This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CYBERNETICS 1 Designing Hyperchaotic Cat Maps With Any Desired Number of Positive Lyapunov Exponents Zhongyun Hua, Member, IEEE, Shuang Yi, Student Member, IEEE, Yicong Zhou, Senior Member, IEEE, Chengqing Li, Senior Member, IEEE, and Yue Wu, Member, IEEE Abstract—Generating chaotic maps with expected dynamics of users is a challenging topic. Utilizing the inherent relation between the Lyapunov exponents (LEs) of the Cat map and its associated Cat matrix, this paper proposes a simple but effi- cient method to construct an n-dimensional (n-D) hyperchaotic Cat map (HCM) with any desired number of positive LEs. The method first generates two basic n-D Cat matrices iteratively and then constructs the final n-D Cat matrix by performing similarity transformation on one basic n-D Cat matrix by the other. Given any number of positive LEs, it can generate an n-D HCM with desired hyperchaotic complexity. Two illustrative examples of n-D HCMs were constructed to show the effectiveness of the proposed method, and to verify the inherent relation between the LEs and Cat matrix. Theoretical analysis proves that the parameter space of the generated HCM is very large. Performance evalua- tions show that, compared with existing methods, the proposed method can construct n-D HCMs with lower computation com- plexity and their outputs demonstrate strong randomness and complex ergodicity. Index Terms—Cat map, Cat matrix, chaotification, hyper- chaotic behavior, Lyapunov exponent (LE). I. I NTRODUCTION C HAOTIC behaviors can be observed in all kinds of natural and non-natural phenomena, such as weather forecasting in meteorology [1] and population growth in sociology [2]. Dynamic systems are mathematical concepts describing chaotic behaviors, and attract intensive atten- tions [3], [4]. A dynamic system demonstrating chaotic behav- ior has properties of ergodicity, unpredictability, and sensitivity Manuscript received May 20, 2016; revised October 9, 2016; accepted December 8, 2016. This work was supported in part by the Macau Science and Technology Development Fund under Grant FDCT/016/2015/A1, and in part by the Research Committee at University of Macau under Grant MYRG2014- 00003-FST and Grant MYRG2016-00123-FST. This paper was recommended by Associate Editor M. Forti. (Corresponding author: Yicong Zhou.) Z. Hua is with the School of Computer Science and Technology, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China, and also with the Department of Computer and Information Science, University of Macau, Macau 999078, China (e-mail: huazyum@gmail.com). S. Yi and Y. Zhou are with the Department of Computer and Information Science, University of Macau, Macau 999078, China (e-mail: yishuang0227@gmail.com; yicongzhou@umac.mo). C. Li is with the College of Information Engineering, Xiangtan University, Xiangtan 411105, China (e-mail: drchengqingli@gmail.com). Y. Wu is with the Information Sciences Institute, University of Southern California, CA 90292 USA (e-mail: yue_wu@isi.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2016.2642166 to change of initial condition and/or control parameter. So, strong chaotic behavior is very desired in many real applica- tions [5] and Lyapunov exponent (LE) is a widely used indi- cator to quantitatively measure it [6]–[8]. If a dynamic system owns one positive LE, it is considered chaotic. Furthermore, if a high-dimensional (HD) dynamic system has at least two positive LEs, it can demonstrate hyperchaotic behavior, and its attractors irregularly distribute in several dimensions [9]. Thus, its behavior is usually much more complex and it has much unpredictable topological structure than that owning only one positive LE [10], [11], making hyperchaotic sys- tems are more attractive, especially in the field of chaos-based cryptography [12]–[14]. Existing hyperchaotic systems can be classified into two categories: 1) discrete-time system and 2) continuous-time sys- tem. A discrete-time system is commonly defined by a differ- ence equation and it can be implemented through an iterative procedure. In contrast, a continuous-time system is usually represented by a partial and/or ordinary differential equation. In the past decades, a wide body of research has been devoted to developing continuous-time hyperchaotic systems using var- ious strategies: state feedback control [15], [16], linear or nonlinear coupling [17], [18], and other techniques [19]–[23]. It deserves noting that Shen et al. [20], [21] proposed a sys- tematic methodology for constructing hyperchaotic systems with multiple positive LEs and further developed a simple model to design hyperchaotic systems with any desired num- ber of positive LEs. Using the methods given in [20] and [21], one can construct a continuous-time hyperchaotic system with multiple positive LEs, and thus can customize it with the expected complex behavior. Compared with continuous-time systems, the occurrence of chaotic behaviors of discrete- time systems can be directly observed. Thus, the latter has many advantages in performance analysis and hard- ware/software implementation, making designing discrete- time hyperchaotic systems with multiple positive LEs very attractive [24]. As a special discrete-time chaotic system, Arnold’s Cat map not only has common properties of discrete-time chaotic systems, but also possesses many exclusive characteris- tics, including adaptability to arbitrary finite precision [25], reversibility [26], area preserving [25], Anosov diffeomor- phism and structural stability [27]. Such nice properties let Cat map receive many researchers’ attentions [28]–[30]. It has been used in many applications, such as the cryptographic applications [31]–[33] and steganography [34]. Besides, Cat 2168-2267 c  2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.