Preface ........................................................ ix
I Asymptotic methods solve algebraic and differential
equations ....................................................... 1
1 Perturbed algebraic equations solved iteratively ............. 5
2 Power series solve ordinary differential equations .......... 33
3 A normal form of oscillations illuminates their character ... 63
Part I Summary ................................................ 107
II Center manifolds underpin accurate modeling ............... 109
4 The center manifold emerges ................................ 113
5 Construct slow center manifolds iteratively ................ 169
Part II Summary ............................................... 217
III Macroscale spatial variations emerge from microscale
dynamics ...................................................... 219
6 Conservation underlies mathematical modeling of fluids ..... 223
7 Cross-stream mixing causes longitudinal dispersion along
pipes ...................................................... 243
8 Thin fluid films evolve slowly over space and time ......... 271
9 Resolve inertia in thicker faster fluid films .............. 295
Part III Summary .............................................. 315
IV Normal forms illuminate many modeling issues .............. 317
10 Normal-form transformations simplify evolution ............. 323
11 Separating fast and slow dynamics proves modeling .......... 341
12 Appropriate initial conditions empower accurate forecasts .. 377
13 Subcenter slow manifolds are useful but do not emerge ...... 405
Part IV Summary ............................................... 441
V High-fidelity discrete models use slow manifolds ........... 443
14 Introduce holistic discretization on just two elements ..... 447
15 Holistic discretization in one space dimension ............. 471
Part V Summary ............................................... 505
VI Hopf bifurcation: Oscillations within the center
manifold ...................................................... 507
16 Directly model oscillations in Cartesian-like variables .... 511
17 Model the modulation of oscillations ....................... 529
Part VI Summary ............................................... 567
VII Avoid memory in modeling nonautonomous systems,
including stochastic .......................................... 569
18 Averaging is often a good first modeling approximation ..... 575
19 Coordinate transforms separate slow from fast in
nonautonomous dynamics ........................................ 585
20 Introducing basic stochastic calculus ...................... 625
21 Strong and weak models of stochastic dynamics .............. 685
Part VII Summary .............................................. 721
Bibliography .................................................. 725
Index ......................................................... 743
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