Suppose that all dimensionless parameters of the medium given by (44)
are of the order of 1, and so 
is the only small parameter of the
problem. Note that, in particular, 
requires that 
.
One can see that due to smallness of when 
,
boundary conditions (47) can be satisfied only with small
and large 
and with the tip 
lying at the
I-branch. Then
With this precision, curvature distribution 
is the same as that
of the involute of a circle. If substituted into equation (41),
these asympotics make terms yy'' and 
much less than others,
and can be obtained as a solution of this equation with these terms
left out. Note that neglecting these very terms corresponds to
independence of the normal front velocity on front curvature, which is
natural for very small curvature, and consistent with the spiral being
an involute of a circle.
If we look for the solution in the form
boundary conditions become
and so
Our solution has a physical sense only at positive and 
. Hence,
the following inequalities should be fulfilled:
Inequality (57) means
i.e. spiral wave solution can be found in this way only if the original
half-wave solution of the unperturbed system is growing but not
shrinking. And (58) shows that the spiral wave solutions
are found only to one side of the manifold 
of the
parametric space, which in our notations is determined by equation
.
Finally, the dimensional parameters of the spiral wave are
