  The Takens-Bogdanov bifurcation is a codimension two bifurcation that
provides a key to the presence of complex dynamics in many systems of physical
interest. When the system is translation-invariant in one spatial dimension
with no left-right preference the imposition of periodic boundary conditions
leads to the Takens-Bogdanov bifurcation with O(2) symmetry. This bifurcation,
analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322,
243 (1987), describes the interaction between steady states and traveling and
standing waves in the nonlinear regime and predicts the presence of modulated
traveling waves as well. The analysis reveals the presence of several global
bifurcations near which the averaging method (used in the original analysis)
fails. We show here, using a combination of numerical continuation and the
construction of appropriate return maps, that near the global bifurcation that
terminates the branch of modulated traveling waves, the normal form for the
Takens-Bogdanov bifurcation admits cascades of period-doubling bifurcations as
well as chaotic dynamics of Shil'nikov type. Thus chaos is present arbitrarily
close to the codimension two point.
