Views
5 years ago

## At O(ǫ) we get ∂ˆtu1

At O(ǫ) we get ∂ˆtu1 − v1 = −∂Tu0 ≡ I1 = −∂TAe iω0 ˆt + c.c ∂ˆtv1 + ω 2 0 u1 = − � ∂Tv0 + βv0 + αu 3 0 − f cosωˆt � ≡ I2 iω0ˆt = e The solvability condition is given by � −iω0∂TA − βiω0A − 3|A| 2 Aα + 1 2 feiΩT � 2π ω 0 0 � 1, − i ω0 � � −iω0ˆt I1 e I2 Only the terms ∝ e iω0 ˆt in Ii contribute to this condition Thus � dˆt = 0 0 = −2∂TA − βA + 3i α |A| ω0 2 A − i fe 2ω0 iΩT ∂TA = − 1 3α βA + i |A| 2 2ω0 2 A − i fe 4ω0 iΩT We can remove the explicit T-dependence in the equation by writing yielding � ∂T A = − 1 � β − iΩ 2 A = Ae iΩT � + [. . .] e 3iω0 ˆt + c.c. A + i 3α |A| 2ω0 2 A − i f (10) 4ω0 To compare with the previous result for the Duffing oscillator introduce amplitude and phase A(T) = 1 2 R(T)e−iφ(T) which results in ∂TR + (iΩ − i∂Tφ)R = − 1 3iα βR + R 2 8ω0 3 − and yields after splitting into real and imaginary parts in agreement with our previous result (7,8). Notes: ∂TR = − 1 f βR + sin φ 2 2ω0 ΩR − ∂TφR = 3 α R 8 ω0 3 − f cosφ 2ω0 30 if e 2ω0 iφ � �� � f sin φ− 2ω0 if cos φ 2ω0

• using the other left 0-eigenvector would result in an equivalent solvability condition leading to the complex conjugate of (10). The complex amplitude equation (10) suggests that in general the complex amplitude satisfies an equation of the form A ′ = µA − γ|A| 2 A + νf (11) where µ ≡ µr + iµi, γ ≡ γr + iγi, ν ≡ νr + iνi are complex coefficients. Notes: • Comparing (33) with (10) shows that the Duffing oscillator does not lead to the most general amplitude equation for a forced oscillator: – for the Duffing oscillator one has γr = 0: no nonlinear dissipation of the oscillation amplitude, only linear damping ⇒ the saturation of the oscillation amplitude occurs through a change of the natural frequency of the oscillator with increasing amplitude, which renders the forcing less effective with increasing oscillation amplitude • The fact that νr = 0 in (10) is of no significance: for arbitrary ν ≡ ¯νe iδ with ˆν ∈ R replacing A → Âeiδ leads to 1.3 Symmetries 1.3.1 No Forcing A ′ = µA − γ|A| 2 A + ˆνf (12) In the absence of forcing (4) is invariant under arbitrary time translations t → t + ∆t i.e. if y(t) is a solution to (4) so is y(t + ∆t). This invariance must be reflected in the resulting amplitude equation (12). However, A does not depend on t. How is it affected by translations in the fast time t? Consider a solution y(t) and the time-shifted solution y(t + ∆t) and their expansions in terms of complex amplitudes A and B, respectively, The expansion implies: y(t) = A(T)e iωt + c.c. + h.o.t. y(t + ∆t) = B(T)e iωt + c.c. + h.o.t. • if y(t) is a solution of the original equation then A(T) is a solution of the amplitude equation and vice versa 31

Page 2 Lecture Notes in Computer Science 4475 Commenced ...