## 2 Description of the Model
The durations of power-on phase and power-off phase are fixed at 10 \(\upmu\)s.  
The derivative of gas voltage is expressed as  
\[\frac{\partial V_{\rm g}}{\partial t}=\frac{d_{\rm g}}{d_{\rm g}+2d_{\rm s}/\varepsilon _{\rm r}}\frac{\mathrm{d}V_{\rm a}}{\mathrm{d}t}-\frac{2d_{s}}{\varepsilon_{ \rm 0}\varepsilon_{\rm r}\big{(}d_{\rm g}+2d_{\rm s}/\varepsilon_{\rm r}\big{)}} \int_{0}^{d_{\rm g}}j_{\rm g}\mathrm{d}x \tag{7}\]  
The memory voltage \(V_{\rm m}(t)\), which is the total voltage fall across the two dielectric barriers, is given by  
\[V_{\rm m}=V_{\rm a}-V_{\rm g} \tag{8}\]  
In this paper, the CO\({}_{2}\) discharge is generated and sustained in a parallel-plate DBD structure, and each electrode is covered by a dielectric barrier with a thickness of 1.0 mm and a permittivity of 6.1. At the boundary of the discharge gap, the following surface reactions are considered in this model: excited CO\({}_{2}\) molecules are quenched to their ground state and ions are neutralized to their corresponding neutral species [43]. In addition, the boundary conditions for the flux of various particles and electron energy are adopted as follows [44; 45]:  
\[\Gamma_{\rm e}\cdot n=\frac{1}{4}N_{\rm e}\sqrt{\frac{8k_{\rm b}T_{\rm e}}{ \pi m_{\rm e}}}-\alpha N_{\rm e}\mu_{\rm e}E\cdot n-\sum_{k}\gamma_{k}(\Gamma_ {k}\cdot n) \tag{9}\]  
\[\Gamma_{k}\cdot n=\frac{1}{4}N_{\rm e}\sqrt{\frac{8k_{\rm b}T_{k}}{\pi m_{k}}} +\alpha\mathrm{sgn}(q_{k})q_{k}N_{k}\mu_{k}E\cdot n \tag{10}\]  
\[n\cdot\Gamma_{\rm e}=\frac{5}{6}N_{\rm e}\sqrt{\frac{8k_{\rm b}T_{\rm e}}{\pi m _{\rm e}}}-\alpha N_{e}\mu_{e}E\cdot n-\sum_{k}\gamma_{k}\varepsilon_{k}( \Gamma_{k}\cdot n) \tag{11}\]  
where the indices \(\rm e\), \(k\), and \(\varepsilon\) represent electrons, heavy particles, and electron energy, respectively. \(q_{k}\) is the charge number of the particle \(k\), and \(\mathrm{sgn}(q_{k})\) is the sign of \(q_{i}\), which is positive (+) for positive ions, negative (-) for electrons and negative ions, and nil for neutral particles, respectively. The secondary electron emission is considered in this model, and the emission coefficient \(\gamma_{k}\) and the initial energy \(\varepsilon_{k}\) of secondary electrons are 0.03 and 2.5 eV, respectively. \(n\) is the normal vector pointing to the dielectric barrier. \(\alpha\) is a switch function depending on the product of \(E\) and \(n\), which is defined as  
\[\alpha= \begin{cases}0&(\mathrm{sgn}(q_{k})E\cdot n\leqslant 0)\\ 1&(\mathrm{sgn}(q_{k})E\cdot n>0)\end{cases}  \tag{12}\]  
To more accurately clarify the mechanism of temporal nonlinear behaviors in CO\({}_{2}\) DBDs under Martian conditions, based on graph theory proposed by [PERSON] and [PERSON]  
Figure 1: Waveform of applied voltage and corresponding current density in the simulation. rakami [46; 47], the reaction set of CO\({}_{2}\) is selected using a 0D kinetic model. According to Ref. [38], six neutral species and seven charged species are considered. Moreover, an electronically excited state of CO\({}_{2}\) (i.e., CO\({}_{2}\)e\({}_{1}\)), four lower symmetric stretching and bending mode of CO\({}_{2}\) (denoted as CO\({}_{2}\)v\({}_{a}\),CO\({}_{2}\)v\({}_{b}\),CO\({}_{2}\)v\({}_{c}\), and CO\({}_{2}\)v\({}_{1a}\)), and eight asymmetric stretching mode of CO\({}_{2}\) (symbolized as CO\({}_{2}\)v\({}_{1}\), CO\({}_{2}\)v\({}_{2}\), CO\({}_{2}\)v\({}_{3}\), CO\({}_{2}\)v\({}_{4}\), CO\({}_{2}\)v\({}_{5}\), CO\({}_{2}\)v\({}_{6}\), CO\({}_{2}\)v\({}_{7}\), and CO\({}_{2}\)v\({}_{8}\)) are also taken into consideration. These species taken into account in the chemical model are given in Table 1. The chemical reactions and the corresponding reaction rate coefficients are obtained from the previous investigation in Ref. [38]. The reaction rate coefficient for electron impact processes and the transport coefficients for electrons are calculated using BOLSIG+ based on the cross sections obtained from the IST-Lisbon database available on the open-access LXCat website [48; 49].