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lecnotes5 - 5 5.1 Forced oscillators and limit cycles...

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5 Forced oscillators and limit cycles 5.1 General remarks How may we describe a forced oscillator? The linear equation β ¨ + ρβ ˙ + γ 2 β = 0 (3) is in general inadequate. Why? Linearity if β ( t ) is a solution, then so is �β ( t ), real. This is incompatible with bounded oscillations (i.e., β max < α ). We therefore introduce an equation with a nonlinearity; and an energy source that compensates viscous damping. 5.2 Van der Pol equation Consider a damping coefficient ρ ( β ) such that ρ ( β ) > 0 for β large | | ρ ( β ) < 0 for β small | | Express this in terms of β 2 : β 2 ρ ( β ) = ρ 0 β 2 1 0 where ρ 0 > 0 and β 0 is some reference amplitude. Now, obviously, ρ > 0 for β 2 > β 0 2 ρ < 0 for β 2 < β 0 2 28
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Substituting ρ into (3), we get d 2 β β 2 d β d t 2 + ρ 0 β 0 2 1 d t + γ 2 β = 0 This equation is known as the van der Pol equation . It was introduced in the 1920’s as a model of nonlinear electric circuits used in the first radios. In van der Pol’s (vaccum tube) circuits, high current = positive (ordinary) resistance; and low current = negative resistance. The basic behavior: large oscillations decay and small oscillations grow. We shall examine this system in some detail. First, we write it in non- dimensional form. We define new units of time and amplitude: unit of time = 1 unit of amplitude = β 0 . We transform t t β β β 0 where β and t are non-dimensional. Substituting above, we obtain d 2 β β β 0 2 d β γ 2 d t 2 β 0 + ρ 0 β 0 1 d t γβ 0 + γ 2 β β 0 = 0 Divide by γ 2 β 0 : d 2 β ρ 0 2 d β 2 + β 1 + β = 0 d t γ d t 29
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Now define the dimensionless control parameter ρ 0 π = > 0 .
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