Chaos in the BMN matrix model

Abstract

We study classical chaotic motions in the Berenstein-Maldacena-Nastase (BMN) matrix model. For this purpose, it is convenient to focus upon a reduced system composed of two-coupled anharmonic oscillators by supposing an ansatz. We examine three ansätze: 1) two pulsating fuzzy spheres, 2) a single Coulomb-type potential, and 3) integrable fuzzy spheres. For the first two cases, we show the existence of chaos by computing Poincaré sections and a Lyapunov spectrum. The third case leads to an integrable system. As a result, the BMN matrix model is not integrable in the sense of Liouville, though there may be some integrable subsectors.

Figures (4)

• Figure 1: Poincaré sections and a Lyapunov spectrum. Different trajectories are indicated by dots with different color, depending on initial conditions.
• Figure 2: Poincaré sections and a Lyapunov spectrum with the ansatz \ref{['ansatz1']}.
• Figure 3: Poincaré sections and a Lyapunov spectrum with the ansatz \ref{['ansatz2']}. In Fig. (a), two KAM tori are contiguous. In Fig. (b), local chaos appears around the boundary.
• Figure 4: Time evolution of a trajectory and a deviation vector set along it.