Views
5 years ago

time translations induce

time translations induce the action t → t + 2π ω ⇒ A → Aeiω 2π ω = A i.e. the amplitude is unchanged by such translations. Thus • with 1:1-forcing any polynomial in A and A ∗ is allowed by symmetries A ′ = a00 + a10A + a01A ∗ + a20A 2 + . . . Why did we not obtain terms like A ∗ or A 2 in our direct derivation of the amplitude equation? Scaling We also assumed weak forcing. To include this information it is useful to consider an extended dynamical system in which the forcing is considered a dynamical variable rather than an external force ¨y + ˆ β ˙y + ω 2 0 y + ˆαy3 − ˆ f = 0 (14) ¨ˆf + ω 2 ˆ f = 0 (15) where we assume ˆ β, y (instead of ˆα), and ˆ f are small. At this point it is not clear how these quantities scale with each other. Expand now Note: y = A(T)e iω0t + A(T) ∗ e −iω0t + . . . ˆf = F(T)e iωt + F(T) ∗ e −iωt • the expansion for ˆ f does not have any higher order terms since its evolution (15) is linear and is not coupled to y Time translations act on the amplitude A(T) and F(T) as Note: t → t + ∆t ⇒ A → Ae iω0∆t F → Fe iω∆t • since the forcing is now part of the dynamical system this extended dynamical system is invariant under any time translations. The expansion will lead to solvability conditions of the type A ′ = � aklmnA k A ∗l F m F ∗n F ′ = � klmn klmn fklmnA k A ∗l F m F ∗n For the 1:1-resonance, ω = ω0, A and F transform the same way under time translations. The selection rule is k − l + m − n = 1 What are the lowest-order terms? 34

• a nonlinear saturating term needs to be retained. Retain a term that is saturating also without forcing. The leading-order saturating term is |A| 2 A • the leading-order forcing term is F • to balance these two essential terms we have F ∼ A 3 ⇒ to leading order the only term containing the forcing is F To leading order we then get A ′ = a1000A + a0010F + a2100|A| 2 A in agreement with (12) with a1000 = µ, a0010 = ν, a2100 = γ. Consistent scaling requires µ = O(A 2 ), whereas ν, γ = O(1). Summary: • oscillation amplitude A and forcing amplitude F are each associated with their respective frequencies ω0 and ω • in terms of Fourier modes all terms in the resulting amplitude equation have to correspond to the same frequency, which for the A-equation is ω0. • no attention has to be paid to the equation for the forcing amplitude since the equation (15) is not coupled to the oscillation amplitude 1.3.3 3:1 Forcing Consider now ω = 3ω0 and use the expansion It induces the action Selection Rule for the equation for A y = A(T)e iω0t + A(T) ∗ e −iω0t + . . . ˆf = F(T)e 3iω0t + F(T) ∗ e −3iωt t → t + ∆t ⇒ A → Ae iϕ F → Fe 3iϕ with ϕ = ω0∆t A k A ∗l F m F ∗n → A k A ∗l F m F ∗n e iϕ(k−l+3(m−n)) k − l + 3 (m − n) = 1 ⇒ k − l = 1 − 3 (m − n) Identify the lowest-order terms in the forcing F: m − n = 1 ⇒ k − l = −2 k = 0 l = 2 ⇒ FA ∗2 35

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