and} 
\qquad
r_{max} = \sqrt{\frac{2}{{\tilde\omega_c}}(b+a)}.$$

Note that the equations of motion are nonlinear here, even in the walls’ absence. The effect of collision with the wall (or centrifugal barrier) is simply to reverse the direction of motion: $$\label{eq:x}
\left( \begin{array}{c}
r(\tau_c^+) \\
p_r(\tau_c^+)
\end{array} \right) = 
\left( \begin{array}{cc}
1 & 0 \\
0 & -1
\end{array} \right)
\left( \begin{array}{c}
r(\tau_c^-) \\
p_r(\tau_c^-)
\end{array} \right),$$ where $\tau_c$ is the time of collision with the wall (or barrier). Finally, the effects of the kicks at $\tau=n$ are obtained by integrating the equations of motion over an infinitesimal duration around $n$: $$\label{eq:y}
\left( \begin{array}{c}
r(n^+) \\
p_r(n^+)
\end{array} \right) = 
\left( \begin{array}{cc}
1 & 0 \\
\eta r & 1 
\end{array} \right)
\left( \begin{array}{c}
r(n^-) \\
p_r(n^-)
\end{array} \right).$$ If we denote the mapping due to the ‘free’ evolution of the particle under the influence of $H_{dc}$ by $M_0$ (equations (\[eq:u\])), that due to the walls by equation by $M_{wall}$ (equations (\[eq:x\]), and the mapping due to the kick by $M_{kick}$ (equations (\[eq:y\])), then the complete one-period map is typically given by the product of several $M$’s for a given energy, [*i.e.*]{}, $$M_{T} = \left( M_0\cdot M_{wall}\right)^N \cdot M_{kick}.$$ In general, the map 