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INDEX

a-limit set 150 Diffeomorphism 3 Attracting set 13 Difference Equations 39-58 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 195-210 Eigenspace 145 - of flow 195-200 Eigenvalues 69 - of map 209-210 - complex 69 - Flip 210 - real 69 - Fold 209 Eigenvectors 69 - Hopf 200-204 Elementary Catastrophes 232 - Pitchford 198 Equations

- saddle node 197 - Bernouilli 11 - supercritical 198 - characteristic 70 - sub critical 197 - difference 39 - transcritical 197 - differential 5 Biological control 293 - Lienard 155 Biology 280 - logistic 212 Blue Sky catastrophe 220 - Van de Pol 154

Equilibrium point 8

Catastrophe Theory 226 - Exchange of Stability 195, 196 - Feigenbaum number 214

- fold 232 - Floquet Theory 185 - cusp 233 Centre Manifold Theorem 187-191 Chaos 211-226 First return map 183

- in map 212-216 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

Hamiltonian flow Systems 171 171

Cusp 233 Hamiltonian function 170

Cycle 149 Hamiltonian system 170-175 Hartman-Grobman Theorem 135 Homeomorphism 160

Degenerate 195,266 Homoclinic bifurcation 219 Determinants 65-67 Homoclinic tangle 218

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

200 218 136

195 104 81

217,218 34, 107

79

167 29

215

221 101 101 101 193 193 155 149

134-146 187, 188

157 219 245 245

Multiplier-accelerator models 54

Nilpotent matrix 87, 91 Normal forms 191 Nonhyperbolic fixed points 187,195 Nonlinear Systems 133-161

Optimal Control 245 Optimal Economic growth 258

Peixoto Theorem 160 Poincare-Bendixson Theorem 151 Poincare map 183, 184 Poincare section 183

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, 226 140, 285

13

218 197 212

217,218 221, 225 195, 241

218 218 228 253

159 100 100 160

Tatonnement Model 277 Transversality Conditions 248

Unfolding 229, 241 Unimodal Map 212 Universal Constant: Feigenbaum 214

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