Mathematical Problems in Engineering

Volume 2015, Article ID 121359, 12 pages

http://dx.doi.org/10.1155/2015/121359

## A Generalized Stability Theorem for Discrete-Time Nonautonomous Chaos System with Applications

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received 17 April 2015; Accepted 7 July 2015

Academic Editor: Ricardo Aguilar-López

Copyright © 2015 Mei Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Firstly, this study introduces a definition of generalized stability (GST) in discrete-time nonautonomous chaos system (DNCS), which is an extension for chaos generalized synchronization. Secondly, a constructive theorem of DNCS has been proposed. As an example, a GST DNCS is constructed based on a novel 4-dimensional discrete chaotic map. Numerical simulations show that the dynamic behaviors of this map have chaotic attractor characteristics. As one application, we design a chaotic pseudorandom number generator (CPRNG) based on the GST DNCS. We use the SP800-22 test suite to test the randomness of four 100-key streams consisting of 1,000,000 bits generated by the CPRNG, the RC4 algorithm, the ZUC algorithm, and a 6-dimensional CGS-based CPRNG, respectively. The numerical results show that the randomness performances of the two CPRNGs are promising. In addition, theoretically the key space of the CPRNG is larger than 2^{1116}. As another application, this study designs a stream avalanche encryption scheme (SAES) in RGB image encryption. The results show that the GST DNCS is able to generate the avalanche effects which are similar to those generated via ideal CPRNGs.

#### 1. Introduction

Chaos, characterized by its deterministic, unpredictable features and extremely sensitive dependence on initial conditions, stems from nonlinear systems (e.g., see [1–3]). During the last three decades, chaos theory has been developed to model complex nature and social phenomena by using quite simple mathematical models. Thus it has captured much attention of the scientific community for predicting the behavior of systems in the real world.

Nonautonomous discrete systems were introduced in [4]; as we can see, they also appear connected to some nonautonomous difference equations (e.g., see [5, 6]). A lot of natural questions concerned the nonautonomous dynamical systems; therefore the research of the nonautonomous system is recently very intensive (e.g., see [7–9]). However, there is no much research on discrete nonautonomous chaotic systems.

Mathematically chaos synchronization (CS) means that the trajectories of two different chaotic systems exhibit identical phenomena with time evolution. Synchronization phenomenon is a kind of typical collective behaviors that could be found in many physical, biological, and engineering systems (e.g., see [10–16]).

Chaos generalized synchronization (CGS) means that, with time evolution, the trajectories of two different chaotic systems tend to become identical with respect to a transformation in a specific domain. Therefore CGS has more general meaning than CS. The study of CGS has also attracted much attention (e.g., see [17–25]).

Generally speaking, there are different methods such that two systems achieve generalized synchronization such as design control laws to force coupled systems to satisfy a prescribed functional relation [26–28]. In series papers [29–32], we have studied the general representations of two systems to achieve GS.

Since the pioneer work of Pecora and Carroll on CS secure communication [33], CS and CGS have been used as new tools in secure communications and have been used as designs of pseudorandom number generators (e.g., [17–25, 34–37]). Research along this line is promising.

This paper extends the concept of the generalized synchronization [38] to generalized stability (GST) for discrete-time nonautonomous chaos system (DNCS), and then we propose a corresponding GST theorem. Using the GST theorem helps design a novel nonautonomous chaotic discrete system and construct a chaos-based pseudorandom number generator (CPRNG). We test the randomness of the CPRNG, the RC4 algorithm, the ZUC algorithm [39], and a 6-dimensional CGS-based CPRNG2 [40] by the SP800-22 test suite of the INST [41], respectively. At last, as an application, by using the CPRNG and the SAES a RGB image has been encrypted in communication.

The rest of this paper is organized as follows. Section 2 introduces the definition and the theorem of GST. Section 3 presents a novel 4-dimensional nonautonomous chaotic discrete system and an 8-dimensional GST system and simulates the dynamic behaviors of the GST system. Section 4 designs the CPRNG and makes the statistic tests for the CPRNG, the RC4 algorithm, the ZUC algorithm, and the 6-dimensional CGS-based CPRNG2, respectively. An image encryption example of the CPRNG and the SAES is introduced in Section 5. Finally, Section 6 presents some concluding remarks.

#### 2. Definition and Theorem of GST

A point of view states that two events with relationship of cause and effect might be described via CGS for two systems. Motivated by CGS, let us introduce the concept of GST.

*Definition 1. *Consider two systemswhereIf there exists a transformation , wherefor , there exist and such that all trajectories of (1) and (2) with initial conditions satisfyThen the systems in (1) and (2) are said to be in GST with respect to the transformation . System (1) is called the driving system; system (2) is said to be the driven system.

In order to construct the novel DNCS with the GST property, we present the following.

Theorem 2. *Let , and be defined by (3); . Suppose that**If two systems (1) and (2) are in GST via the transformation , if and only if, the function given in (2) has the following form:**and the function**guarantees that the zero solution of the following error equation is stable on the open set defined by Definition 1:*

*Proof. *DenoteThenTherefore, two dynamic systems (1) and (2) are in GST via the transformation , if and only if the function makes the zero solution of the error equation (9) stable. This completes the proof.

*3. A Novel GST DNCS*

*The construction of the novel GST DNCS is divided into 3 steps: (1) construct a chaotic system ; (2) introduce a transformation ; (3) construct a system such that guarantees the zero solution of the error equation (9) stable.*

*Step 1. *Construct a novel 4-dimensional nonautonomous chaotic system as follows:The calculated Lyapunov exponents of system (12) are ; therefore, system (12) is a chaotic system.

*Now, select the following initial condition:*

*The chaotic orbits of the state variables for the first 5000 iterations are shown in Figure 1. The evolution of state variables, , , , and , is shown in Figure 2. Observe that the dynamic behaviors of the chaotic map demonstrate chaotic attractor characteristics.*