Designing Hyperchaotic Cat Maps with any Desired Number of Positive Lyapunov Exponents

Introduction

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 efficient 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 evaluations show that, compared with existing methods, the proposed method can construct n-D HCMs with lower computation complexity and their outputs demonstrate strong randomness and complex ergodicity.

Proposed n-D Cat map generation method

First, one can construct basic n-D Cat matrix by iteratively expanding an initial matrix with a 2×2 matrix. The construction procedure is described as follows:

  • Step1: Set the initial matrix as a 1×1 special matrix if n is odd, otherwise set it as the parametric 2-D Cat matrix
  • Step2: Place the initial matrix and another parametric 2-D Cat matrix in the main diagonal of a 2×2 block matrix. To increase the parameter space of the obtained Cat matrix, the locations of the two matrices are assigned randomly.
  • Step 3: As for the other two matrix blocks in the antidiagonal direction, randomly select one and set it of fixed value zero. Then, elements of the other matrix block are assigned with any integer randomly.
  • Step4: Set the current composite matrix as the initial matrix;
  • Step 5: Repeat Step 2 through Step 4 [(𝑛−1)/2]−1 times.

Then, the final n-D Cat matrix can be obtained by performing similarity transformation on a basic n-D Cat matrix with another.

Performance evaluations

Information entropy can be used to test the randomness of outputs of Cat maps generated by different generation methods. Figure 1 shows the average information entropy values of n-D Cat maps generated by different methods.  It displays that the proposed method can generate

n-D Cat maps with much larger average information entropy values than other existing methods. This further verifies that the proposed method can generate n-D Cat maps with extremely good randomness.

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Figure 1. Average information entropy values of n-D Cat maps generated by different methods. (a) dimension n=3 and the number of finite states Nϵ{3,4,…,15}; (b) the number of finite states N=3 and dimension nϵ{3,4,…,10}.

Correlation dimension describes the space dimensionality of a set of points as a type of fractal dimension. For a dynamic system, its attractor strangeness or degrees of freedom can be measured by correlation dimension. Figure 2(a) shows the average correlation dimension values of 3-D Cat maps generated by different methods. As can be seen from the figure, the 3-D Cat maps generated by the proposed method have trajectories with higher correlation dimension values than the other eight methods. Figure 2(b) depicts their average values. The results further prove that the proposed method can generate n-D Cat maps with bigger correlation dimensions and better ergodicity.

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Figure 2. Average information entropy values of n-D Cat maps generated by different methods. (a) dimension n=3 and the number of finite states Nϵ{3,4,…,15}; (b) the number of finite states N=3 and dimension nϵ{3,4,…,10}.