelerator models 54 Potential functions Prey-predator models Repeller Saddle loop connection Saddle node Schwarzian derivative Sensitive dependence on initial conditions Silnikov Theory Singularity Smale-Birkhoff Smale horseshoe Splitting Lemma Stabilization Control models Stability - asymptotic - local asymptotic - global asymptotic - structural 163, , , , , Tatonnement Model 277 Transversality Conditions 248 Unfolding 229, 241 Unimodal Map 212 Universal Constant: Feigenbaum 214 Nilpotent matrix 87, 91 Normal forms 191 Nonhyperbolic fixed points 187,195 Nonlinear Systems Optimal Control 245 Optimal Economic growth 258 Peixoto Theorem 160 Poincare-Bendixson Theorem 151 Poincare map 183, 184 Poincare section 183

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