Chaos in two-dimensional φ³ theory with oscillator modes

A classical scalar field in a box with a periodic boundary is approximately described as a superposition of the spatially homogeneous mode and the lowest oscillator mode. This approximation reduces the scalar field theory to a four-dimensional nonlinear system with three constants, the total energy E, the "angular momentum" ℓ, and the wave number k of the oscillator mode. In (k, ℓ)-space, the parameter combinations which yield chaos are those for which (i) 0.3 ≲ k ≲ 0.9 and 0 ≤ ℓ ≲ 0.1, and those in (ii) the arm-shaped region that ranges from (k, ℓ) ≃ (0.9, 0.0) to (0.7, 0.4). Stochasticity is most conspicuous when E takes its maximum value. As E decreases, the stochasticity is rapidly lost, and when E becomes below roughly 60% of the maximum value, the system behaves deterministically, for any choice of k, ℓ and the initial conditions. Stochasticity is lost also in the large ℓ, large k and small k limits. There is no (k, ℓ, E) combination that yields chaos for (almost) all initial conditions. In the present paper, these results are confirmed numerically. Some of the types of behavior can be explained in terms of the curvature of the potential surface, weak coupling areas, and the shape of the kinetic region.