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236 5. Some worked examples arising from physical problems

E5.16 Josephson junction
The Josephson junction between two superconductors, which are separated by a thin
insulator, can produce an AC current when a DC voltage is applied across the junc-
tion (this by virtue of the tunnelling effect). An equation that models an aspect of this
phenomenon (Sanders, 1983) is

where a and b are given constants, and We will construct the
asymptotic solution, using the method of multiple scales, for Note that,
in the absence of the term then is a solution of the complete problem.
We anticipate that the presence of will force a non-zero solution which, if it
remains bounded, should be for all time (t); thus we write Further, we
introduce

and so satisfies the equation

We assume a bounded, periodic solution can be written as

and so we may expand

Thus we obtain the set of equations

and so on. These equations follow the pattern for a nearly linear oscillator; see §4.1.
The general solution of (5.98a) is

with initial conditions