The kicked rotor (KR) is perhaps the simplest physical model to illuminate the transition from regular to chaotic motion in classical mechanics. It is also widely applied as a model of light-matter interactions. In the conventional treatment, the infinitesimal width of each kick allows an immediate integration of the equations of motion. This in turn allows a full description of the dynamics via a discrete mapping, the standard map, if one looks at the dynamics only stroboscopically. In this work, we present a generalized kicked rotor (GKR) model with finite-width periodic forcing. We derive a family of maps that depend on the parameter shift $\tau$, valid for small to intermediate kick strength $K$ and width $\mathrm{\Delta }$, that allows one to capture the motion in both the driven and kicked regimes. The fixed points and symmetry of the mapping are shown analytically to depend on the value of the shift parameter, but the structure of regular vs chaotic motion is unaffected. We then formulate a series of higher-order corrections (HGKR) that extends the mapping to large $K$ and $\mathrm{\Delta }$. The HGKR mapping is in excellent agreement with full numerical results. We observe momentum localization for sufficiently large $K$ and $\mathrm{\Delta }$, an effect whose physical origin can be understood from the analytic structure of the mapping equations.

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• Chaos
• Classical mechanics
• Hamiltonian systems
• Multiple time scale dynamics