Figure 1

The upper left panel is the figure of potential with ${\Lambda }_{*}=0.1$, $\frac{m}{\sqrt{{\Lambda }_{*}}}=1$ and $\Lambda =0.05$. The upper right panel is the figure of $\ln a$ with respect to time. The dashed and solid line are the cases that the height ${\Lambda }_{*}$ of potential are taken as $0.07,0.04$, respectively, where $\Lambda =0.005$ is taken. The lower panels are the figures of field with respect to time, where $\sigma =1$ is taken. We find that for different cycles of universe the field oscillates in different minima.Reuse & Permissions

Figure 2

The left side panels are the figures of field with respect to time, while the right side panels are the figures of its potential energy (solid line) and kinetic energy (dashing line) in time interval (81, 89). The upper right panel is the case that the field strides over two hills and one valley and the lower right panel is the case of one hill and one valley.Reuse & Permissions

Figure 3

The 2D landscape. The potential is plotted as a function of two scalar fields. The dashed line is the evolution of total kinetic energy of fields during some cycle.Reuse & Permissions

Figure 4

When the minima of valleys which the field rolls to finally are dS minima, the cycle will generally cease, (or occur in subsequently nucleated bubbles). To make the universe cycle more effectively, the potential (solid line) in the figure is a feasible example. In almost all cases the field enters into negative potential not by tunnelling over hill (dashed line) but by rolling, which is locally just the case in the cyclic model [26]. But different from that, here the universe will be in different regions of potential of fields in each cycle. Not in all cycles are there such local potentials making field show a quintessencelike behavior at present as in cyclic model [13]. However, when the number of cycles is large or approaches infinity, some cycles similar to the universe which we live in will be possible.Reuse & Permissions