  Studying Yang-Mills field and gravitational field in class A Bianchi
spacetimes, we find that chaotic behavior appears in the late phase (the
asymptotic future). In this phase, the Yang-Mills field behaves as that in
Minkowski spacetime, in which we can understand it by a potential picture,
except for the types VIII and IX. At the same time, in the initial phase (near
the initial singularity), we numerically find that the behavior seems to
approach the Kasner solution. However, we show that the Kasner circle is
unstable and the Kasner solution is not an attractor. From an analysis of
stability and numerical simulation, we find a Mixmaster-like behavior in
Bianchi I spacetime. Although this result may provide a counterexample to the
BKL (Belinskii, Khalatnikov and Lifshitz) conjecture, we show that the BKL
conjecture is still valid in Bianchi IX spacetime. We also analyze a
multiplicative effect of two types of chaos, that is, chaos with the Yang-Mills
field and that in vacuum Bianchi IX spacetime. Two types of chaos seem to
coexist in the initial phase. However, the effect due to the Yang-Mills field
is much smaller than that of the curvature term.
