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Interpolating vector fields for near identity maps and averaging

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Figure

Figure 1: Dynamics of a symplectic map on a four-dimensional cylinder TNote that all points are in a small neighbourhood of a two-dimensional torus
Figure 2: Iterates of the standard map Mϵ (left) and of the time-ϵ map ΦϵXn associated to theinterpolating vector field (with n = 5) (right)
Figure 3: ϵ = 0.1. Level plots of log10 |ΦϵXn(x0) − Mϵ(x0)| for x0 ∈ [−π, π] × [−2π, 2π], andfour different values of n.
Figure 4: ϵΦ = 0.5. Left: Iterates of the standard map Mϵ. Right: Iterates of the time-ϵ mapϵXn (with n = 10) associated to the interpolating vector field.
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