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## 1 Forced Oscillators

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

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