# Chaotic Communication System with Symmetry-Based Modulation
## 2 Materials and Methods
### Semi-Implicit Symmetric Integration Method and Variable Symmetry Approach  
The basic concept of diagonally-implicit numerical integration methods, or shortly D-methods, was reported in present works [18; 19; 20]. Such methods hold the computational efficiency of explicit methods possessing higher stability and precision, as well as preservation of some geometrical properties of the solution while simulating conservative systems. The mentioned features were first achieved in Euler-Cromer and Stormer-Verlet semi-implicit algorithms, though they were created for solving Hamiltonian systems. The diagonal implicitness can be resolved by the simple iterations method [20], which makes it possible to expand the applicability of D-methods to a broader class of dynamical systems. The only limitation here is that D methods exist only for systems of dimension \(N\geq 2\), degenerating for first-order systems to explicit and implicit Euler methods.  
The composition D-method, known as the self-adjoint semi-implicit method or composition D-method (CD), is a particular case of diagonally implicit methods. The CD method proved to be well-suited for solving chaotic systems since it prevents chaotic trajectories of the discrete system from slipping to periodic or quasi-chaotic orbits during long-term simulation. Moreover, it does not suppress chaos in both conservative and dissipative systems [20].  
Let us consider the CD method \(\Psi\) with integration step size \(h\):  
\[\Psi_{h}=\Phi_{h/2}\circ\Phi_{h/2}^{*}. \tag{7}\]  
It is a composition of a pair of basic adjoint D-methods \(\Phi_{h/2}\) and \(\Phi_{h/2}^{*}\) taken with halved step size \(h/2\). In a discrete moment of time \(t_{n}\), where the solution \(x_{n}\) is already known, one can apply the pair of methods \(\Phi_{h/2}\) consequently to obtain \(x_{n+1}\).  
Having a dynamical system of order \(N\geq 2\)  
\[\dot{\mathbf{x}}=f(\mathbf{x}),\mathbf{x}=(x_{1},x_{2},\dots,x_{N})^{\top}, \tag{8}\]  
one should split it into \(N\) parts, for example for case \(N=2\):  
\[\begin{array}{l}\dot{u}=f_{u}(u,w);\\ \dot{w}=f_{w}(u,w).\end{array} \tag{9}\]  
Then, the first adjoint method \(\Phi_{h/2}\) is:  
\[\begin{array}{l}u_{n+\frac{1}{2}}=u_{n}+\frac{h}{2}\cdot f_{u}(u_{n},w_{n}); \\ w_{n+\frac{1}{2}}=w_{n}+\frac{h}{2}\cdot f_{w}(u_{n+\frac{1}{2}},w),\end{array} \tag{10}\]  
and the second adjoint method \(\Phi_{h/2}^{*}\) is:  
\[\begin{array}{l}w_{n+1}=w_{n+\frac{1}{2}}+\frac{h}{2}\cdot f_{w}(u_{n+\frac{ 1}{2}},w_{n+1});\\ u_{n+1}=u_{n+\frac{1}{2}}+\frac{h}{2}\cdot f_{u}(u_{n+1},w_{n+1}).\end{array} \tag{11}\]  
The first adjoint method is fully explicit, and the second adjoint method contains implicitness in the diagonal elements of the system matrix.  
Note that the division of time step \(h\) directly by \(2\) in Equation (7) is just a special case of more generalized integrator. Let us introduce symmetry coefficient \(s\in(0,1)\) and apply it to step size \(h\) to split it into two arbitrary parts:  
\[\begin{array}{l}h_{1}=h\cdot s;\\ h_{2}=h\cdot(1-s).\end{array} \tag{12}\]  
Then, a family of adjoint semi-implicit methods with variable symmetry, or VSCD methods, appears:  
\[\Psi_{h,s}=\Phi_{h_{1}}\circ\Phi_{h_{2}}. \tag{13}\]  
These methods retain some of the basic properties of the CD method and provide the affine transform of the simulated system phase space without breaking the chaotic regime. Graphical interpretation of a certain VSCD method is presented in Figure 1.