On Chaos in the Fractional-Order Discrete-Time Unified System and Its Control Synchronization

In this paper, we propose a fractional map based on the integer-order unified map. The chaotic behavior of the proposed map is analyzed by means of bifurcations plots, and experimental bounds are placed on the parameters and fractional order. Different control laws are proposed to force the states to zero asymptotically and to achieve the complete synchronization of a pair of fractional unified maps with identical or nonidentical parameters. Numerical results are used throughout the paper to illustrate the findings.


Introduction
Chaotic dynamical systems have attracted a considerable level of attention over the last three decades due to the wide range of applications. Chaotic systems can be divided into two main categories: continuous-time and discrete-time. In this paper, we are interested in discrete-time chaotic systems, which are also referred to as chaotic maps. Over the years, a number of different chaotic maps have been proposed in the literature and applied in different fields including the Henon map [1], the Lozi system [2], the generalized Henon map [3], the Baier-Klein system [4], the Stefanski map [5], the Rössler map [6] and the Wang map [7]. Since the Henon and Lozi maps are the earliest discrete-time chaotic systems, they have been studied extensively by a vast number of researchers. One of the interesting studies is that of Zeraoulia and Sprott [8], where the authors proposed a new chaotic map as a combination of the the Henon and Lozi maps named the unified map. In this unified map, they included a parameter that can be varied between zero and one to alter the dynamics of the map with one end of the spectrum belonging to the Henon map and the other to the Lozi map. A summary of the unified map's bifurcation and dynamics will be presented later on in Section 2.1.
The general idea of a fractional derivative for a continuous-time real function can be attributed to Gottfried Leibniz in a letter he wrote back in 1695. The proper definition of such a derivative, however, was not formulated until the late 19th Century as a result of the collective works of Liouville, Grunwald, Letnikov and Riemann. Fractional calculus has since been shown to be useful in the

Results
Since the subject of fractional discrete calculus is still relatively new and the notation has not yet been settled, we start with a general description of the notation and stability results that will aid the reader in understanding the analysis to come. We note that similar to continuous-time fractional calculus, where numerous definitions exist for the fractional derivative of a function, the Caputo one is the most used today. Throughout this paper, the notation C ∆ υ a X (t) is employed to denote the υ-Caputo type delta difference of a function X (t) : N a → R with N a = {a, a + 1, a + 2, ...} [12] defined as: where υ ∈ N is the fractional order, t ∈ N a+n−υ , and n = [υ] + 1. In (11), the υ-th fractional sum of ∆ n s X (t) is defined similar to [11] as: with υ > 0, σ(s) = s + 1. The term t υ denotes the falling function defined in terms of the Gamma function Γ as: The following two theorems describe the numerical formula for a fractional discrete map, as well as the fractional discrete direct Lyapunov method, respectively. In order to use Theorem 2 later on, we will require the inequality described by Lemma 1 below. Theorem 1. [36] For the delta fractional difference equation: the equivalent discrete integral equation can be obtained as: where: Theorem 2. [14] If there exists a positive definite Lyapunov function V (X (t)) such that: then the trivial solution of the system is asymptotically stable. Lemma 1. [14] For X(t) = (x 1 (t), ..., x n (t)) T , 0 < υ ≤ 1 and ∀t ∈ N a+1−υ , the following inequality holds: The following subsections describe the main findings of our study. Section 2.1 describes the signal model of the proposed fractional unified map and investigates its entropy and chaotic dynamics by means of phase portraits and bifurcation plots. Sections 2.2 and 2.3 describe the proposed stabilization and synchronization schemes.

Chaos in the Fractional-Order Unified Map
In [8], Zeraoulia and Sprott proposed a new unified piecewise chaotic map: x (n + 1) = 1 − 1.4 f α (x (n)) + y (n) , y (n + 1) = 0.3x (n) (9) where the bifurcation parameter α is in the range 0 ≤ α ≤ 1 and function f α is defined as: The importance of this unified map stems from the fact that setting α to zero yields the classical Hénon map depicted in Figure 1a and given by: On the other hand, when α = 1, we end up with the classical Lozi map depicted in Figure 1b and defined as: x (n + 1) = 1 − 1.4 |x (n)| + y (n) , y (n + 1) = 0.3x (n) . What is more interesting is that they demonstrated how the system exhibits a chaotic behavior for 0 ≤ α ≤ 1. Figure 2 shows the bifurcation diagram with the critical parameter α being changed in steps of ∆α = 0.001. System (9) may be rewritten in the form: Then, using the discrete fractional calculus described in the previous section, we may define the fractional unified map as: for t ∈ N a+1−υ , where C ∆ υ a x (t) and C ∆ υ a y (t) are the Caputo-like delta differences of states x (t) and y (t), respectively, and 0 < υ ≤ 1 is the fractional order.
Following Theorem 2, using the discrete kernel function: and assuming that a = 0, the numerical formulas for the fractional map (14) may be obtained as: In [8], it was reported that while the bifurcation parameter α is close to zero, the function f α defined in (10) behaves similar to the term x 2 (n). and when α is close to one it behaves similar to the absolute function |x (n)|. Since the values α = 0 and α = 1 belong to the fractional Hénon and Lozi maps, which have been studied previously in the literature, we choose to ignore them and investigated the numerical Formula (16) over the range 0 < α < 1. Considering the values α = 0.8 and α = 0.2, Figure   With the same parameters and the same initial condition, using numerical Formula (16), the fractional map is shown in Figure 6 for different fractional orders υ. As υ decreases, the phase plane of the fractional maps changes its shape until it completely disappears. For α = 0.2, the minimum fractional order that produced a bounded attractor is υ = 0.88. On the other hand, when α = 0.8, we see that we obtain a bounded attractor as far as υ = 0.4. The "approximate entropy" was proposed by Pincus to present the complexity of time series [37,38]. Approximate entropy measurement is helpful and applied to discover different chaotic systems [39,40]. The brief computation of the approximate entropy is presented as follows. By selecting a sequence of N data samples s (1) , s (2) , ..., s (N), a sequence of vectors is constructed: in which m is the embedding dimension. The distance between the vector S (i) and the vector S (j) is d (S (i) , S (j)). It is noted that we choose the following threshold [37,38]: where std(s) denotes the standard deviation of the data s. As a result, we can calculate the number of vectors (K) having distance d (S (i) , S (j)) ≤ r [37,38]. The approximate entropy is: in which φ m (r) is given by: and C m i (r) is defined by: We have calculated the approximate entropy (ApEn) for the fractional-order unified map, and the results are reported in Table 1. For α = 0.2, the complexity of the fractional-order unified map is reduced when the value of υ is decreased to 0.88. For α = 0.8, when reducing the value of υ from 0.98 to 0.88, the complexity of the map varies. The results display the changes of the phase portraits in Figure 6. To investigate the chaotic behavior of the fractional map, we study the bifurcation of the parameter α with the step size ∆α = 0.001. Figure 7 illustrates the results. The figure demonstrates clearly the chaotic behavior of the fractional map. When υ = 1, chaos is observed in the interval [0, 1]. A decrease in the fractional order υ leads to a decrease in the interval, where chaos is apparent. As shown in Figure 8, for υ = 0.4, chaos is seen in the interval α ∈ [0.799, 0.95]. A better understanding of the route to chaos can be seen in Figure 9, where the range of α is increased to 1.8. However, for α ∈ [0, 1.8], a border-collision bifurcation scenario is observed. The map begins with a fully-developed chaotic regime, and increasing α leads to the disappearance of the chaotic band and the appearance of a four-period orbit. Experiments have also shown that even with fractional orders υ less than one, the fractional map still behaves in a similar manner to the standard case with the one exception that the chaotic interval varies with υ.

Control of the Fractional-Order Unified Map
In this section, we are interested in a one-dimensional adaptive control law that forces the states of the proposed two-dimensional fractional unified system to zero asymptotically. Chaos control is an important subject, particularly in engineering, as it may have applications in stabilizing a chaotic dynamical system, e.g., a two degree of freedom robot arm. Theorem 3. The 2D fractional-order unified chaotic map can be controlled under the 1D control law: Proof. The controlled fractional-order unified chaotic map can be described as follows: Substituting the proposed control law (22) in (23) leads to the new dynamics: Since the aim of the control law (22) is to force the two states towards zero asymptotically, what we want to do is to show that the zero solution of this resulting controlled system dynamics is globally asymptotically stable. In order to do so, we employ the fractional discrete Lyapunov method described in Theorem 2. We propose the Lyapunov function: leading to: Using the inequality of Lemma 1, we see that: Therefore, by Theorem 2, we conclude that the zero solution of System (23) is in fact globally asymptotically stable, and thus, the fractional unified map is controlled.
In order to put Theorem 3 to the test, a MATLAB script was run taking υ = 0.95 and a = 0 and following the time evolution of the states of (23). The result in Figure 10 clearly shows how the states progress towards zero.

Synchronization
Perhaps synchronization is the most interesting aspect of chaotic dynamical systems in general, as it allows many applications of chaos. In this section, we present two synchronization schemes related to the proposed fractional unified system. The first deals with two fractional identical unified maps and the second with different maps. The terms identical and different in this context refer to the parameter α having the same values or different values, respectively.

Synchronization of Identical Fractional Unified Maps
Let us consider the master system described for t ∈ N a+1−υ by: . (28) Note that the subscript m in the states refers to the master. As for the slave system, we choose the exact same map, but use a subscript s instead and with the addition of a controller for the first state, i.e.,: where u is a controller to be determined later. The aim of synchronization is to force the error system: to zero asymptotically. The following theorem presents the proposed control law.
Theorem 4. Subject to: where |x s (t)| = |x m (t)| ≤ l, the master and slave pair is globally synchronized.
Proof. The fractional difference equations related to error System (30) can be given by: Substituting the control law (31) yields the simplified system: Using the same Lyapunov functional from Theorem 3 and employing Lemma 1, we have: It then follows directly from Theorem 2 that the zero solution of error System (33) is globally asymptotically stable. Hence, regardless of the initial conditions, the errors are guaranteed to converge towards zero asymptotically, meaning that the master and slave maps are synchronized.
The control law (31) was implemented in MATLAB with υ = 0.95 and a = 0. Over time, the synchronization error depicted in Figure 11 can be seen to converge towards zero.

Synchronization of Different Fractional Unified Maps
Let us, now, consider master System (28) with the slave: for t ∈ N a+1−υ and with β = α. The error system is defined in the same way as in (30). Theorem 5 below presents the one-dimensional control law that will generally synchronize the slave to the master regardless of the values of α and β.
Proof. This result can be proven in much the same way as Theorem 4 was using the same Lyapunov function and the error dynamics: leading to: Using the same previous Lyapunov function, we have: Again, it is easy to conclude that the zero error of (38) is globally asymptotically stable and that the master and slave are synchronized. Figure 12 shows the time evolution of the synchronization errors for the fractional maps with different parameters. We assumed a = 0 and υ = 0.95. Again, the errors can be easily shown to converge towards zero asymptotically.

Discussion
In this paper, we have proposed a fractional-order unified map based on the integer-order unified map developed by Zeraoulia and Sprott in [8] as a cross-over system between the Hénon and Lozi maps. The dynamics and bifurcations of the proposed map are discussed. Based on the results, we see that the bifurcation parameter α has an effect on the range of fractional orders for which chaos is observed. The lower α is, the lower the range becomes. In addition, we see that the fractional order also has an impact on the existence and shape of the chaotic behavior. This is what was referred to in [18] as an added degree of freedom. For instance, if the map is to be used in a secure communications or encryption setting, then it will allow for a wider range of pseudo-random keys. We have also calculated and presented the approximate entropy values of the fractional-order map for various values of the fractional order. Again, we see that the values of α and υ both have an impact on the entropy. This paper also proposed a one-dimensional adaptive control strategy that forces the states towards zero asymptotically. The convergence of the states to zero was established by means of the Lyapunov method and verified by means of numerical results. Furthermore, two one-dimensional synchronization schemes were proposed for master and slave fractional unified maps with identical or nonidentical parameters. Numerical results were presented to confirm the success of these synchronization schemes.