One-Dimensional Nonlinear Model for Producing Chaos

Motivated by the concept of circuit design in digital circuit, this paper proposes a one-dimensional (1D) nonlinear model (1D-NLM) for producing 1D discrete-time chaotic maps. Our previous works have designed four nonlinear operations of generating new chaotic maps. However, they focus only on discussing individual nonlinear operations and their properties, but fail to consider their relationship among these operations. The proposed 1D-NLM includes these existing nonlinear operations, develops two new nonlinear operations, discusses their relationship among different nonlinear operations, and investigates the properties of different combinations of these operations. To show the effectiveness of 1D-NLM in generating new chaotic maps, as examples, we provide four new chaotic maps and study their dynamics properties from following three aspects: equilibrium point, stability, and bifurcation diagram. Performance evaluations are provided using the Lyapunov exponent, Shannon entropy, correlation dimension, and initial state sensitivity. The evaluation results show that these new chaotic maps have more complex chaotic behaviors than existing ones. To demonstrate the performance of 1D-NLM in practical applications, we use a pseudo-random number generator (PRNG) to compare new and existing chaotic maps. The randomness test results indicate that new chaotic map generated by 1D-NLM shows better performance than existing ones in designing PRNG.