# Diffusion in three-dimensional Liouvillian maps. Academic Article

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### abstract

• It is shown that chaotic trajectories in volume-preserving flows, r\ifmmode \dot{}\else \.{}\fi{}${=\mathrm{u}}_{\mathrm{\ensuremath{\epsilon}}}$(x,y,z,t), which are arbitrarily close to integrability, 0<\ensuremath{\epsilon}\ensuremath{\ll}1, can be either trapped or diffusive throughout the available space. A classification of these flows is proposed which both distinguishes and predicts the appropriate type of behavior. In the unbounded case, a new mechanism of diffusion is found which combines motion on the resonances with an adiabatic drift. This process is reminiscent of Arnold diffusion.

### publication date

• January 1, 1988