Some Properties of a Discrete Lorenz System Obtained by Variable Midpoint Method and Its Application to Chaotic Signal Modulation

Various discretization effects caused by applying numerical integration techniques to continuous chaotic systems are broadly studied in nonlinear science. Along with the negative impact on the precision of the various finite-difference schemes, such effects may have surprisingly fruitful practical applications, e.g. pseudo-random number generation, image encryption with improved diffusion and confusion properties, chaotic path planning, and many others. One such application is chaos-based communication systems which gained attention in recent decades due to their high covertness and broadband transmission capability. A crucial problem in the design of chaotic communication systems is the modulation of carrier signals. Due to the noise-like properties of chaotic signals, they can barely be modulated using the same methods as conventional harmonic signals. Thus, developing new modulation techniques is of great interest in the field of chaotic communications. In this study, we investigate the discrete model of the Lorenz oscillator obtained using controllable midpoint numerical integration and develop a novel modulation technique for chaos-based communication systems. We discover and analyze the multistability phenomenon in the dynamics of the investigated finite-difference Lorenz model through bifurcation, the basin of attraction, and Lyapunov spectrum analysis procedures. Using a specially designed testbench, we explicitly show that the proposed modulation method outperforms commonly used parametric modulation and is nearly equal to the state-of-the-art symmetry-based modulation in terms of covertness and noise resistivity. In addition, the proposed modulation technique is much easier to implement using computer arithmetics, especially in fixed-point hardware. The reported results may be efficiently applied to designing advanced chaos-based communications systems or improving the characteristics of existing communication system architectures.