Dynamical systems: an introduction with applications in economics and biology
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin u.a.
Springer
1994
|
| Ausgabe: | 2., rev. and enl. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. [295] - 311 |
| Beschreibung: | XVIII, 314 S. zahlr. graph. Darst. |
| ISBN: | 3540576614 0387576614 |
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| 100 | 1 | |a Tu, Pierre N. V. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Dynamical systems |b an introduction with applications in economics and biology |c Pierre N. V. Tu |
| 250 | |a 2., rev. and enl. ed. | ||
| 264 | 1 | |a Berlin u.a. |b Springer |c 1994 | |
| 300 | |a XVIII, 314 S. |b zahlr. graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. [295] - 311 | ||
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| 650 | 4 | |a Mathématiques économiques | |
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| 650 | 4 | |a Economics, Mathematical | |
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Datensatz im Suchindex
| _version_ | 1804123960588107776 |
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| adam_text | Contents
Preface v
1 Introduction 1
2 Review of Ordinary Differential Equations 5
2.1 First Order Linear Differential Equations 6
2.1.1 First Order Constant Coefficient Linear Differential Equations 7
2.1.2 Variable Coefficient First Order Linear Differential Equations 9
2.1.3 Equations Reducible to Linear Differential Equations 11
2.1.4 Qualitative Solution: Phase Diagrams 12
2.1.5 Some Economic Applications 14
1. Walrasian Tatonnement Process 14
2. The Keynesian Model 15
3. Harrod Domar s Economic Growth Model 16
4. Domar s Debt Model (1944) 17
5. Profit and Investment 17
6. The Neo Classical Model of Economic Growth 18
2.2 Second and Higher Order Linear Differential Equations 19
2.2.1 Particular Integral (xp or xe) where d(t) = d Constants .... 24
2.2.2 Particular Integral (xp) when d = g(t) is some Function of t . 25
1. The Undetermined Coefficients Method 25
2. Inverse Operator Method 26
3. Laplace Transform Method 29
2.3 Higher Order Linear Differential Equations with Constant Coefficients 31
2.4 Stability Conditions 33
2.5 Some Economic Applications 34
1. The IS LM Model of the Economy 34
2. A Continuous Multiplier Acceleration Model 35
3. Stabilization Policies 35
4. Equilibrium Models with Stock 37
2.6 Conclusion 38
3 Review of Difference Equations 39
3.1 Introduction 39
3.2 First Order Difference Equations 40
3.2.1 Linear Difference Equations 40
X
3.2.2 Non linear Difference Equations and Phase Diagram 42
3.2.3 Some Economic Applications 43
1. The Cobweb Cycle 43
2. The Dynamic Multiplier Model 44
3. The Overlapping Generations Model 44
3.3 Second Order Linear Difference Equations 46
3.3.1 Particular Integral 46
3.3.2 The Complementary Functions xc(t) 47
3.3.3 Complete Solution and Examples 49
3.4 Higher Order Difference Equations 51
3.5 Stability Conditions 51
3.5.1 Stability of First Order Difference Equations 52
3.5.2 Stability of Second Order Difference Equations 52
3.5.3 Stability of Higher Order Difference Equations 53
3.6 Economic Applications 54
3.6.1 Samuelson s (1939) Business Cycle 54
3.6.2 Hick s (1950) Contribution to the Theory of Trade Cycle ... 55
3.7 Concluding Remarks 57
4 Review of Some Linear Algebra 59
4.1 Vector and Vector Spaces 59
4.1.1 Vector Spaces 60
4.1.2 Inner Product Space 61
4.1.3 Null Space and Range, Rank and Kernel 62
4.2 Matrices 63
4.2.1 Some Special Matrices 63
4.2.2 Matrix Operations 64
4.3 Determinant Functions 65
4.3.1 Properties of Determinants 65
4.3.2 Computations of Determinants 66
4.4 Matrix Inversion and Applications 68
4.5 Eigenvalues and Eigenvectors 69
4.5.1 Similar Matrices 72
4.5.2 Real Symmetric Matrices 73
4.6 Quadratic Forms 74
4.7 Diagonalization of Matrices 76
4.7.1 Real Eigenvalues 76
4.7.2 Complex Eigenvalues and Eigenvectors 78
4.8 Jordan Canonical Form 79
4.9 Idempotent Matrices and Projection 81
4.10 Conclusion 82
xi
5 First Order Differential Equations Systems 83
5.1 Introduction 83
5.2 Constant Coefficient Linear Differential Equation (ODE) Systems . . 83
5.2.1 Case (i). Real and Distinct Eigenvalues 84
5.2.2 Case (ii). Repeated Eigenvalues 87
5.2.3 Case (iii). Complex Eigenvalues 88
5.3 Jordan Canonical Form of ODE Systems 89
Case (i) Real Distinct Eigenvalues 90
Case (ii) Multiple Eigenvalues 91
Case (iii) Complex Eigenvalues 93
5.4 Alternative Methods for Solving x = Ax . 95
5.4.1 Sylvester s Method 95
5.4.2 Putzer s Methods (Putzer 1966) 96
5.4.3 A Direct Method of Solving x = Ax 97
5.5 Reduction to First Order of ODE Systems 98
5.6 Fundamental Matrix 98
5.7 Stability Conditions of ODE Systems 100
5.7.1 Asymptotic Stability 100
5.7.2 Global Stability: Liapunov s Second Method id
5.8 Qualitative Solution: Phase Portrait Diagrams 102
5.9 Some Economic Applications 107
5.9.1 Dynamic IS LM Keynesian Model 107
5.9.2 Dynamic Leontief Input Output Model 109
5.9.3 Multimarket Equilibrium Ill
5.9.4 Walras Cassel Leontief General Equilibrium Model 112
6 First Order Difference Equations Systems 115
6.1 First Order Linear Systems 115
6.2 Jordan Canonical Form 117
Case (i). Real Distinct Eigenvalues 118
Case (ii). Multiple Eigenvalues 119
Case (iii). Complex Eigenvalues 120
6.3 Reduction to First Order Systems 121
6.4 Stability Conditions 123
6.4.1 Local Stability 123
6.4.2 Global Stability 125
6.5 Qualitative Solutions: Phase Diagrams 126
6.6 Some Economic Applications 128
1. A Multisectoral Multiplier Accelerator Model 128
2. Capital Stock Adjustment Model 129
3. Distributed Lags Model 129
4. Dynamic Input Output Model 130
xii
7 Nonlinear Systems 133
7.1 Introduction 133
7.2 Linearization Theory 134
7.2.1 Linearization of Dynamic Systems in the Plane 136
7.2.2 Linearization Theory in Three Dimensions 144
7.2.3 Linearization Theory in Higher Dimensions 145
7.3 Qualitative Solution: Phase Diagrams 147
7.4 Limit Cycles 149
Economic Application I: Kaldor s Trade Cycle Model 152
7.5 The Lienard Van der Pol Equations
and the Uniqueness of Limit Cycles 154
Economic Application II: Kaldor s Model
as a Lienard Equation 156
7.6 Linear and Nonlinear Maps 157
7.7 Stability of Dynamical Systems 159
7.7.1 Asymptotic Stability 159
7.7.2 Structural Stability 160
7.8 Conclusion 161
8 Gradient Systems, Lagrangean and Hamiltonian Systems 163
8.1 Introduction 163
8.2 The Gradient Dynamic Systems (GDS) 163
8.3 Lagrangean and Hamiltonian Systems 167
8.4 Hamiltonian Dynamics 170
8.4.1 Conservative Hamiltonian Dynamic Systems (CHDS) 171
8.4.2 Perturbed Hamiltonian Dynamic Systems (PHDS) 174
8.5 Economic Applications , . . . 176
8.5.1 Hamiltonian Dynamic Systems (HDS) in Economics 176
8.5.2 Gradient (GDS) vs Hamiltonian (HDS) Systems in Economics 177
8.5.3 Economic Applications: Two State Variables Optimal
Economic Control Models 178
8.6 Conclusion 181
9 Simplifying Dynamical Systems 183
9.1 Introduction 183
9.2 Poincare Map 183
9.3 Floquet Theory 185
9.4 Centre Manifold Theorem (CMT) 187
9.5 Normal Forms 191
9.6 Elimination of Passive Coordinates 192
9.7 Liapunov Schmidt Reduction 193
9.8 Economic Applications and Conclusions 194
10 Bifurcation, Chaos and Catastrophes in Dynamical Systems 195
10.1 Introduction 195
10.2 Bifurcation Theory (BT) 195
xiii
10.2.1 One Dimensional Bifurcations 197
10.2.2 Hopf Bifurcation 200
10.2.3 Some Economic Applications 204
1. The Keynesian IS LM Model 204
2. Hopf Bifurcation in an Advertising Model 205
3. A Dynamic Demand Supply Model 207
4. Generalized Tobin s Model of Money and Economic Growth 208
10.2.4 Bifurcations in Discrete Dynamical Systems 209
1. The Fold of Saddle Node Bifurcation 209
2. Transcritical Bifurcation 210
3. Flip Bifurcation 210
4. Logistic System 210
10.3 Chaotic or Complex Dynamical Systems (DS) 211
10.3.1 Chaos in Unimodal Maps in Discrete Systems 212
10.3.2 Chaos in Higher Dimensional Discrete Systems 216
10.3.3 Chaos in Continuous Systems 216
10.3.4 Routes to Chaos 217
1. Period Doubling and Intermittency 217
2. Horseshoe and Homoclinic Orbits 218
10.3.5 Liapunov Characteristic Exponent (LCE)
and Attractor s Dimension 221
10.3.6 Some Economic Applications 222
1. Chaotic Dynamics in a Macroeconomic Model 222
2. Erratic Demand of the Rich 224
3. Structure and Stability of Inventory Cycles 224
4. Chaotic economic Growth with Pollution 225
5. Chaos in Business Cycles 226
10.4 Catastrophe Theory (C.T.) 226
10.4.1 Some General Concepts 227
10.4.2 The Morse and Splitting Lemma 228
10.4.3 Codimension and Unfolding 229
10.4.4 Classification of Singularities 231
10.4.5 Some Elementary Catastrophes 232
1. The Fold Catastrophe 232
2. The Cusp Catastrophe 233
10.4.6 Some Economic Applications 235
1. The Shutdown of the Firm (Tu 1982) 235
2. Kaldor s Trade Cycle 236
3. A Catastrophe Theory of Defence Expenditure 238
4. Innovation, Industrial Evolution and Revolution 240
10.4.7 Comparative Statics (C.S.), Singularities and Unfolding . . . .241
10.5 Concluding Remarks 243
xiv
11 Optimal Dynamical Systems 245
11.1 Introduction 245
11.2 Pontryagin s Maximum Principle 245
11.2.1 First Variations and Necessary Conditions 248
11.2.2 Second Variations and Sufficient Conditions 252
11.3 Stabilization Control Models 253
11.4 Some Economic Applications 256
1. Intergenerational Distribution of Non renewable Resources . . 256
2. Optimal Harvesting of Renewable Resources 256
3. Multiplier Accelerator Stabilization Model 257
4. Optimal Economic Growth (OEG) 258
11.5 Asymptotic Stability of Optimal Dynamical Systems (ODS) 260
11.6 Structural Stability of Optimal Dynamical Systems 263
11.6.1 Hopf Bifurcation in Optimal Economic Control Models and
Optimal Limit Cycles 263
Two State Variable Models 264
Multisectoral OEG Models 265
11.6.2 Chaos in Optimal Dynamical Systems (ODS) 267
11.7 Conclusion 268
12 Some Applications in Economics and Biology 271
12.1 Introduction 271
12.2 Economic Applications of Dynamical Systems 271
12.2.1 Business Cycles Theories 271
1. Linear Multiplier Accelerator Models 272
2. Nonlinear Models 273
2.1. Flexible Multiplier Accelerator Models 273
2.2. Kaldor s Type of Flexible Accelerator Models ... 275
2.3. Goodwin s Class Struggle Model 275
3. Optimal Economic Fluctuations and Chaos 276
12.2.2 General Equilibrium Dynamics 276
Tatonnement Adjustment Process 277
Non Tatonnement Models 278
12.2.3 Economic Growth Theories 279
1. Harrod Domar s Models 279
2. Neo Classical Models 279
2.1. Two Sector Models 280
2.2. Economic Growth with Money 281
2.3. Optimal Economic Growth Models 282
2.4. Endogenous Economic Growth Models 282
12.3 Dynamical Systems in Biology 284
12.3.1 One Species Growth Models 284
12.3.2 Two Species Models 285
1. Predation Models 285
2. Competition Models 288
12.3.3 The Dynamics of a Heartbeat 288
XV
12.4 Bioeconomics and Natural Resources 290
12.4.1 Optimal Management of Renewable
and Exhaustible Resources 290
12.4.2 Optimal Control of Prey Predator Models 292
(i) Control by an Ideal Pesticide 292
(ii) Biological Control 293
12.5 Conclusion 294
Bibliography 295
|
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| dewey-tens | 000 - Computer science, information, general works |
| discipline | Informatik Mathematik Wirtschaftswissenschaften |
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| language | German |
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| publisher | Springer |
| record_format | marc |
| spelling | Tu, Pierre N. V. Verfasser aut Dynamical systems an introduction with applications in economics and biology Pierre N. V. Tu 2., rev. and enl. ed. Berlin u.a. Springer 1994 XVIII, 314 S. zahlr. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. [295] - 311 Biomathématiques Differentiaalvergelijkingen gtt Dynamique différentiable Dynamische systemen gtt Mathématiques économiques Differentiable dynamical systems Economics, Mathematical Biomathematics Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Mathematisches Modell (DE-588)4114528-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006358284&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tu, Pierre N. V. Dynamical systems an introduction with applications in economics and biology Biomathématiques Differentiaalvergelijkingen gtt Dynamique différentiable Dynamische systemen gtt Mathématiques économiques Differentiable dynamical systems Economics, Mathematical Biomathematics Mathematisches Modell (DE-588)4114528-8 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4114528-8 (DE-588)4013396-5 |
| title | Dynamical systems an introduction with applications in economics and biology |
| title_auth | Dynamical systems an introduction with applications in economics and biology |
| title_exact_search | Dynamical systems an introduction with applications in economics and biology |
| title_full | Dynamical systems an introduction with applications in economics and biology Pierre N. V. Tu |
| title_fullStr | Dynamical systems an introduction with applications in economics and biology Pierre N. V. Tu |
| title_full_unstemmed | Dynamical systems an introduction with applications in economics and biology Pierre N. V. Tu |
| title_short | Dynamical systems |
| title_sort | dynamical systems an introduction with applications in economics and biology |
| title_sub | an introduction with applications in economics and biology |
| topic | Biomathématiques Differentiaalvergelijkingen gtt Dynamique différentiable Dynamische systemen gtt Mathématiques économiques Differentiable dynamical systems Economics, Mathematical Biomathematics Mathematisches Modell (DE-588)4114528-8 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Biomathématiques Differentiaalvergelijkingen Dynamique différentiable Dynamische systemen Mathématiques économiques Differentiable dynamical systems Economics, Mathematical Biomathematics Mathematisches Modell Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006358284&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT tupierrenv dynamicalsystemsanintroductionwithapplicationsineconomicsandbiology |