- Text
- Solution,
- Integral,
- Equation,
- Forcing,
- Amplitude,
- Term,
- Integration,
- Solutions,
- Consider,
- Maximum,
- Lecture,
- Note,
- Sketches,
- Hermann,
- Riecke,
- Home,
- People.esam.northwestern.edu

Lecture Note Sketches Hermann Riecke - ESAM Home Page

1 Forced Oscillators Resonances in forced oscillations are important in many areas • dangerous resonances: stability of structures • useful resonances: amplification of signals, e.g. – electronic circuits – double amplification in the ear: two staged oscillators Understanding of externally forced oscillators provides also insight into resonantly coupled oscillators, e.g., • laser arrays • heart cells • neurons – synchrony can carry additional informaiton – too much synchrony may amount to seizures Types of forcing • parametric forcing: a parameter of the system is modified in time Simple example: Pumping on a swing ml¨ θ + mg sin θ = 0 ⇒ ü + g sin θ = 0 l(t) with l being the distance of the center of mass to the pivot. By shifting his/her center of mass the person changes the effective length l of the pendulum • non-parametric forcing: the forcing introduces an additional term in the equation Simple example: Pushing on a swing ü + g sin θ = F(t) l Useful asymptotic expansions can be obtained for weak forcing near and away from resonances. The expansions and results depend on the type of resonance and the type of forcing, which often reflect the symmetries of the overall system. 1.1 Parametrically Forced Oscillators Consider second-order differential equations with periodically varying coefficients: parametrically forced oscillators. 6

1.1.1 The Mathieu Equation Consider linear ode describing a forced harmonic oscillator ü + (δ + ǫ cos 2t)u = 0 which would model the swing for small angle θ. Instead of varying the forcing frequency with fixed natural frequency of the unforced oscillator we keep here the forcing frequency fixed, ω = 2, and vary the natural frequency √ δ. Expect: the resonant forcing drives the amplitude of the oscillator to large values. Goal: find the curves δ(ǫ), 0 ≤ ǫ ≪ 1, for which the Mathieu equation has a periodic solution (with period 2π), i.e. twice the period of the forcing. Expand: Collect O(1) : O(ǫ): O(ǫ 2 ): For the solution to have period 2π we need **Note**: δ = δ0 + ǫδ1 + ǫ 2 δ2 + . . . u = u0 + ǫu1 + ǫ 2 u2 + . . . ü0 + δ0u0 = 0 ü1 + δ0u1 = −u0 (δ1 + cos 2t) ü2 + δ0u2 = −δ2u0 − u1 (δ1 + cos 2t) δ0 = n 2 n = 0, 1, 2, . . . • for n = 0 the unforced solution is constant, its value is arbitrary. • for n ≥ 2 the minimal period of the unforced solution is 2π/n. i) Case n = 0, i.e. δ0 = 0 O(1): δ = ǫδ1 + ǫ 2 δ2 + . . . u0 = c0 ≡ 1 we can choose the constant amplitude arbitrarily. 7

- Page 1 and 2: Lecture Note Sketches Perturbation
- Page 3 and 4: 4 Fronts and Their Interaction 87 4
- Page 5: References [1] P. Coullet, C. Elphi
- Page 9 and 10: • case δ1 = + 1 2 � � 1 ü2
- Page 11 and 12: • we can assume from the start th
- Page 13 and 14: 1.1.2 Floquet Theory In the discuss
- Page 15 and 16: • it is also convenient to introd
- Page 17 and 18: Goal: determine α(δ, ǫ). For a r
- Page 19 and 20: Forcing Strength ε 1 0.5 0 -0.5 n=
- Page 21 and 22: 10 5 −10 0 0 −5 t 50 100 150 20
- Page 23 and 24: O(ǫ): with Note: y0(ψ, T) = R(T)
- Page 25 and 26: i.e. Now the nonlinear problem: Fix
- Page 27 and 28: • α < 0: Notes: - in-phase solut
- Page 29 and 30: Since L is singular we expect that
- Page 31 and 32: • using the other left 0-eigenvec
- Page 33 and 34: Thus, m = n + 1 or αmn = 0 Alterna
- Page 35 and 36: • a nonlinear saturating term nee
- Page 37 and 38: • Non-resonant forcing does not i
- Page 39 and 40: with O(1) forcing (cf. in Sec.1.2.1
- Page 41 and 42: Thus, if ∆ > 0 both solutions R2
- Page 43 and 44: Here we can get an integral express
- Page 45 and 46: which diverge at the upper limit fo
- Page 47 and 48: as in the case a > 0. But: now the
- Page 49 and 50: • if the integration by parts lea
- Page 51 and 52: First consider maximum at the lower
- Page 53 and 54: Is this series for I(x; ǫ) asympto
- Page 55 and 56: • since we kept only the first no
- Page 57 and 58:
It looks like a case for Watson’s

- Page 59 and 60:
Extend integration to (−∞, +∞

- Page 61 and 62:
Bound the integral term ���

- Page 63 and 64:
π i to ensure decay of the exponen

- Page 65 and 66:
using � ∞ 0 e−u ln u du = γ

- Page 67 and 68:
using the series expansion from the

- Page 69 and 70:
Example: Behavior near the saddle p

- Page 71 and 72:
Limiting behavior of the contours v

- Page 73 and 74:
Write xρ = (X + iY ) ρ ≡ Φ + i

- Page 75 and 76:
3 Nonlinear Schrödinger Equation C

- Page 77 and 78:
3.1 Some Properties of the NLS Cons

- Page 79 and 80:
Notes: Thus: ˙x = 1 2 m ˙x2 + V (

- Page 81 and 82:
• the boost velocity c or the bac

- Page 83 and 84:
and Note: ∂φ0 � i 1 2 λ20 ψ0

- Page 85 and 86:
• with increasing amplitude the p

- Page 87 and 88:
4 Fronts and Their Interaction Cons

- Page 89 and 90:
a) • this equation can be read as

- Page 91 and 92:
• the coefficient of ψ is chosen

- Page 93 and 94:
Using that ψL,R satisfy the O(ǫ 0

- Page 95 and 96:
Analogously for x > xm: 0 = ǫ∂Tx

- Page 97 and 98:
- L = 0 corresponds to a pure gas p