## 6 Methodology
The operator \(T\) represents the transposition of a vector or matrix. These control functions \(f,\ g\) are not accessible publicly but, based on the phase-state vectors, can be determined as extreme modifications of the trajectory curve in the 3D coordinate system set in the solar system. The position vector represents the trajectory and velocity of Juno in the solar reference system \(\overline{r_{JUN}}\) and \(\overline{r_{JUN}}\), respectively. Similarly, the trajectory and velocity of Jupiter are represented by \(\overline{r_{JUP}}\) and \(\overline{r_{JUP}}^{\prime}\), respectively. It is evident that:  
\[\frac{d}{dt}\overline{r_{JUN}}=\overline{r_{JUN}}^{\prime}. \tag{4}\]  
And  
\[\frac{d}{dt}\overline{r_{JUP}}=\overline{r_{JUP}}^{\prime}. \tag{5}\]  
Because Juno is a spinning orbiter with high stability, it needs higher-order position vector derivatives to detect the object's status change. Similar to the Formulae (4) and (5), the acceleration, jerk and snap of Juno are second, third and fourth derivative of the position. Because we have a discrete sampling period in the scale of one minute, we have to consider the average of these quantities. The instantaneous average acceleration (_a_), jerk (_j_) and snap (s) of Juno are determined with the following formulae:  
\[\overline{\mu_{IUN}}=\frac{\Delta\overline{\psi_{IUN}}}{\Delta t}. \tag{6}\]  
\[\overline{j_{IUN}}=\frac{\Delta\overline{\pi_{IUN}}}{\Delta t}. \tag{7}\]  
\[\overline{s_{IUN}}=\frac{\Delta\overline{j_{IUN}}}{\Delta t}. \tag{8}\]  
The sampling period of our dataset is \(\Delta t=60\ s\). The position and velocity values were downloaded from the data source, but the acceleration, jerk and snap were calculated based on the velocity and time period. The acceleration, jerk and snap values are not instantaneous but average values (see Formulae (6)-(8)) in each time period \(\Delta t\). Because the value of \(\Delta t\) is very small relative to the overall movement process (~11.5 years), we consider these average ratios as instantaneous values. All the 3D vectors are continuous in time, but we have \(N\) discrete-time samplings with a period of one minute downloaded from the database of NASA. This makes us use discrete vectors \(\overline{X}\in\mathbb{R}^{3 xN}\), \(\overline{v}\in\mathbb{R}^{3 xN}\), \(\overline{a}\in\mathbb{R}^{3 xN}\), \(\overline{j}\in\mathbb{R}^{3 xN}\) and \(\overline{s}\in\mathbb{R}^{3 xN}\) with \(N=5,583,178\) samplings in the time interval between the launch time on 8 August 2011 of Juno (SOI) and the end of the accessible data on 4 January 2023 (EOI). The use of the higher-order velocity derivatives makes it possible to detect EEs more accurately. We had to distinguish three phases in time of the trajectory during the analysis:  
1. OP1: Before the JOI phase, Juno's trajectory is a smoothed 3D curve in the solar system. This phase lasts approximately five years after the launch from the Earth. The reference system is the solar system. 2. OP2: In the JOI phase, Juno approaches Jupiter and, based on the gravity effects, starts, step-by-step, to move around it. The reference system remains the solar system. 3. OP3: After the JOI phase in the solar reference system, Juno has a helicoid orbit in 3D with Jupiter in the gravity center (see Figure 3). In a time of weeks, the trajectory of Juno around Jupiter is an ellipse, with one of the focal points being Jupiter. It should be mentioned that this trajectory modifies slowly on the scale of several months or years, and this relative trajectory is slowly modifying over time, too. An overview of the applied methods is given in Table 2. Evaluation of the acceleration, jerk, and snap is considered in each operation phase to be relative to the barycenter of the solar system. Trajectory modification is considered an extreme event (EE). A powerful event happens when the complex logical expression of the EE condition is true. These conditions are logical functions of the logical variables \(A_{1},A_{2},\ldots,S_{2},S_{3}\). These