Variable-Step Semi-Implicit Solver with Adjustable Symmetry and Its Application for Chaos-Based Communication

In this article, we introduce a novel approach to numerical integration based on a modified composite diagonal (CD) method, which is a variation of the semi-implicit Euler–Cromer method. This approach enables the finite-difference scheme to maintain the dynamic regime of the solution while adjusting the integration time step. This makes it possible to implement variable-step integration. We present a variable-step MCD (VS-MCD) version with a simple and stable Hairer step size controller. We show that the VS-MCD method is capable of changing the dynamics of the solution by changing the symmetry coefficient (reflecting the ratio between two internal steps within the composition step), which is useful for tuning the dynamics of the obtained discrete model, with no influence of the appropriate step size. We illustrate the practical application of the developed method by constructing a direct chaotic communication system based on the Sprott Case S chaotic oscillator, demonstrating high values in the largest Lyapunov exponent (LLE). The tolerance parameter of the step size controller is used as the modulation parameter to insert a message into the chaotic time series. Through numerical experiments, we show that the proposed modulation scheme has competitive robustness to noise and return map attacks in comparison with those of modulation methods based on fixed-step solvers. It can also be combined with them to achieve an extended key space.