2.2.2. Lyapunov Exponent

One of the most important features of a chaotic system is a strong sensitivity to initial values. The Lyapunov exponent (LE) [26] provides a quantitative description of the initial state sensitivity of a chaotic system. A maximum Lyapunov exponent of the chaotic map greater than 0 indicates that the system is in a chaotic state. For a two-dimensional chaotic system, if the system's two Lyapunov exponents are greater than 0, then the system is in a hyperchaotic state.

In Figure 4a–c, the Lyapunov exponents of TDSCL, DLCL, and LSMCL are calculated. From these diagrams, TDSCL displays hyperchaotic behavior when approximately *μ* ∈ (0.05, 1]. When *μ* = 1, the maximum Lyapunov exponent of TDSCL is close to 9.2. Therefore, compared with the other two maps, TDSCL not only has a larger chaotic state interval, but also a larger Lyapunov exponent in a large continuous interval. Compared with DLCL and LSMCL, TDSCL is more sensitive to small changes in the initial value of the system and has better unpredictability.

**Figure 4.** Lyapunov exponent: (**a**) TDSCL, (**b**) DLCL, and (**c**) LSCML.

#### 2.2.3. Permutation Entropy

The permutation entropy can be used to measure the complexity of chaotic sequences [27]. For a given chaotic system, an entropy of the generated chaotic sequence close to 1 indicates that the chaotic system has unpredictability. As shown in Figure 5, the PE of DLCL is close to 1, only when *μ* in the interval of [0.7, 1], and the permutation entropy of LSMCL is always less than 0.8. The permutation entropy value of TDSCL is very close to 1 when *μ* ∈ [0.1, 1]. This indicates that the chaotic sequences generated by TDSCL have more complex dynamic behaviour.

**Figure 5.** Permutation entropy for (**a**) TDSCL, (**b**) DLCL, and (**c**) LSCML.
