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## F ∗ : m − n = −1

F ∗ : m − n = −1 ⇒ k − l = 4 k = 4 l = 0 ⇒ F ∗ A 4 Balance saturation and forcing A 3 ∼ FA 2 ⇒ F ∼ A With this scaling terms of order O(F 2 A) would be of the same order, i.e. m + n = 2, m = 2 n = 0 ⇒ k − l = −5 ⇒ F 2 A ∗5 m = 1 n = 1 ⇒ k − l = 1 ⇒ |F | 2 A m = 0 n = 2 ⇒ k − l = 7 ⇒ F ∗2 A 7 To leading order symmetry and scaling results in an equation of the form Notes: A ′ = � µ + β|F | 2� A − γ|A| 2 A + δFA ∗2 • Through the term β|F | 2 the forcing modifies the linear coefficient of the equation – depending on the sign of βr the forcing can enhance or reduce the damping – through βi|F | 2 the frequency of small-amplitude oscillations are modified by the forcing • The forcing will lead to qualitatively new phenomena only through the term involving A ∗2 because only it breaks the symmetry A → Ae iϕ for ϕ �= 2π 3 . • For consistent scaling we need again µ = O(A 2 ) 1.3.4 Non-resonant Forcing Consider ω = αω0 with α not an integer. Selection Rule lowest-order term k − l + α (m − n) = 1 ⇒ m = n k = l + 1 |F | 2 A ⇒ the forcing appears only through |F | 2 , i.e. the phase of the forcing does not play a role and there is no resonance between the oscillator and the forcing Balance saturation and forcing resulting in the amplitude equation Notes: A 3 ∼ F 2 A ⇒ F ∼ A A ′ = � µ + |F | 2� A − γ|A| 2 A 36 (16)

• Non-resonant forcing does not introduce new terms in the equation of the unforced oscillator (at any order), it only modifies its coefficients. None of the terms are phasesensitive. – in principle, all coefficients depend on |F | 2 – the strongest effect of the forcing is on the bifurcation parameter µ because it is small – only for the linear term is the shift of the coefficient of the same order as the coefficient itself and therefore relevant at leading order • In the resonant cases - as those discussed before - all the coefficients depend also on |F | 2 , but again most effects are of higher order. More relevant are, however, the new terms that are phase-sensitive. • Resonant forcing with higher resonances (m : 1 with m ≥ 4) do not lead to additional terms in the lowest order amplitude equation (see homework) – but they introduce new higher-order terms that are phase-sensitive – to capture aspects of their impact on the system in a leading-order amplitude equation one may have to consider singular limits like |γ| ≪ 1, i.e. consider higher singular points. 1.4 Forced Duffing Oscillator II Consider now explicitly the Duffing oscillator with forcing near 3ω0 with initial conditions y(0) = δ, ˙y(0) = 0. ¨y + ǫβ ˙y + ω 2 0y + ǫαy 3 = f cosωt To get an overview of possible resonance consider first general ω and O(1)-forcing O(1): O(ǫ): Insert y0 into f1 using y0(t) = δ cosω0t + ¨y0 + ω 2 0 y0 = f cosωt f ω2 (cosωt − cosω0t) 0 − ω2 ¨y1 + ω 2 0y1 = −β ˙y0 − y 3 0 ≡ f1 K ≡ f ω 2 0 − ω 2 f1 = β {ω0 sin ω0t + K [ω sin ωt − ω0 sin ω0t]} − {K cosωt + (1 − K) cosω0t} 3 = βω0 (1 − K)sin ω0t + βω sin ωt − − � K 3 cos 3 ωt + 3K 2 (1 − K) cos 2 ωt cosω0t + 3K (1 − K) 2 cos ωt cos 2 ω0t + (1 − K) 3 cos 3 ω0t � 37

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