## 3 Results and Discussion
The temporal evolutions of the current density and the corresponding trajectories in phase space with the gap widths of 5.0, 5.1, and 5.16 mm are given in Figure 3. As shown in Figure 3a, the discharge enters a reverse chaotic discharge, which is demonstrated in Figure 3b, where the trajectories in phase space are no longer periodic. With the further increasing gap width to 5.1 mm, the Reverse Period-two (RP2) discharge is obtained in Figure 3c. Similar to P2, the repetition period of discharge current doubles the voltage periods. The two larger loops in the third quadrant of Figure 3d clearly show that the discharge maintains a stable RP2 discharge. As shown in Figure 3e, when the gap width is increased to 5.16 mm, the discharge transits into the Reverse Period-one (RP1) discharge through an inverse period-doubling bifurcation, the corresponding trajectory in phase space presents only one loop in Figure 3f. From the foregoing discussion, there is only a large positive current pulse during the power-on phase and the discharge is a stable P1 state when the width gap is 4.0 mm. Figure 4 plots the temporal evolutions of current density, applied voltage, gas voltage, and memory voltage in P1 discharge with a width gap of 4.0 mm, where the black solid line, green dotted line, red dashed line, and blue dot-dashed line denote the current density, applied voltage, gas voltage, and memory voltage, respectively. During the voltage rise phase, the gas voltage increases with the applied voltage. The CO\({}_{2}\) gas is ignited when the gas voltage reaches 750 V, a lot of charged particles are generated, and a large current pulse with a peak value of 10.9 mA/cm\({}^{2}\) is obtained. At the same time, the charged particles are driven to the surface of the dielectric barriers by the applied electric field, building a reverse induced electric field, which rapidly improves the memory voltage and reduces the gas voltage, resulting in the extinguishing of the discharge. During the voltage fall phase, the reverse electric field could lead to a weak reverse discharge, and the peak current density is only \(-1.2\) mA/cm\({}^{2}\). Figure 3: Temporal evolutions of discharge current density with the gap widths of 5.0 mm (**a**), 5.1 mm (**c**), and 5.16 mm (**e**), together with the corresponding trajectories in phase space with the gap widths of 5.0 mm (**b**), 5.1 mm (**d**), and 5.16 mm (**f**). To investigate the spatiotemporal behavior of the electrons and heavy ions produced by the CO\({}_{2}\) discharge, Figure 5 shows the spatial distributions of positive charge density, negative charge density, electron density, and electric field at several key moments in P1 discharge, and these key moments are marked in Figure 4. The black solid line, red dashed line, and blue dotted line in Figure 5 denote the positive charge density, negative charge density (i.e., the sum of negative ion density and electron density), and electron density, respectively, and the green dot-dashed line represents the electric field in the discharge gap. At the moment a when the applied voltage starts to rise, as shown in Figure 5a, the charged particle density is very low, and the electric field is uniformly distributed throughout the whole discharge gap, with a smaller value of \(-0.55\) kV/cm. This indicates that the charged particles produced by the previous discharge event have completely dissipated. Then, at 2.8 \(\upmu\)s after the voltage application, the gas is ignited and the discharge current density rapidly reaches the positive peak value. Figure 5b shows the corresponding spatial distributions of charged particle density and electric field, where the discharge operates in glow mode. In Figure 5b, the cathode fall region, negative glow region, and positive column region could be clearly identified. A strong electric field and large density of positive charges in the cathode fall region are obtained, and the positive column region is quasi-neutrality with almost equal densities of positive and negative charges.