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| Marotto F. — Introduction to Mathematical Modeling Using Discrete Dynamical Systems |
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| Предметный указатель |
Markov process 61—66 76 256—262
Markov process, regular 76
Markov process, state vector 257
Markov process, state/transition diagram 61
Markov process, steady-state 76 257
Markov process, three-state 256—262
Markov process, three-state, reducing to two variables 259 260
Markov process, transition matrix 257
Markov process, two-state 61—66 76
Markov process, two-state, reducing to one variable 63
Mathematica 33 315 320—326
Mathematical model 2 see
Matrix 211
Matrix coefficient, linear system 219 255
Matrix coefficient, simultaneous equations 214
Matrix determinant 215
Matrix eigenvalues see "Eigenvalue(s)"
Matrix eigenvectors 220—225
Matrix identity 213 214 226 264
Matrix inverse 214 215
Matrix multiplication 212 213
Matrix norm 218
Matrix of partial derivatives 278
Matrix of partial derivatives and dynamics 279 283 290—292
Matrix transition 257—261
May, R.M. 173
Mean value theorem 123
Membrane 25
Microsoft Excel 33 315—320
Migration 24 153 191—193
Models, Bio-medicine, chronic illness 260
Models, Bio-medicine, contagious disease 76 91 269
Models, Bio-medicine, virus growth 48
Models, Business, market share 67 263
Models, Business, profitability 67
Models, Business, unemployment level 76
Models, Ecology, competing species 191—193 267
Models, Ecology, harvesting 24 153 163 191—193
Models, Ecology, prey-predator 190 191 266
Models, Ecology, single species 23 24 89 105 110 192 193 267 268
Models, Economics, demand 193 194 270
Models, Economics, price 25 92 93 193 194 270
Models, Economics, supply 193
Models, Engineering, filtering pollutants 27 38 48 58
Models, Engineering, swaying skyscraper 48
Models, Finance, annuity savings 22 59
Models, Finance, compound interest 13
Models, Finance, credit card debt 77
Models, Finance, loan 22 60
Models, Finance, simple interest 12
Models, Learning Theory, learning a topic 95
Models, Learning Theory, stages of training 64 67 264
Models, Physics, beating drum 10
Models, Physics, bouncing ball 6 9
Models, Physics, coin toss 3 4
Models, Physics, cooling object 27 48
Models, Physics, falling object 3 5
Models, Physics, radioactive decay 48
Models, Physics, swinging pendulum 10 27 48
Models, Physics, weather prediction 301 302
Models, Sociology, education levels 65 67
Models, Sociology, home ownership 67
Models, Sociology, income mobility 261 263
Models, Sociology, spread of technology 95
Models, Sociology, urbanization 61 67 257—259 263
Modulus 241 242 250 278 283 291
Monotonic solution 70 73 74 see
Negative of a vector 210
Newton Root-Finding Method 127—133 154 175 314
Newton's law of cooling 27
Non-autonomous linear equation 28—30
Non-autonomous linear equation, homogeneous 39 40
Non-autonomous linear equation, non-homogeneous 39 49 50
Non-autonomous nonlinear equation 98 99
Non-homogeneous linear equation 39
Non-homogeneous linear equation, autonomous see "Autonomous linear equation"
Non-homogeneous linear equation, non-autonomous 49 50
Non-homogeneous linear system 197 254
Non-homogeneous linear system, fixed point 204—206
Non-homogeneous linear system, rotation/stability 255 256
Nonlinear Competition Model 267
Nonlinear equation 87 96 see
Nonlinear equation, 2-cycle 141 144 146
Nonlinear equation, chaos 176—186 see
Nonlinear equation, fixed point 104 106 119
Nonlinear equation, m-cycle 147—151
Nonlinear equation, solution 101 109
Nonlinear Infection Model I/II 91 156 157
Nonlinear Infection-Recovery Model 269
Nonlinear model see also "Models"
Nonlinear model, differential equation(s) 3 5 6 133 134 140
Nonlinear model, differential equation(s), system of 275 301 302
Nonlinear model, equation 88—93 95 99 105 110 153 163
Nonlinear model, system 266—270 274 276
Nonlinear Overlapping-Generations Model 268
Nonlinear Population Model I/II 89 97 155 156
Nonlinear Prey-Predator Model 266
Nonlinear Price Model I/II/III 93
Nonlinear Price-Demand Model 270
Nonlinear regression 96
Nonlinear system 265 270 see
Nonlinear system of differential equations 275 301 302
Nonlinear system, chaos 295—299 see
Nonlinear system, fixed point 270 271 278 279
Nonlinear system, periodic point/cycle 282 283
Norm of a matrix 218
Normal equations 82 83
Numerical solution 8 32 33
Numerical solution, Mathematica 320—326
Numerical solution, Microsoft Excel 315—320
One repeated real eigenvalue see "Eigenvalue(s)"
One-parameter family 155 156 see
Ordinary annuity 21 22 59
Oscillation see also "Periodic point/cycle"
Oscillation, linear equation 70 73 74
Oscillation, nonlinear equation 118 119 124
Oscillation, periodic 36 99 144 145
Overlapping-generations 192 193 267 268
Overlapping-generations, Linear Overlapping-Generations Model I/II 192 193
Overlapping-generations, Nonlinear Overlapping-Generations Model 268
PARAMETER 21 155 287
Parameter space 289—295
Parameter, birth rate 23
Parameter, carrying capacity 89 267 268
Parameter, death rate 23
Parameter, growth rate 23 88 89 190—193 266—268
Parameter, harvesting rate 24 153 163 191—193
Parameter, immigration rate 24 191—193
Parameter, infection rate 90 99 268 269 288
Parameter, interest rate 11—13 21 22
Parameter, migration rate 24 153 191—193
Parameter, recovery rate 90 268 269
Parameterized family 155
Parameterized family, fixed point 157
Parameterized family, fixed point, bifurcation 166
Parameterized family, fixed point, interval of existence/stability 157—159
Parameterized family, period-doubling cascade 171—173
Parameterized family, periodic point/cycle 159 160
Parameterized family, periodic point/cycle, bifurcation 169—171
Parameterized family, periodic point/cycle, interval of existence/stability 159—162
Parameterized system 287—295
Parameterized system, fixed point 287—293
Parameterized system, fixed point, bifurcation 290—295
Parameterized system, fixed point, interval/region of existence/stability 287—289
Parameterized system, period-doubling cascade 290 291
Parameterized system, period-multiplying cascade 293—295
Parasite 190
Partial derivative(s) 80—82 278
Partial derivative(s) and normal equations 81 82
Partial derivative(s), matrix of see "Matrix of Partial Derivatives"
| Partial product 42
Partial sum 31
Pendulum 10 27 48
Period three (and chaos) 184 296
Periodic oscillation 36 144 145
Periodic oscillation, infection rate 99
Periodic point/cycle see also "2-cycle" "m-cycle"
Periodic point/cycle, irrationally 249 292
Periodic point/cycle, nonlinear equation 141—152 177
Periodic point/cycle, nonlinear system 282—285 290—295
Periodic point/cycle, parameterized see "Parameterized family/system"
Periodic rotation 247—249 292
Phase plane graph 202
Polar coordinates 240 292
Pollution 27 38 48 58
Population (single species) 23 24 88 89 192 193 267 268 see
Population (single species), carrying capacity 89 267 268
Population (single species), density dependence 88 89 97 101 267 268
Population (single species), differential equation 134
Population (single species), equation, Linear Immigration/Migration/Harvesting Model 24
Population (single species), equation, Linear Population Model 23
Population (single species), equation, logistic 89 97 155 156
Population (single species), equation, Nonlinear Population Model I/II 89 97 98 155 156
Population (single species), equation, other 105 153 163
Population (single species), generation 23
Population (single species), growth rate 23 24 88 89 192 267 268
Population (single species), system, Linear Overlapping-Generations Model I/II 192 193
Population (single species), system, Nonlinear Overlapping-Generations Model 267 268
Predator/Prey 190 266
Predator/Prey, Linear Prey-Predator Model I/II 190 191
Predator/Prey, Nonlinear Prey-Predator Model 266
Price 24 25 91—93 193 194 269 270
Price, equation, Linear Price Model 25
Price, equation, Nonlinear Price Model I/II/III 92 93
Price, system, Linear Price-Demand Model 194
Price, system, Linear Price-Demand-Supply Model 193
Price, system, Nonlinear Price-Demand Model 270
probability distribution 4
PRODUCT 41—43
Profit 67
Qualitative behavior/dynamics 15—18 35 68 75 197 201
Radioactive decay 48
Random process see "Stochastic process"
Real axis 240
Real eigenvalue(s) see "Eigenvalue(s)"
Real part 238
Recovery 90 91 268 269
Recursion, recursive equation 9
Refining a model 4 5
Region of existence/stability 289—295
Regression 78—84 96 196
Regular Markov process 76
Relative error 137 138
Rent (annuity) 59—61
Renters 67
Repeated real eigenvalue(s) see "Eigenvalue(s)"
Repeller 106 see
Repeller, fractal 304—307
Repeller, source 205 271
Root see "Newton Root-Finding Method"
Rotation, linear system 247—249 251 256
Rotation, nonlinear system 279 292 293
Saddle 206 271 see
scalar 210
Scaling factor 156
Scrambled set, nonlinear equation 176—179
Scrambled set, nonlinear system 186 296 302—305
Second-order equation 192 267 268
Second-order equation, converting to 197 198
Second-order equation, solving 231—233 245—247
Sensitive dependence on initial conditions 179—182 296 304
Separating variables 134
Sierpinski triangle 308 311
Sierpinski triangle, using Mathematica 325 326
Sierpinski triangle, using Microsoft Excel 319 320
Simple interest 11—12
Simple Interest Formula 12
Simple Interest Model 12 21 28
Simple interest, linear growth of 46
Simultaneous equations 213—215
Single species see "Population"
Sink 205 271 see
Skyscraper 48
Smale, S. 183 297
Snap-back repeller 184 185 298 299
Software see "Computer Software"
Solution space graph 201
Solution space graph and linear dynamics 205 206
Solution space graph and nonlinear dynamics, chaos 273 295 296 299
Solution space graph and nonlinear dynamics, stability 271—273 292 293
Solution space graph, using Mathematica 323—325
Solution space graph, using Microsoft Excel 319
Solution vector 219
Solution vector, length 222—224 234 250 251
Solution vector, rotation 247—251 see
Solution, formula for exact 12
Solution, linear equation, autonomous, homogeneous 43
Solution, linear equation, autonomous, non-homogeneous 54
Solution, linear equation, non-autonomous, homogeneous 39 40
Solution, linear system, homogeneous, complex eigenvalues 245—247
Solution, linear system, homogeneous, one repeated eigenvalue 231—233
Solution, linear system, homogeneous, two real eigenvalues 223—225
Solution, nonlinear equation 101 109
Solution, nonlinear system 275
Solution, nontrivial 215 216
Solution, quantitative, vs. qualitative behavior 15—17 68 69
Solution, trivial 215
Source 205 271 see
Spread of technology 95
Stability see also "Parameterized family/system"
Stability, differential equation 137
Stability, linear equation, autonomous 72 74
Stability, linear system, homogeneous 205 206
Stability, linear system, homogeneous, complex eigenvalues 250
Stability, linear system, homogeneous, one repeated eigenvalue 234
Stability, linear system, homogeneous, two real eigenvalues 223
Stability, linear system, non-homogeneous 205 206 255 256
Stability, nonlinear equation, 2-cycle 143—146
Stability, nonlinear equation, fixed point 106 119 124
Stability, nonlinear equation, m-cycle 148—151
Stability, nonlinear system, fixed point 271 279
Stability, nonlinear system, invariant curve 273 295
Stability, nonlinear system, irrationally periodic cycle 292 293
Stability, nonlinear system, m-cycle 283
State (Markov process) 61
State diagram 61
State vector 257
Steady-state 76 257
Stochastic process 2 4 61—66 177 256—262 307—309
Strange Attractor 186 296 305
Substitution 55 74 155 255
SUM 30—32
Superstable 126 163
Supply 24 91 92 193 194 269
Supply, Linear Price-Demand-Supply Model 193
System of differential equations 275 301 302
System of simultaneous equations 213—216
System, discrete dynamical 6
System, dynamical 2
System, linear 189 197
Taylor series 52 239 244
Telescoping product 41
Temperature 27 48 140
Term (of compounding) 13
terminal velocity 19 140
Testing a model 4 5
Threshold 75 102 110 164 173 276
Time-series graph 35
Time-series graph and cobweb graph 111 113 114
Time-series graph and linear dynamics 74 199—202
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