# Coherent Chaotic Communication Using Generalized Runge-Kutta Method
## 2 Materials and Methods  
### Second-Order Runge-Kutta Methods  
There are many known methods belonging to the Runge-Kutta family that possess second-order algebraic accuracy [44]. A famous example of a second-order method with two stages is the explicit midpoint method:  
\[y_{n+1}=y_{n}+hf(t_{n}+\frac{h}{2},y_{n}+\frac{h}{2}f(t_{n},y_{n})). \tag{1}\]  
Another well-known example is the explicit trapezoidal rule, also known as [PERSON]'s method [45]:  
\[y_{n+1}=y_{n}+\frac{h}{2}f(t_{n},y_{n})+\frac{h}{2}f(t_{n}+h,y_{n}+hf(t_{n},y _{n})). \tag{2}\]  
Also, [PERSON]'s method [46] is broadly used:  
\[y_{n+1}=y_{n}+\frac{h}{4}f(t_{n},y_{n})+\frac{3h}{4}f(t_{n}+\frac{2h}{3},y_{n} +\frac{2h}{3}f(t_{n},y_{n})). \tag{3}\]  
All these methods are explicit second-order Runge-Kutta ODE solvers. Let us write the generalized formula for the second-order explicit Runge-Kutta method with newly introduced parameter \(\alpha\):  
\[y_{n+1}=y_{n}+h((1-\frac{1}{2\alpha})f(t_{n},y_{n})+\frac{1}{2\alpha}f(t_{n}+ \alpha h,y_{n}+\alpha hf(t_{n},y_{n}))). \tag{4}\]  
The Butcher table for the generalized second-order explicit Runge-Kutta method is as follows:In two-stage form, the method holds:  
\[\begin{split} k_{1}&=f(t_{n},y_{n});\\ k_{2}&=f(t_{n}+ah,y_{n}+ahf(y_{n}));\\ y_{n+1}&=y_{n}+h(\alpha_{1}k_{1}+\alpha_{2}k_{2}); \end{split} \tag{5}\]  
where \(\alpha_{1}=1-\frac{1}{2\alpha}\), \(\alpha_{2}=\frac{1}{2\alpha}\).  
In one of our previous studies [47], we investigated the case of controlling the midpoint position in the midpoint method. Note that while it is reasonable to select a range of midpoint positions for the variable \(m\in[0,1]\), choosing midpoint values outside this range may result in additional changes in the dynamics and lead to interesting new behavior not typical for the prototype system. However, it is important to note that selecting a midpoint value other than 0.5 could lead to a decrease in the stability and convergence of the finite-difference model, which reduces the practical applicability of a controllable midpoint position for chaotic signal modulation tasks. Nevertheless, the proposed RK-alpha method is free from the above-mentioned shortcomings. According to its decomposition into Taylor series,  
\[k_{2}=f(t_{n},y_{n})+ahf^{\prime}(t_{n},y_{n})+O((ah)^{2}) \tag{6}\]  
One can substitute (6) into (5) to obtain:  
\[\begin{split} y_{n+1}=y_{n}+h((1-\frac{1}{2\alpha})f(t_{n},y_{n} )+\frac{h}{2\alpha}f(t_{n},y_{n})+\frac{h^{2}}{2}f^{\prime}(t_{n},y_{n})+O(( \alpha h)^{2}))=\\ =y_{n}+hf(t_{n},y_{n})+\frac{h^{2}}{2}f^{\prime}(t_{n},y_{n})+O( (\alpha h)^{2}).\end{split} \tag{7}\]  
As one can see from (7), a variable \(\alpha\) does not present in the final series expansion and influences only the truncation error. Thus, it is generally valid to select the value of \(\alpha\) arbitrarily. Usually, it is supposed that \(\alpha\in(0,1]\), which corresponds well with the geometrical interpretation of the method. Nevertheless, it is not necessary to choose \(\alpha\) from this range to preserve the second order of accuracy. The dynamics exhibited by a discrete system at \(\alpha\) values beyond the range of \((0,1]\) are of particular interest. Interactions with the adaptive discrete operator can generate some nonlinear phenomena in the discrete model behavior that are not present in its continuous prototype. For example, in [18], the authors described the possibility of inducing artificial multistability in a discrete Chen system using composition diagonally implicit integration methods with controllable symmetry. Therefore, the hypothesis of the current study is that changing the parameter \(\alpha\) can lead to similar changes in system dynamics.