a vertical segment that contains the target point. Fortunately the evolution of a vertical segment can be written concretely as a (1+2)-dimensional piece-wise polynomial skew-product map system.

Thus here we introduce an auxiliary “segment map” system, given as $\tilde M\ |\ \tilde \Omega\rightarrow\tilde \Omega,\ \tilde
\Omega=T^1([0,1))\times({\mathbf R}\times{\mathbf R}_+), {\mathbf
    R}_+=\{x\in{\mathbf R}|x\geq 0\}$, $\tilde
  M(\theta_n,z_n,w_{n})=(\theta_{n+1},z_{n+1},w_{n+1})$ $$\begin{aligned}
\theta_{n+1} &=& (\theta_n + \omega) \ \mbox{mod}\ 1,\cr
z_{n+1} &=& \left\{
\begin{array}{ll}
a - z_n^2 -w_n^2 + \epsilon \cos(2\pi\theta_n),& (|z_n|\geq w_n)\cr
a - (|z_n|+w_n)^2/2 + \epsilon \cos(2\pi\theta_n),& (|z_n|<w_n)
\end{array}
\right.\cr
w_{n+1} &=&  \left\{
\begin{array}{ll}
2|z_n|w_n,& \ (|z_n|\geq w_n)\cr
(|z_n|+w_n)^2/2,& \ (|z_n|<w_n)
\end{array}
\right.\end{aligned}$$ Here $\theta$ is the “phase” of the external force as in the QPLM(\[EQ:QPLM\]). $z$ and $w$ correspond to the center position and the half-length of the segment respectively. Although the derivative of this map has discontinuity on $z=0$, we have observed no symptom of pathological phenomena due to this discontinuity.

Note that, the subspace of zero-length segments ($\tilde\Omega_0=\{w=0\}\subset\tilde\Omega$) is kept invariant by the map $\tilde M$, and the restriction of $\tilde M$ on $\tilde\Omega_0$ could be naturally identified with QPLM ($M$ on $\Omega$).

We sometimes call a subset of the phase space with common $\theta$ value as a “fiber” (both for QPLM 