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## • if y(t+∆t) is a

• if y(t+∆t) is a solution of the original equation then B(T) is a solution of the amplitude equation and vice versa Moreover, there is an explicit relation between A(T) and B(T) ⎧ ⎨ y(t + ∆t) = ⎩ B(T)e iωt A(T)e iω(t+∆T) = A(T)e iω∆t e iωt Therefore one obtains the following commutative diagram ⇒ B(T) = A(T)e iω∆t y(t) solves the original equation ⇔ A(T) solves the amplitude equation ⇕ ⇕ y(t + ∆t) solves the original equation ⇔ A(T)e iω∆t solves the amplitude equation One says: • Translations ∆t in time induce an action on the amplitude: t → t + ∆t ⇒ A(T) → A(T)e iω∆t In this case the action corresponds to a phase shift by an arbitrary amount ∆φ = ω∆t. • The amplitude equation must be equivariant under that action: all terms of the amplitude equation must transform the same way under that operation Selection rule Since the amplitude equation arises in an expansion in terms of the complex amplitude it has the general form d dT A = F(A, A∗ ) = � αmnA m A ∗n (13) If A(T) is a solution to (13) so must be A(T)e i∆φ for arbitrary ∆φ. Thus Inserting dA/dT from (13) we get m,n d dT Aei∆φ = F(Ae i∆φ , A ∗ e −i∆φ ) = � � m,n Equating like powers of A and A ∗ m,n αmnA m A ∗n e i(m−n)∆φ αmnA m A ∗n e i∆φ = � αmnA m A ∗n e i(m−n)∆φ m,n αmn = αmne i∆φ(m−n−1) 32 for all ∆φ

Thus, m = n + 1 or αmn = 0 Alternatively, one can express this result also as: The action induced by the time-translation symmetry transforms the terms in the expansion as A n A ∗m → A n A ∗m e iϕ(n−m) • Equivariance of the amplitude equation under this action requires that for all terms in the amplitude equation the difference n − m must be the same: thus n − m = k for some k ∈ N. • Since the amplitude equation has a term d A in it one has k = 1. dT Thus, the only terms allowed are of the form Scaling |A| 2l A 0 ≤ l ∈ N In the weakly nonlinear regime A(T) ≪ 1. To leading order in the amplitudes one therefore gets A ′ = µA − γ|A| 2 A implying the scaling Notes: d dT ∼ µ ∼ |A|2 • the symmetry condition allows us to write down the form of the resulting amplitude equation without performing the nonlinear expansion in detail. • of course, to obtain the values of the coefficients one still has to do the algebra 1.3.2 1:1 Forcing We considered forcing near the 1:1-resonance: ω = ω0 + ǫΩ. The small detuning Ω can be captured through the dependence of the complex amplitude A on the slow time T. Therefore consider a system exactly at the 1:1-resonance. With 1:1-resonance the system is not invariant for arbitrary time shifts, but for shifts With the expansion t → t + 2π ω y(t, T) = A(T)e iωt + A(T) ∗ e −iωt + O(ǫ) 33

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