1. Main Results

• Derivation of the Setler Dynamics: The Setler dynamics are derived by transforming continuous formulas into discrete forms suitable for our model. The continuous form of our dynamical system is defined as:

  $${\displaystyle \frac{d\alpha }{d\tau }}=\lambda sin\left(\alpha \right)cos\left(\delta \right)+\beta sin\left(\omega \tau \right),$$

  $${\displaystyle \frac{d\delta }{d\tau }}=\lambda cos\left(\alpha \right)sin\left(\delta \right)+\gamma cos\left(\omega \tau \right),$$

  $${\displaystyle \frac{dr}{d\tau }}=\lambda {(sin\left(\delta \right)cos\left(\alpha \right))}^2+\delta sin\left(\omega \tau \right).$$
  Here, $\alpha$, $\delta$, and r represent the positional coordinates, while $\lambda$, $\beta$, $\gamma$, and $\omega$ are parameters governing the dynamics. By applying a discrete time step $\Delta \tau$, the discrete form of the dynamics is:

  $${\mathbf{x}}_{n+1}={\mathbf{x}}_n+\Delta \tau ·\mathbf{F}({\mathbf{x}}_n,{\tau }_n),$$

  where $\mathbf{x}=(\alpha ,\delta ,r)$ and $\mathbf{F}$ is the function derived from the continuous system. This transformation allows for the analysis of chaotic behavior and entropy in a discrete framework.
• Entropy Growth in Stellar Dynamics: The Perelman entropy for the Setler system exhibits exponential growth over time. For the unperturbed case ($R=0$), the entropy increases as $\mathcal{W}(g,f,\tau )\sim \mathcal{O}\left(e^{{\kappa }_1\tau }\right)$, where ${\kappa }_1$ is the largest growth rate from the exponential terms in the potential function $f\left(\tau \right)$. In the perturbed case ($R\ne 0$), while the scalar curvature term R slightly modifies the entropy, the overall growth remains exponential, similar to the unperturbed case.
• Complexity in Predicting Stellar Positions: The increasing entropy suggests that as time progresses, predicting the exact location of stars becomes increasingly complex. The exponential increase in entropy reflects growing disorder, making it progressively harder to accurately predict stellar positions with certainty over long timescales. This result indicates that while deterministic dynamics govern the system, the inherent complexity of celestial positional systems grows over time.

2. Introduction

3. Discrete Dynamical System for Stellar Positions

4. Analyzing the Lyapunov Exponents

The Lyapunov exponents (${\lambda }_i$) are calculated to measure the sensitivity to initial conditions in the dynamical system. The following algorithm outlines the procedure used to compute the Lyapunov exponents:

5. Bifurcation Analysis and Chaotic Dynamics

6. Comparison between the Lorenz Attractor and the Attractor of the New Dynamics System

6.2. Comparison of Attractors

Both attractors exhibit the characteristic double-lobed structure, indicative of chaotic dynamics. However, the Setler Position attractor shows additional complexity, suggesting that the specific combination of parameters chosen contributes to a more intricate dynamical structure. This complexity could be due to the astrophysical interpretation of the parameters, which introduce unique interactions not present in the Lorenz system.

The similarities in attractor shapes between the Setler Position and Lorenz systems suggest that the new system operates under similar chaotic mechanisms. However, the distinct features observed in the Setler Position attractor highlight its unique chaotic properties, which may warrant further investigation into the role of its specific parameters.

Figure 4. Lorenz Attractor with parameters $\sigma =10$, $\rho =28$, $\beta ={\displaystyle \frac{8}{3}}$.

Figure 4. Lorenz Attractor with parameters $\sigma =10$, $\rho =28$, $\beta ={\displaystyle \frac{8}{3}}$.

Figure 5. Setler Position Attractor with parameters $\lambda =0.5$, $\beta ={\displaystyle \frac{8}{3}}$, $\gamma ={\displaystyle \frac{28}{3}}$, $\delta =10$, $\omega =0.1$.

Figure 5. Setler Position Attractor with parameters $\lambda =0.5$, $\beta ={\displaystyle \frac{8}{3}}$, $\gamma ={\displaystyle \frac{28}{3}}$, $\delta =10$, $\omega =0.1$.

Sensitivity Analysis

Sensitivity analysis is crucial in understanding how small changes in parameters or initial conditions affect the overall dynamics. In this subsection, we explore both parametric and geometric sensitivity using computational tools.

• Parameter Sensitivity: The parameters $\lambda$, $\beta$, $\gamma$, and $\omega$ play critical roles in determining the behavior of $\alpha \left(\tau \right)$, $\delta \left(\tau \right)$, and $r\left(\tau \right)$. To analyze sensitivity, we varied these parameters slightly and observed their impact on the system’s trajectory.

Using Mathematica, we implemented the continuous dynamical system and performed numerical simulations to observe how small changes in $\lambda$ or $\beta$ influence the evolution of $\alpha \left(\tau \right)$. The following plot, see Figure 6 illustrates the divergence of trajectories under small parameter perturbations.

Figure 6. Sensitivity of $\alpha \left(\tau \right)$ to parameter $\lambda$, for ${\lambda }_1=10$ (red) and ${\lambda }_2=17.2$ (blue).

Figure 6. Sensitivity of $\alpha \left(\tau \right)$ to parameter $\lambda$, for ${\lambda }_1=10$ (red) and ${\lambda }_2=17.2$ (blue).

Figure Analysis:

As depicted in the figure, we observe the sensitivity of $\alpha$ to two different values of $\lambda$. The red curve corresponds to ${\lambda }_1=10$ and the blue curve to ${\lambda }_2=17.2$, both representing the system’s response over time $\tau$.

The plot reveals that both trajectories exhibit periodic oscillations with nearly identical amplitudes, oscillating between approximately $-0.3$ and $0.3$. The primary distinction between the two trajectories is a slight phase shift, where the red curve (${\lambda }_1$) lags behind the blue curve (${\lambda }_2$). This suggests that while changes in $\lambda$ do not significantly affect the amplitude of $\alpha \left(\tau \right)$, they do influence the timing of the oscillations.

This phase shift indicates that increasing $\lambda$ causes the system to oscillate more rapidly, leading to an observable shift in the peaks and troughs. The system is thus **moderately sensitive** to changes in $\lambda$ in terms of timing, but it demonstrates **robustness** in terms of amplitude stability. This insight is essential for systems where synchronization or timing is a critical factor.

The analysis shows that small variations in $\lambda$ primarily affect the phase of oscillations rather than the magnitude. This type of sensitivity is particularly relevant when analyzing systems that rely on precise timing, as even minor parameter changes could lead to synchronization issues or phase misalignment.

Sensitivity Analysis

Sensitivity analysis is crucial in understanding how small changes in parameters or initial conditions affect the overall dynamics. In this subsection, we explore both parametric and geometric sensitivity using computational tools.

• Parameter Sensitivity: The parameters $\lambda$, $\beta$, $\gamma$, and $\omega$ play critical roles in determining the behavior of $\alpha \left(\tau \right)$, $\delta \left(\tau \right)$, and $r\left(\tau \right)$. To analyze sensitivity, we varied these parameters slightly and observed their impact on the system’s trajectory.

Second Case: Sensitivity to Parameter $\omega$:

In the second case, we explored the sensitivity of $\alpha$ to changes in $\lambda$ for ${\lambda }_1=1000$ and ${\lambda }_2=0.00007$, with the parameter $\omega$ constrained in the range $(0,1)$. The system was numerically solved for these values, and the resulting plot is as follows:

Figure 7. Sensitivity of $\alpha \left(\tau \right)$ to parameter $\lambda$, with ${\lambda }_1=1000$ (red) and ${\lambda }_2=0.00007$ (blue).

Figure 7. Sensitivity of $\alpha \left(\tau \right)$ to parameter $\lambda$, with ${\lambda }_1=1000$ (red) and ${\lambda }_2=0.00007$ (blue).

Figure Analysis:

In this second case, we observe that the sensitivity of $\alpha \left(\tau \right)$ to $\lambda$ shows a more significant impact, especially when considering larger changes in $\lambda$ values. The red curve corresponding to ${\lambda }_1=1000$ exhibits a rapid, non-oscillatory growth in $\alpha \left(\tau \right)$, reaching a peak value of around 70 before slightly decreasing. On the other hand, the blue curve (${\lambda }_2=0.00007$) remains nearly constant at very low values throughout the time range.

This behavior highlights the **high sensitivity** of the system to $\lambda$ when $\omega$ is constrained to small values within the range $(0,1)$. With smaller values of $\omega$, the influence of $\lambda$ on the system dynamics becomes more pronounced, as evidenced by the stark contrast between the two trajectories. The red curve, representing a large $\lambda$, causes the system to diverge rapidly, while the small $\lambda$ results in nearly no change in the state variable $\alpha \left(\tau \right)$.

This analysis suggests that $\omega$ plays a critical role in modulating the sensitivity to $\lambda$. When $\omega$ is kept within a limited range, the system’s sensitivity to $\lambda$ can lead to vastly different outcomes, emphasizing the importance of careful tuning of these parameters, especially in systems where large parameter shifts could induce rapid divergence.

Manifold Structure and Continuous-Time Transformation

7. Metric and Geometry

8. Ricci Flow and Curvature Evolution in Stellar Dynamics

9. Ricci Flow and Curvature Evolution in Stellar Dynamics

10. Entropy Calculation for the Setler System Using F as a Solution

11. Numerical Solution of the Setler Dynamical System Using the Runge-Kutta Method

11.2. Numerical Analysis of the Setler Dynamical System Using Runge-Kutta Method (First Case)

In this subsection, we solve the Setler dynamical system numerically using the Runge-Kutta Method (RKM) and analyze the behavior of the system under specific parameter values. The parameters used in this first case are based on a modified form of the Lorenz system, with the following values:

$$\lambda =1,\beta ={\displaystyle \frac{23}{8}},\gamma ={\displaystyle \frac{8}{3}},\omega =0.5.$$

The initial conditions for this system are set as:

$$\alpha \left(0\right)=0.1,\delta \left(0\right)=0.2,r\left(0\right)=0.3.$$

Using these parameters, we solve the system of differential equations:

$$\begin{array}{cc}\hfill {\alpha }^{\prime }\left(\tau \right)& =\lambda sin\left(\alpha \right(\tau \left)\right)cos\left(\delta \right(\tau \left)\right)+\beta sin\left(\omega \tau \right),\hfill \\ \hfill {\delta }^{\prime }\left(\tau \right)& =\lambda cos\left(\alpha \right(\tau \left)\right)sin\left(\delta \right(\tau \left)\right)+\gamma cos\left(\omega \tau \right),\hfill \\ \hfill r^{\prime }\left(\tau \right)& =\lambda {(sin\left(\delta \left(\tau \right)\right)cos\left(\alpha \left(\tau \right)\right))}^2+\delta \left(\tau \right)sin\left(\omega \tau \right).\hfill \end{array}$$

We use Mathematica’s ‘NDSolve’ function to compute the numerical solution over the interval $\tau \in [0,10]$.

Figure 8 illustrates the evolution of $\alpha \left(\tau \right)$, $\delta \left(\tau \right)$, and $r\left(\tau \right)$ over time, showing the distinct behaviors of the variables under the influence of the chosen parameters. The plot reveals that the solutions exhibit a smooth yet non-linear evolution, with each variable following a different pattern influenced by the initial conditions and parameters.

Figure 8. Numerical solutions of the Setler dynamical system using the Runge-Kutta Method (First Case). Parameter values: $\lambda =1$, $\beta ={\displaystyle \frac{23}{8}}$, $\gamma ={\displaystyle \frac{8}{3}}$, $\omega =0.5$. The initial conditions are $\alpha \left(0\right)=0.1$, $\delta \left(0\right)=0.2$, and $r\left(0\right)=0.3$.

Figure 8. Numerical solutions of the Setler dynamical system using the Runge-Kutta Method (First Case). Parameter values: $\lambda =1$, $\beta ={\displaystyle \frac{23}{8}}$, $\gamma ={\displaystyle \frac{8}{3}}$, $\omega =0.5$. The initial conditions are $\alpha \left(0\right)=0.1$, $\delta \left(0\right)=0.2$, and $r\left(0\right)=0.3$.

The Runge-Kutta Method [21] provides an accurate approximation for the solutions of this system. In particular, it can be seen that the entropy calculations from the previous section can be further refined using the precise solutions derived from the RKM. This numerical approach not only improves the accuracy but also highlights the complex interplay between the variables under the Ricci flow, which was initially introduced in our Perelman entropy computations.

In future sections, we will explore other parameter configurations and analyze their impact on the dynamic behavior of the system.

11.3. Numerical Analysis with Increased Time Interval (Second Case)

In this second case, we extend the time interval to $\tau \in [0,\mathrm{100,000}]$ while keeping the same parameter values:

$$\lambda =1,\beta ={\displaystyle \frac{23}{8}},\gamma ={\displaystyle \frac{8}{3}},\omega =0.5.$$

The initial conditions remain:

$$\alpha \left(0\right)=0.1,\delta \left(0\right)=0.2,r\left(0\right)=0.3.$$

As seen in Figure 9, the system’s behavior drastically changes compared to the first case. Notably, $\alpha \left(\tau \right)$ (blue curve) decreases to negative infinity as $\tau$ increases, while both $\delta \left(\tau \right)$ (red curve) and $r\left(\tau \right)$ (green curve) increase to positive infinity. This divergence suggests that the system becomes increasingly unstable over time.

Figure 9. Numerical solutions of the Setler dynamical system using the Runge-Kutta Method (Second Case). The time interval has been extended to $\tau \in [0,\mathrm{100,000}]$, with parameter values: $\lambda =1$, $\beta ={\displaystyle \frac{23}{8}}$, $\gamma ={\displaystyle \frac{8}{3}}$, $\omega =0.5$. Initial conditions are $\alpha \left(0\right)=0.1$, $\delta \left(0\right)=0.2$, and $r\left(0\right)=0.3$.

Figure 9. Numerical solutions of the Setler dynamical system using the Runge-Kutta Method (Second Case). The time interval has been extended to $\tau \in [0,\mathrm{100,000}]$, with parameter values: $\lambda =1$, $\beta ={\displaystyle \frac{23}{8}}$, $\gamma ={\displaystyle \frac{8}{3}}$, $\omega =0.5$. Initial conditions are $\alpha \left(0\right)=0.1$, $\delta \left(0\right)=0.2$, and $r\left(0\right)=0.3$.

The significant divergence of $\alpha \left(\tau \right)$, $\delta \left(\tau \right)$, and $r\left(\tau \right)$ at large times suggests that the system’s solutions approach asymptotic behavior, where the nonlinear terms dominate. In particular, the form of $F\left(\tau \right)$, which governs the long-term evolution of the system, becomes crucial for understanding the system’s behavior. Based on the trends in the plot, we propose the following approximate form for the solutions at large $\tau$:

$$F\left(\tau \right)\sim c_1{e}^{{\kappa }_1\tau }+c_2{e}^{{\kappa }_2\tau },$$

where ${\kappa }_1$ and ${\kappa }_2$ are constants determined by the parameters $\lambda$, $\beta$, $\gamma$, and $\omega$, and $c_1$ and $c_2$ depend on the initial conditions.

This form reflects the exponential growth (or decay) of the variables observed in the plot. The rapid growth of $r\left(\tau \right)$ and $\delta \left(\tau \right)$ indicates that ${\kappa }_1>0$, while the decay of $\alpha \left(\tau \right)$ suggests ${\kappa }_2<0$. Further analysis of these constants could yield a more precise closed-form expression for $F\left(\tau \right)$ in this regime.

In this extended time case, the system’s entropy is expected to grow exponentially, given the divergence of the variables, suggesting a connection with chaotic or unstable behavior as $\tau$ increases.

12. Computation of Perelman Entropy for the Setler Dynamical System Usig $f\left(\tau \right)$

13. Conclusions

14. Future Research

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