J.A. Sanders F. Verhulst J. Murdock Averaging Methods in Nonlinear Dynamical Systems Second Edition 4y Springer List of Figures 0.1 The map of the book xxiii 2.1 Phase orbits of the Van der Pol equation x + x = e(l — x2)x ... 23 2.2 Solution x(t) of x + x = ^x2 cos(i), x(0) = 0, x(0) = 1 26 2.3 Exact and approximate solutions of x + x = ex 27 2.4 'Crude averaging' of x + 4ecos2(t)i; + x = 0 28 2.5 Phase plane for x + 4ecos2(t)x + x = 0 29 2.6 Phase plane of the equation x + x ~ ex2 = e2(l — x2)x 42 4.1 F(t) = J2™=i sin(t/2") as a function of time 85 4.2 The quantity 5x/(eM) as a function of £ 86 5.1 Phase plane for the system without interaction of the species. .. 94 5.2 Phase plane for the system with interaction of the species 95 5.3 Response curves for the harmonically forced Duffing equation. 98 5.4 Solution x starts in x(0) and attracts towards 0 101 5.5 Linear attraction for the equation x + x + ex3 + 3e2x = 0 110 6.1 Connection diagram for two coupled Duffing equations 116 6.2 Separation of nearby solutions by a hyperbolic rest point 121 6.3 A box neighborhood 124 6.4 A connecting orbit 130 6.5 A connecting orbit 131 6.6 A dumbbell neighborhood 132 6.7 A saddle connection in the plane 140 7.1 Oscillator attached to a flywheel 154 8.1 Phase flow of 4> + e/3{0) sin(0) = ea(0) 173 8.2 Solutions x = x2(t) based on equation (8.5.2) 179 10.1 One normal mode passes through the center of the second one. 219 List of Figures 10.2 The normal modes are linked 219 10.3 Poincare-map in the linear case 219 10.4 Bifurcation diagram for the 1 : 2-resonance 226 10.5 Poincare section for the exact 1 : 2-resonance 226 10.6 Projections for the resonances 4:1,4:3 and 9:2 235 10.7 Poincare map for the 1 : 6-resonance of the elastic pendulum. 236 10.8 Action simplex 242 10.9 Action simplex for the the 1:2: 1-resonance 251 10.10Action simplex for the discrete symmetric 1:2: 1-resonance. 252 lO.HAction simplex for the 1:2: 2-resonance normalized to H1 253 10.12Action simplex for the 1:2: 2-resonance normalized to H2 254 10.13Action simplex for the 1:2: 3-resonance 256 10.14The invariant manifold embedded in the energy manifold 256 10.15Action simplex for the 1:2: 4-resonance for A > 0 258 10.16Action simplices for the 1:2: 5-resonance 261.