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Unformatted text preview: 5 Forced oscillators and limit cycles 5.1 General remarks How may we describe a forced oscillator? The linear equation + + 2 = (3) is in general inadequate. Why? Linearity if ( t ) is a solution, then so is ( t ), real. This is incompatible with bounded oscillations (i.e., max < ). 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 coecient ( ) such that ( ) > for large   ( ) < for small   Express this in terms of 2 : 2 ( ) = 2 1 where > and is some reference amplitude. Now, obviously, > for 2 > 2 < for 2 < 2 28 Substituting into (3), we get d 2 2 d d t 2 + 2 1 d t + 2 = 0 This equation is known as the van der Pol equation . It was introduced in the 1920s as a model of nonlinear electric circuits used in the first radios. In van der Pols (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 = . We transform t t / where and t are nondimensional. Substituting above, we obtain d 2 2 d 2 d t 2 + 1 d t + 2 = 0 Divide by 2 : d 2 2 d 2 + 1 + = 0 d t d t 29 Now define the dimensionless...
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 Fall '06
 DanielRothman

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