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Studying Yang-Mills field and gravitational field in class A Bianchispacetimes, we find that chaotic behavior appears in the late phase (theasymptotic future). In this phase, the Yang-Mills field behaves as that inMinkowski 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 (nearthe initial singularity), we numerically find that the behavior seems toapproach the Kasner solution. However, we show that the Kasner circle isunstable and the Kasner solution is not an attractor. From an analysis ofstability and numerical simulation, we find a Mixmaster-like behavior inBianchi I spacetime. Although this result may provide a counterexample to theBKL (Belinskii, Khalatnikov and Lifshitz) conjecture, we show that the BKLconjecture is still valid in Bianchi IX spacetime. We also analyze amultiplicative effect of two types of chaos, that is, chaos with the Yang-Millsfield and that in vacuum Bianchi IX spacetime. Two types of chaos seem tocoexist in the initial phase. However, the effect due to the Yang-Mills fieldis much smaller than that of the curvature term.Comment: 15 pages, 8 figure

Chaos of Yang-Mills field in class A Bianchi spacetimes