A novel method based on the pseudo-orbits to calculate the largest Lyapunov exponent from chaotic equations

Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps-Introduction to the Method

Controlling stability of dynamical systems is one of the most important challenges in science and engineering. Hence, there appears to be continuous need to study and develop numerical algorithms of control methods. One of the most frequently applied invariants characterizing systems' stability are Lyapunov exponents (LE). When information about the stability of a system is demanded, it can be determined based on the value of the largest Lyapunov exponent (LLE). Recently, we have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. The present article introduces new methods of LLE estimation for continuous systems and maps. We have shown that application of our approaches will introduce significant improvement of the efficiency. We have also proved that our approach is simpler and more efficient than commonly applied algorithms. Moreover, as our approach works in the case of dynamical maps, it also enables an easy application of this method in noncontinuous systems. We show comparisons of efficiencies of algorithms based our approach. In the last paragraph, we discuss a possibility of the estimation of LLE from maps and for noncontinuous systems and present results of our initial investigations.

Image encryption based on the pseudo-orbits from 1D chaotic map

Chaotic systems have been extensively applied in image encryption as a source of randomness. However, dynamical degradation has been pointed out as an important limitation of this procedure. To overcome this limitation, this paper presents a novel image encryption scheme based on the pseudo-orbits of 1D chaotic maps. We use the difference of two pseudo-orbits to generate a random sequence. The generated sequence has been successful in all NIST tests, which implies it has adequate randomness to be employed in encryption process. Confusion and diffusion requirements are also effectively implemented. The usual low key space of 1D maps has been improved by a novelty procedure based on multiple perturbations in the transient time. A factor using the plain image is one of the perturbation conditions, which ensures a new and distinct secret key for each image to be encrypted. The proposed encryption scheme has been efficaciously verified using the Lena, Baboon, and Barbara test images.