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## Turning points of

Turning points of R∞(Ω) correspond to dΩ dR∞ Amplitude f 12 10 8 6 4 2 f=50 beta=0.5 dΩ1,2 dR∞ 0 -50 0 Frequency 50 = 0 = 3 4 αR∞ ± −2f2 � f2 4 − β2 Figure 7: Dependence of the amplitude of the phase-locked solution on the forcing frequency with damping (f = 50, β 2 = 0.5). 1.2.2 Duffing Oscillator as a System of Equations For simplicity demonstrate the approach with the Duffing oscillator again near the 1:1resonance ¨y + ω 2 0 + ǫ ˙y + ǫαy 3 = ǫf cos ωt ω = ω0 + ǫΩ Rewrite in terms of a system of first-order equations with Thus i.e. � u L v R 2 ∞ u = y v = ˙y 1 R 3 ∞ ˙u − v = 0 ˙v + ω 2 0 u = −ǫ � βv + αu 3 − f cosωt � � � ∂t −1 ≡ Expanding as usually � u v ω 2 0 ∂t � � � u v � U = A(T) V � � = 0 −ǫ {βv + αu 3 − f cosωt} � e iω0t + O(ǫ) + c.c. We need to determine an evoluation equation for the complex amplitude A(T). We have from the terms proportional to eiω0t � � � U iω0 L0 ≡ V � � � −1 U = 0 V ⇒ ω 2 0 iω0 28 � U V � � � � 1 = iω0

Since L is singular we expect that at O(ǫ) secular terms will arise, which will imply a solvability condition. Previously, this condition simply amounted to setting the terms proportional to cosω0t and to sin ω0t to 0. Dealing with a system of equations it seems at first glance as if there are too many solvability conditions for this single complex amplitude: there are two complex equations for each Fourier mode. We need to invoke explicitly the Fredholm Alternative Theorem, for which we need the left zero-eigenvector (U + (t), V + (t)) of L. For that we need to define a scalar product. For yi(t) ≡ (ui(t), vi(t)) choose 〈y1(t),y2(t)〉 ≡ � 2π ω 0 The left zero-eigenvector (U + (t), V + (t)) is defined by for any � u(t) v(t) � . Use integration by parts � 2π ω 0 0 0 = = = Thus � U + Note: 0 � � � + + ∗ ∂t −1 U (t), V (t) ω2 0 ∂t � � (u1(t) ∗ , v1(t) ∗ � u2(t) ) v2(t) � � u(t) v(t) U +∗ (∂ˆtu − v) + V +∗ � ω 2 0 u + ∂ˆtv � dˆt � dt � = 0 −u∂ˆtU +∗ − U +∗ V + V +∗ ω 2 0 U − V ∂ˆtV +∗ dˆt � �� −∂ˆt ω 2 0 V + � = −1 −∂ˆt � U + 0 V + 0 � � U + V + � ∗� t � u(ˆt) v(ˆt) � e ±iω0ˆt � 1 = ± i ω0 � dˆt � ±iω0ˆt e • L is not symmetric, therefore the left and the right 0-eigenvectors differ from each other. • There are two left 0-eigenvectors, which are complex conjugate of each other because the original equation is real. Including the slow time T the expanded Duffing equations becomes ∂ˆtu − v = −ǫ∂Tu ∂ˆtv + ω 2 0u = −ǫ � ∂Tv + βv + αu 3 − f cosωˆt � 29

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