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## Notes:

Notes: • for non-zero damping – the oscillation amplitude does not diverge at the resonance ω0 = ω – the maximal amplitude is attained at a frequency below the natural frequency ω0 • with decreasing damping the resonance peak of the response curve becomes sharper and taller • the maximum in the response disappears for ˆ β 2 > 2ω 2 0 1.2.1 1:1 Forcing Consider frequencies near 1:1 resonance: ω ≈ ω0 For simplicity take ω0 = 1 Use multiple scales Insert into O(1): ω = ω0 + ǫΩ = 1 + ǫΩ T = ǫt y(t) = y0(˜t, T) + ǫy1(˜t, T) + . . . ¨y + ǫβ ˙y + ω 2 0y + ǫαy 3 = ǫf cosωt ∂2y0 ∂˜t 2 + y0 = 0 y0(0, 0) = δ, ∂yy(0, 0) = 0 ∂t y0(˜t, T) = R(T) cos � ˜t + Φ(T) � We expect that there will be a locking regime in which y oscillates with the same frequency as the forcing ⇒ write Initial conditions Φ(T) = ΩT − φ(T) y0(˜t, T) = R(T) cosψ(˜t, T) with ψ(˜t, T) = ˜t + ΩT − φ(T) (6) Write y(˜t, T) in terms of ψ and T: then φ(0) = 0 R(0) = δ y(˜t, T) = y(ψ, T) ∂ ∂t = ∂˜t + ǫ∂T = (∂˜tψ + ǫ (Ω − ∂Tφ)) ∂ψ + ǫ∂T = (1 + ǫ (Ω − ∂Tφ)) ∂ψ + ǫ∂T � 2 + O(ǫ ) ∂ 2 ∂t 2 = ∂2 ψ + ǫ � 2 (Ω − ∂Tφ)∂ 2 ψ + 2∂2 ψT 22

O(ǫ): with Note: y0(ψ, T) = R(T) cosψ ∂ 2 ψ y1 + y1 = −2 (Ω − ∂Tφ)∂ 2 ψ y0 − 2∂ 2 ψT y0 − β∂ψy0 − αy 3 0 + f cos (ψ + φ) ≡ f1(ψ, T) • the linear operator ∂ 2 ψ Consider f1 y1(0, 0) = 0 ∂ty1 = −∂Ty0 + 1 is singular with zero-modes cosψ and sin ψ. f1(ψ, T) = 2 (Ω − ∂Tφ) R cosψ + 2R ′ sin ψ + βR sin ψ − αR 3 cos 3 ψ + f cos (ψ + φ) where R ′ = ∂TR Since ∂ 2 ψ i.e.. = 2 (Ω − ∂Tφ) R cosψ + 2R ′ sin ψ + βR sin ψ − αR 33 4 +f cos ψ cosφ − f sin ψ sin φ � = cosψ 2 (Ω − ∂Tφ)R + f cosφ − 3 4 αR3 � cosψ − αR31 cos 3ψ 4 + sin ψ {2R ′ + βR − f sin φ} − 1 4 αR3 cos 3 ψ + 1 is singular we get again solvability conditions to avoid secular terms 2 (Ω − φ ′ ) R + f cosφ − 3 4 αR3 = 0 2R ′ + βR − f sin φ = 0 Rφ ′ − with initial conditions R(0) = δ, φ(0) = 0. Notes: R ′ + 1 1 βR = f sin φ (7) � 2 2 Ω − 3 8 αR2 � R = 1 f cosφ (8) 2 • Eqs.(7,8) for the amplitude and phase of the forced oscillator were derived under the assumptions – weak damping – weak forcing – small coefficient of the nonlinear term, which is equivalent to small amplitudes (eliminate the ǫ in the nonlinear term by rescaling the ampliutde y → √ ǫy 23

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