312 Index coupled harmonic oscillators, 50 critical point, 20 Dahlquist root condition, 172 Dahlquist’s Equivalence Theorem, 298 defect, 221 dense output, 221 differential equation, 1, 5 discretization methods, 133 discretization parameter, 134 dynamical systems theory, 15 E. Noether’s Principle, 81 ecological models, 56 eigenvalue, 260 elementary differentials, 268 embedded, 165 embedded methods, 180 equation of exponential growth, 37 equation of simple harmonic motion, 37 equations of evolution, 1 equilibrium point, 20 Euler’s Method, 16, 17, 144 Euler-Lagrange Equations, 68, 70 exercises, 2 Existence Theorem, 8 explicit method, 134, 159 exponential growth, 15 extrapolation, 151 extremal, 72 fixed point, 20 fixed-point iteration, 160 flow, 15 flow-box coordinates, 252 forced harmonic oscillator, 55 formal accuracy, 142 Foucault pendulum, 103 geodesics, 86 global error, 18, 145, 219 Gram-Schmidt Procedure, 231 Green’s operator, 56 Gronwall’s Inequality, 22 Hamilton’s Equations, 86 harmonic oscillator, 16, 112, 181 heat equation, 187 Heun’s Method, 162, 265 homoclinic tangle, 31 Hooke’s Law, 38 horseshoe map, 31 hyperbolic, 46, 197 ignorable coordinate, 73 implicit method, 134, 159 implicit midpoint method, 169 improved Euler Method, 162 inhomogeneous linear ODE, 52 initial value problem, 1, 7 inner-product space, 228 invariance properties of flows, 33 Inverse Function Theorem, 240 isolated system, 91 IVP, 7 Jordan canonical form, 44 KdV equation, 198 Lagrange’s Equations, 86 law of evolution, 1 leapfrog method, 154 linear m-step method, 135 linearized Korteweg-deVries equation, 198 local constant of the motion, 253 local error, 146, 214 local truncation error, 213, 215, 284 local truncation errors, 146 logistic equation, 6, 59 Lorenz attractor, 28 Lyapounov exponent, 31 Maximal Solution Theorem, 13 method of small vibrations, 128 method of successive approximations, 10 metric space, 225 midpoint method, 151, 265 Milne device, 178 Milne’s corrector, 289 model parameter, 140 modified trapezoidal method, 162 multistage, 134 multistep, 134
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