A Family of Generalized Henon Maps Possessing Strange Attractors
Jim Howard
CIPS, CU Boulder
We describe a family of Generalized Henon Maps Tn:
x' = y - k + ε xn
y' = -bx
where n>1 is an integer, k is an arbitrary real number, ε = 1 or -1,
and b2 ≤ 1. Such mappings are interesting for study as tractable limiting
cases of more general polynomial automorphisms of the plane and reveal
important differences between maps of even and odd degree. Symmetries and
reversors are described in the symplectic case (b=1) and used to follow
period doubling sequences. For even n simple expressions are derived
for saddle-center and period doubling bifurcations of the fixed points
and period-two orbits. For odd n, with ε = +1 period doubling of fixed
points cannot occur, while for ε = -1 saddle-center bifurcations are
forbidden. Period doubling sequences are worked out in a few cases and
exhibit possible anomalies. In the dissipative case (b2 ≠ 1) strange
attractors are found for maps of even degree but apparently do not occur
for odd degree. The complex processes responsible for the birth and death
of strange attractors are explored.