## Attractors, Bifurcations, & Chaos: Nonlinear Phenomena in EconomicsThe present book relies on various editions of my earlier book "Nonlinear Economic Dynamics", first published in 1989 in the Springer series "Lecture Notes in Economics and Mathematical Systems", and republished in three more, successively revised and expanded editions, as a Springer monograph, in 1991, 1993, and 1997, and in a Russian translation as "Nelineynaia Economicheskaia Dinamica". The first three editions were focused on applications. The last was differ ent, as it also included some chapters with mathematical background mate rial -ordinary differential equations and iterated maps -so as to make the book self-contained and suitable as a textbook for economics students of dynamical systems. To the same pedagogical purpose, the number of illus trations were expanded. The book published in 2000, with the title "A ttractors, Bifurcations, and Chaos -Nonlinear Phenomena in Economics", was so much changed, that the author felt it reasonable to give it a new title. There were two new math ematics chapters -on partial differential equations, and on bifurcations and catastrophe theory -thus making the mathematical background material fairly complete. The author is happy that this new book did rather well, but he preferred to rewrite it, rather than having just a new print run. Material, stemming from the first versions, was more than ten years old, while nonlinear dynamics has been a fast developing field, so some analyses looked rather old-fashioned and pedestrian. The necessary revision turned out to be rather substantial. |

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### Contents

1 Introduction | 1 |

12 Linear Versus Nonlinear Modelling | 2 |

13 Modelling Nonlinearity | 4 |

15 Perturbation Analysis | 6 |

16 Numerical Experiment | 7 |

17 Structural Stability | 8 |

19 Chaos and Fractals | 9 |

110 Layout of the Book and Reading Strategies | 10 |

82 The Original Model | 308 |

83 Nonlinear Investment Functions and Limit Cycles | 309 |

Existence | 312 |

Asymptotic Approximation | 315 |

Transients and Stability | 320 |

87 The TwoRegion Model | 325 |

88 The Persistence of Cycles | 326 |

89 Perturbation Analysis of the Coupled Model | 328 |

Ordinary | 13 |

22 Linear Systems | 20 |

23 Structural Stability | 28 |

24 Limit Cycles | 32 |

25 The Hopf Bifurcation | 37 |

26 The SaddleNode Bifurcation | 39 |

PoincareLindstedt | 41 |

TwoTiming | 47 |

Lyapunovs Method versus Linearisation | 53 |

210 Forced Oscillators Transients and Resonance | 56 |

van der Pol | 60 |

Duffing | 69 |

213 Chaos | 76 |

214 Poincare Sections and Return Maps | 79 |

215 A Short History of Chaos | 90 |

Partial | 95 |

32 Time and Space | 96 |

dAlamberts Solution | 97 |

34 Initial Conditions | 99 |

35 Boundary Conditions | 101 |

Variable Separation | 103 |

37 The General Solution and Fouriers Theorem | 106 |

38 Friction in the Wave Equation | 109 |

39 Nonlinear Waves | 111 |

Gradient and Divergence | 114 |

311 Line Integrals and Gausss Integral Theorem | 118 |

Eigenfunctions | 124 |

313 The Square | 127 |

314 The Circular Disk | 132 |

315 The Sphere | 136 |

316 Nonlinearity Revisited | 141 |

317 Tessellations and the EulerPoincare Index | 143 |

318 Nonlinear Waves on the Square | 145 |

319 Perturbation Methods for Nonlinear Waves | 150 |

4 Iterated Maps or Difference Equations | 161 |

42 The Logistic Map | 162 |

43 The Lyapunov Exponent | 171 |

44 Symbolic Dynamics | 174 |

45 Sharkovskys Theorem and the Schwarzian Derivative | 178 |

46 The Henon Model | 180 |

47 Lyapunov Exponents in 2D | 184 |

48 Fractals and Fractal Dimension | 187 |

49 The Mandelbrot Set | 192 |

410 Can Chaos be Seen? | 196 |

411 The Method of Critical Lines | 199 |

412 Bifurcations and Periodicity | 209 |

5 Bifurcation and Catastrophe | 217 |

51 History of Catastrophe Theory | 218 |

52 Morse Functions and Universal Unfoldings in 1 D | 219 |

53 Morse Functions and Universal Unfoldings in 2 D | 223 |

Fold | 228 |

Cusp | 229 |

Swallowtail and Butterfly | 232 |

Umblics | 235 |

6 Monopoly | 239 |

62 The Model | 241 |

63 Adaptive Search | 244 |

64 Numerical Results | 246 |

65 Fixed Points and Cycles | 248 |

66 Chaos | 252 |

67 The Method of Critical Lines | 254 |

68 Discussion | 259 |

7 Duopoly and Oligopoly | 261 |

72 The Cournot Model | 262 |

73 Stackelberg Equilibria | 265 |

74 The Iterative Process | 266 |

75 Stability of the Cournot Point | 269 |

76 Periodic Points and Chaos | 271 |

77 Adaptive Expectations | 275 |

78 The Neimark Bifurcation | 276 |

79 Critical Lines and Absorbing Area | 283 |

710 Adjustments Including Stackelberg Points | 285 |

711 Oligopoly with Three Firms | 287 |

712 Stackelberg Action Reconsidered | 295 |

713 Back to Duopoly | 296 |

714 True Triopoly | 303 |

Continuous Time | 307 |

810 The Unstable Zero Equilibrium | 331 |

811 Other Fixed Points | 333 |

812 Properties of Fixed Points | 337 |

813 The Arbitrary Phase Angle | 338 |

814 Stability of the Coupled Oscillators | 340 |

815 The Forced Oscillator | 342 |

817 The Small Open Economy | 344 |

819 Catastrophe | 346 |

820 Period Doubling and Chaos | 347 |

821 Relaxation Cycles | 351 |

The Autonomous Case | 354 |

The Forced Case | 355 |

Continuous Space | 357 |

92 Interregional Trade | 358 |

93 The Linear Model | 360 |

94 Coordinate Separation | 362 |

95 The Square Region | 364 |

96 The Circular Region | 366 |

97 The Spherical Region | 367 |

98 The Nonlinear Spatial Model | 370 |

99 Dispersive Waves | 372 |

910 Standing Waves | 374 |

911 Perturbation Analysis | 376 |

Discrete Time | 381 |

102 Investments | 382 |

103 Consumption | 384 |

104 The Cubic Iterative Map | 385 |

105 Fixed Points Cycles and Chaos | 386 |

106 Formal Analysis of Chaotic Dynamics | 393 |

108 The Three Requisites of Chaos | 394 |

109 Symbolic Dynamics | 395 |

1010 Brownian Random Walk | 396 |

1011 Digression on Order and Disorder | 400 |

1012 The General Model | 401 |

1013 Relaxation Cycles | 402 |

1014 Lyapunov Exponents and Fractal Dimensions | 405 |

1015 Numerical Studies of the General Case | 408 |

1016 The Neimark Bifurcation | 411 |

1017 Critical Lines and Absorbing Areas | 418 |

The Model | 426 |

Fixed Points | 429 |

Invariant Spaces | 430 |

1021 Processes in Three Dimensions | 437 |

11 Dynamics of Interregional Trade | 443 |

112 The Basic Model | 444 |

113 Structural Stability | 449 |

114 The Square Flow Grid | 451 |

115 Triangular Hexagonal Grids | 454 |

116 Changes of Structure | 457 |

117 Dynamisation of Beckmanns Model | 463 |

118 Stability | 464 |

119 Uniqueness | 467 |

Increasing Complexity | 471 |

121 The Development Tree | 473 |

122 Continuous Evolution | 475 |

123 Diversification | 476 |

124 Lancasters Property Space | 478 |

126 Bifurcations | 479 |

127 Consumers | 481 |

128 Producers | 484 |

129 Catastrophe | 486 |

1210 Simple Branching in 1 D | 487 |

1211 Branching and Emergence of New Implements in 1 D | 489 |

1212 Catastrophe Cascade in 1 D | 492 |

1213 Catastrophe Cascade in 2 D | 494 |

1214 Fast and Slow Processes | 497 |

1215 Alternative Futures | 499 |

Multiple Attractors | 503 |

131 Population Dynamics | 504 |

132 Diffusion | 509 |

133 Stability | 514 |

134 The Dynamics of Capital and Labour | 519 |

529 | |

List of Figures | 535 |

543 | |

### Other editions - View all

Attractors, Bifurcations, & Chaos: Nonlinear Phenomena in Economics Tönu Puu No preview available - 2014 |

Attractors, Bifurcations, & Chaos: Nonlinear Phenomena in Economics Tönu Puu No preview available - 2010 |

### Common terms and phrases

absorbing area Accordingly amplitude attraction basins basins of attraction becomes bifurcation diagram boundary conditions catastrophe catastrophe theory chaos chaotic attractor circle coefficients coexistent complex consider constant convergence coordinates Cournot point critical lines cubic defined denoted derivative determinant differential equations dimension dimensional display divergence dynamical system economics eigenvalues equal to zero equilibrium fact finite fixed point fold fractal fractal dimension frequency function hence income infinity initial conditions integral intersection invariant curve invariant plane iteration Jacobian limit cycle linear linearised logistic logistic map loses stability Lyapunov exponent monkey saddle motion negative Neimark bifurcation node nonlinear nonzero obtain orbits origin oscillator pair parameter value period doubling phase space picture Poincare section positive potential produce quasiperiodic region right hand side saddle points second order simulation sine singular solution solve spatial square structural stability Substituting theorem tion tongues torus trajectories unstable variables vector wave whereas