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where For to be periodic in T, i.e. in we require

which may be integrated to give

where Hence, for any initial amplitude as
the solution exhibits a limit cycle (ultimately an oscillation in T with amplitude 2). This
is the raison d’être of the triode circuit.
If we proceed with the analysis (the details of which are left as an exercise) we find,
first , that

where and are additional arbitrary functions. (The constant is fixed by the
initial data At the next order, we deduce that the amplitude
remains bounded as only if

Another problem with an electrical background, but with a rather different asymp-
totic structure, will now be described.
E5.6 A diode oscillator with a current pump
In this problem, which contains two small parameters (one of which is used to simplify
some of the intermediate results, as expedient), we seek the initial condition which
leads to a periodic solution. The circuit (figure 12) is represented by the equations

and then Kirchhoff’s law gives

which leads to the non-dimensional equation

Typical values of the parametersare: (This equation was brought
to the author’s attention by a colleague, Dr Armstrong.)