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

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

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