## 3 Results and Discussion  
By increasing the gap width from 4 to 5.2 mm under the same simulation conditions, the evolution of temporal nonlinear behaviors in CO\({}_{2}\) discharges is numerically investigated, the DBDs are driven by a tailored sinusoidal voltage with power-on and power-off phases. As shown in Figure 1, only a large current pulse is generated during the power-on phase, and the charged particles produced by the discharge can move toward the surface of dielectric barriers during the power-off phase. In fact, according to previous work [38], when the CO\({}_{2}\) discharge is extinguished, electrons can quickly reach the surface of the dielectric barrier, and the dominant charged particles in the discharge region are heavy CO\({}_{2}^{+}\) and CO\({}_{3}^{-}\) ions, which could slowly migrate to the electrodes. Thus, when the gap width is increased, these ions cannot reach the barriers during the power-off phase and stay in the discharge gap, which affects the electric field for the next discharge. In other words, an increase in the gap width can prolong the time for charged particles to reach the surface of barriers, affecting the density and position of the residual charged particles, which results  
\begin{table}
\begin{tabular}{c c c} \hline
**Radicals and Molecules** & **Charged Species** & **Excited Species** \\ \hline CO\({}_{2}\)CO\({}_{2}\)CO\({}_{2}\)CO\({}_{3}\) & \(\mathrm{e},\mathrm{O}^{-},\mathrm{O}_{2}^{-},\mathrm{CO}_{3}^{-},\mathrm{CO}_{4}^{-}, \mathrm{CO}_{2}^{+},\mathrm{O}_{2}^{+}\) & CO\({}_{2}\)v\({}_{1},\mathrm{CO}_{2}\)v\({}_{2},\mathrm{CO}_{2}\)v\({}_{3},\mathrm{CO}_{in the formation of nonlinear behaviors. The simulation results show that the temporal nonlinear behaviors are very sensitive to the changes in the control parameters and can be maintained over a gap width interval. To more clearly display the effects of gap width and the transformation mechanism of nonlinear behaviors, several typical nonlinear behaviors under different gap widths are selected. For example, when the gap width is increased to about 4.2 mm, the discharge evolves from Period-one (P1) state to Period-two (P2) state, so 4.2 mm is taken as a typical to investigate the formation mechanism of P2 discharge. A more detailed study on the effects of gap width will be presented later in this paper. Figure 2 shows the temporal evolutions of the current density together with the corresponding trajectories constructed by the current density and gas voltage in phase space with the gap widths of 4.0, 4.2, 4.42, and 4.5 mm. As shown in Figure 2a, when the gap width is 4.0 mm, the discharge current pulse is repeated every voltage period, and a stable P1 discharge is obtained. There is only one loop in phase space in Figure 2b, which demonstrates the P1 discharge in Figure 2a. As the gap width is increased to 4.2 mm, the discharge evolves into a P2 state in Figure 2c and the current period is twice the voltage period. It is clear that two different loops in phase space are obtained in Figure 2d. Then, continuing to increase the gap width, the chaotic discharge is obtained in Figure 2e, corresponding to many different loops in phase space in Figure 2f. The chaos lasts over a small gap width interval, and as the gap width increases to 4.5 mm, a clear Period-three (P3) discharge is presented in Figure 2g, where the current density repeats every three voltage periods. In Figure 2h, three different loops in phase space are presented. In fact, the existence of P3 discharge is often accompanied by chaos according to the Li-Yorke theory [58]. Figure 2: Temporal evolutions of discharge current density with the gap widths of 4.0 mm (**a**), 4.2 mm (**c**), 4.42 mm (**e**), and 4.5 mm (**g**), together with the corresponding trajectories in phase space with the gap widths of 4.0 mm (**b**), 4.2 mm (**d**), 4.42 mm (**f**), and 4.5 mm (**h**). Then, as the gap width further increases, the discharge evolves into a reverse discharge with a weak positive discharge and a strong negative discharge.