Algorithm 1 The Lorenz time discretization algorithm

1. Input: vector field $F\left(X\right)$, initial state $X_0$ ∈ ${\mathbb{R}}^3$, step h > 0, number of steps N, and $X_0$ is the initial condition. 2. Initialization: X ← $X_0$; $X_0$ is written for k = 0, 1, …, N − 1; 3. Scheme selection: Euler (explicit, order 1), RK4 (explicit, order 4), and Midpoint/Crank–Nicolson (centroid, implicit, order 2); 4. Iterations (fixed point): Set $X_{\mathrm{k}+1}^{\left(0\right)}$←$X_k$, 3a n = 0, 1, … n to convergence          $X_{(k+1)}^{(n+1)}=X_k+h\mathcal{F}\left({\displaystyle \frac{X_k+X_{(k+1)}^n}{2}}\right)$, 5. Stop criterion: ‖$X_{(k+1)}^{(n+1)}-X_{(k+1)}^n$‖ ≤ tol, where tol is a numerical tolerance threshold that defines the allowable difference between two consecutive states or iterations in the model. 6. It sets the criterion for stopping the iteration process, i.e., when to assume that the system is “sufficiently” stabilized or has reached a stationary solution. 7. Final: Xk+1 ← $X_{(k+1)}^{(n+1)}$; 8. End of cycle. Returning {Xk}. 9. Output: discrete trajectory ${\left\{X_k\right\}}_{\mathrm{k}=0}^{\mathrm{N}}$, where $X_k$=${\varphi }_h$($X_k$) and $\varphi$ е пoтoкa нa системaтa.