  The dependence of the Lyapunov exponent on the closeness parameter,
$\epsilon$, in tangent bifurcation systems is investigated. We study and
illustrate two averaging procedures for defining Lyapunov exponents in such
systems. First, we develop theoretical expressions for an isolated tangency
channel in which the Lyapunov exponent is defined on single channel passes.
Numerical simulations were done to compare theory to measurement across a range
of $\epsilon$ values. Next, as an illustration of defining the Lyapunov
exponent on many channel passes, a simulation of the intermittent transition in
the logistic map is described. The modified theory for the channels is
explained and a simple model for the gate entrance rates is constructed. An
important correction due to the discrete nature of the iterative flow is
identified and incorporated in an improved model. Realistic fits to the data
were made for the Lyapunov exponents from the logistic gate and from the full
simulation. A number of additional corrections which could improve the
treatment of the gates are identified and briefly discussed.
