## 3 Results and Discussion
Figure 8 shows the temporal evolutions of current density, applied voltage, gas voltage, and memory voltage in PR2 discharge with a gap width of 5.1 mm. It can be seen that, during the voltage rise phase, the rapid increase of the memory voltage suppresses the gas voltage, resulting in a weaker positive discharge. However, during the voltage fall phase, a strong negative discharge caused by the surface charge accumulated on the surface of dielectric barriers can be observed. As shown in Figure 8, the breakdown voltages for the first and second voltage periods are \(-\)858 and \(-\)846 V, and the corresponding peak current densities are \(-\)7.2 and \(-\)7.6 mA/cm\({}^{2}\), respectively. In addition, several key moments in RP2 discharge are marked in Figure 8, where a and d are the moments when positive discharge starts (that is, the beginning of the voltage periods) in the first and second voltage periods, b and e are the moments at which the negative discharge begins (i.e., the gas voltage starts to rise in reverse at this moment) in the first and second voltage periods, and c and f are the the moments when the positive current density reaches the maximum values. Figure 8: Temporal evolutions of the current density, applied voltage, gas voltage, and memory voltage in PR2 discharge with a gap width of 5.1 mm. Figure 9 shows the spatial distributions of charged particle density and electric field at the beginning of the positive discharge for two consecutive voltage periods in RP2 discharge. The black and red lines in Figure 9 represent the corresponding spatial distributions at the moments a and d in Figure 8, respectively. As shown in Figure 9, further increasing the gap width allows more charged particles to stay in the discharge gap, and the negative charges are mainly electrons. When the applied voltage begins to rise, these charged particles, especially electrons, are rapidly driven to the surface of the dielectric barriers by the applied electric field to build a reverse electric field, which improves the memory voltage and reduces the gas voltage, resulting in a weak positive discharge and a strong negative discharge. Subsequently, Figure 10 shows the spatial distributions of the charged particle density and electric field at several key moments in the RP2 discharge, where Figure 10a-d plot the corresponding spatial distributions at the moments b, c, e, and f marked in Figure 8, respectively. As shown in Figure 10a, at the moment b, the residual charged particles in the anode sheath and the cathode sheath are CO\({}_{3}^{-}\) and CO\({}_{2}^{+}\) ions with the maximum densities of \(1.1\times 10^{9}\) and \(1.3\times 10^{9}\) cm\({}^{-3}\), respectively. In addition, at the moment e in Figure 10, the densities of residual CO\({}_{3}^{-}\) and CO\({}_{2}^{+}\) ions in the anode sheath and cathode sheath reach \(1.5\times 10^{9}\) and \(1.9\times 10^{9}\) cm\({}^{-3}\), respectively. Obviously, the density of residual ions in the sheath region at the moment e is greater than that at the moment b, thus the discharge of the second voltage period can easily be ignited at a lower voltage. In addition, the discharge current density is determined by the electric field and the charged particle density. According to Figure 10b,d, the higher charged particle density and stronger electric field in the cathode fall region at the moment f in the second voltage period lead to a larger peak current density of the negative discharge. When the gap width is increased to 5.16 mm, the discharge undergoes an inverse period-doubling bifurcation from RP2 state to RP1 state, and the current density is repeated every voltage period. Figure 11 shows the temporal evolutions of the current density, applied voltage, gas voltage, and memory voltage in RP1 discharge.