curves in green are the kinetic energy. []{data-label="fig:hybrid_evol"}](hybrid_evol.eps){width="80mm"}

![The same as Fig. \[fig:hybrid\_evol\], but when the potential of the field is given by Eq.  with $\phi_{*}=9\phi_{\rm R}$, $7\phi_{\rm R}$, $5\phi_{\rm R}$, and $3\phi_{\rm R}$.[]{data-label="fig:chaotic_evol"}](chaotic_evol.eps){width="80mm"}

The above simplest example does not have a transition to the late-time slow-roll phase. Here we introduce two simple toy models, which have the transition from the initial rapid-roll phase to the late-time slow-roll phase, and we discuss the tensor perturbation in these models later.

The first model has the effective potential of the following type: $$\begin{aligned}
\frac{\kappa}{3}V(\phi )=\left( H_*^2-H_{\rm R}^2\right)\exp\Bigl[
\sqrt{2\kappa\epsilon_{*}}\left(\phi -\phi_{*}\right)
\Bigr] +H_{\rm R}^2\,,
\label{eq:exponential-type}\end{aligned}$$ with $H_*\gg H_R$. The potential of a similar type often appears in supergravity and superstring theory. As a concrete example, we will choose the parameters $H_*^2=10^{-3}M_{\rm pl}^2$, $H_{\rm R}^2=10^{-30}M_{\rm pl}^2$ and $\phi_{*}=10M_{\rm pl}$. In Fig. \[fig:hybrid\_evol\], the time evolution of the energy densities for $\epsilon_{*}=0.1$, $0.5$, $1$, $10^2$ and $10^4$ is shown. As is expected, for $\epsilon_{*}<1$ the field starts to decay slowly at $t\approx H_*^{-1}$, for $\epsilon_{*}\geq 1$ it starts to fall down toward the slow-roll plateau at $t=\epsilon_{*}^{-1/2}H_*^{-1}$ and its energy density decays in proportion to $1/a_{\rm R}^2$, i.e. tracking.

The second model is a variant of 