Analyzing the Behavior of a Non-linear Advertising Capital Model;
An Application of Bifurcation Theory, Lyapunov Exponents
and Correlation Dimension
Irma Luhta and Ilkka Virtanen
Abstract
In this study a time delayed feedback model describing the relations between
advertising and goodwill is introduced. The model has its origin in the
classical Nerlove-Arrow advertising capital model (Nerlove and Arrow 1962).
A continuous advertising function is used and the non-linear market effect of
advertising on the dynamics of goodwill is employed. In the case of a lagged
effect of advertising the dimension of the model exceeds unity. This means an
unstable situation in the limiting behavior, i.e. in the bifurcation point an
invariant closed curve is born as the attracting fixed point becomes repelling.
After this standard Hopf bifurcation the behavior of goodwill is periodic or
nearly periodic untill subsequent period doubling bifurcations may happen
leading the system to chaos with increasing parameter values. This period
doubling route to chaos is analyzed numerically by bifurcation diagrams and by
the techniques of the Lyapunov exponents and the correlation dimension. The
stability conditions for the fixed point of the model is determined
analytically.
(The Art and Science of Decision-Making, 138-151)