## 6 Methodology
### State Space Analysis and Discussion  
EEs detected in each of the Juno OP are listed in Table 4. EEs are logical expressions of the elemental event conditions (i.e., an extreme event in OP1 is an AND operation between  
Figure 14: Hilbert transform of the acceleration in OP3, snap. Figure 13: Hilbert transform of the acceleration in OP3, jerk. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Operation Phase** & **OP1** & **OP2** & **OP3** \\ \hline Start of OP & 7 August 2021 17:18:06 & 1 July 2016 00:00:01 & 5 July 2016 03:55:00 \\ \hline End of OP & 1 July 2016 00:00:00 & 5 July 2016 03:54:00 & 4 January 2023 06:03:51 \\ \hline \multirow{3}{*}{Event condition} & \(A_{11}:\)\(HS(a)>800\) & \multirow{3}{*}{\(A_{2}:\)\(HS(a)>4500\)} & \(A_{31}:\)\(\varnothing\) \\  & \(A_{12}:\)\(f(a)>10^{-4}\) Hz & & \(A_{2}:\)\(\varnothing\) \\  & \(J_{1}:\)\(\varnothing\) & & \(J_{2}:\)\(\varnothing\) \\  & \(S_{1}:\)\(\varnothing\) & & \(S_{2}:\)\(\varnothing\) \\ \hline EE condition & \(\mathit{Cond}_{1}=A_{11}\cdot A_{12}\) & \(\mathit{Cond}_{2}=A_{2}\) & \(\mathit{Cond}_{3}=S_{31}\cdot S_{32}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Logical conditions of the Juno orbit modification in each operation phase. event conditions A_11 and A_12, where A_11 means all the time moments when the Hilbert spectrum of the acceleration is greater than 800). A neutral elemental event condition is noted by \(\emptyset\). The dates are in the astronautical format of \"YYYY-MM-DDThh:mm:ss\". A total of 9, 2, and 12 EEs were identified as trajectory modifications in the three different operation phases, OP1, OP2, and OP3. It should be mentioned that these events are closed in time to the well-known maneuvers shared by the project Juno. Modifications to the trajectory were sensed in a larger time difference than the propagation time of the communication radio signal between the Earth and Juno. Even the time difference between the local time on Juno or Earth, or in the barycenter of the solar system, is smaller than the time differences between the maneuvers mentioned by NASA and the EEs detected in this research work. Figures 15-20 show the (v, a, j) and (a, j, s) normalized space state of Juno in the three different operation phases. As shown in Figure 17, jerk, which is the derivative of acceleration, enables and represents variations in body accelerations without considering sensor orientation. The following graph (Figure 18) displays the Juno state space of the normalized acceleration, jerk and snap (a, j, s) during Operation Phase 2. It depicts the average jerk length, which refers to the distance covered by the jerk within a certain time interval. \begin{table}
\begin{tabular}{c c} \hline \hline
**Operation Phase** & **Extreme Event Dates** \\ \hline  & 2012 01 21T02:19:59; 2012 02 05T01:24:59; 2012 \\  & 09 19T23:18:59 \\  & 2012 09 19T23:20:59; 2013 03 11T14:19:59; 2014 \\  & 03 23T10:23:59 \\  & 2014 03 23T10:25:59; 2014 11 15T12:06:00; 2014 \\ \hline  & 2016 07 01T00:00:59 (JOI Start); 2016 07 05T03:53:59 (JOI End) \\ \hline  & 2016 07 13T18:02:00; 2016 10 25T17:59:00; 2017 02 271T:43:59; 2018 09T15:59:59; 2018 \\  & 03 011T17:01:59 \\ (No. of EE = 12) & 2019 08 16T03:26:00; 2019 10 14T21:46:00; 2019 \\  & 2019 10 14T21:52:00; 2019 10 14T21:51:00 \\  & 2019 10 14T21:52:00; 2019 10 14T21:55:00 \\  & 05 04T11:59:59 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Extreme events (EE) of the Juno orbit (TDB time zone). Figure 15: State space of Juno in OP1 (v, a, j): top view. Figure 16: State space of Juno in OP1 (v, a, j): perspective view. Figure 17: State space of Juno in OP2: (v, a, j). Figure 18: State space of Juno in OP2: (a, j, s). Figure 19 shows that in the state space \((v,\ a,\ j)\), Juno executes a chaotic movement in the actual operation phase (OP3). This phenomenon is not observable in the state space (a, j, s) in Figure 20. A possible reason for the chaotic movement of Juno is the magnetic sensor on just one of the propellers, providing no precise balance around the rotation axis.