A novel perturbation method to reduce the dynamical degradation of digital chaotic maps

A chaotic map, which is realized on finite precision device, such as computer, will suffer dynamical degradation. Such chaotic maps cannot be regarded as rigorous chaos anymore, since their chaotic characteristics are degraded, and naturally, these kinds of chaotic maps are not secure enough for cryptographic use. Therefore, in this paper, a novel perturbation method is proposed to reduce the dynamical degradation of digital chaotic maps. Once the state is repeated during the iteration, the parameter and state are both perturbed to make the state jump out from a cycle. This method is convenient to implement without any external sources and can be used for different kinds of digital chaotic maps. The most widely used logistic map is used as an example to prove the effectiveness of this method. Several numerical experiments are provided to prove the effectiveness of this method. Under the same precision, the number of iterations when entering a cycle and the period of the improved map are greater than those of the original one. The complexity analysis shows that the improved map can get an ideal complexity level under a lower precision. All these results prove that this perturbed method can greatly improve the dynamical characteristics of original chaotic map and is competitive with other remedies. Furthermore, we improve this method by using a variable perturbation, where the perturbation is affected according to the number of iteration steps. Numerical experiments further prove that this improved perturbation method has a better performance in suppressing dynamical degradation.