Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization

Ouannas, Adel; Khennaoui, Amina Aicha; Momani, Shaher; Grassi, Giuseppe; Pham, Viet-Thanh

doi:10.1063/5.0004884

Chaotic systems with no equilibrium are a very important topic in nonlinear dynamics. In this paper, a new fractional order discrete-time system with no equilibrium is proposed, and the complex dynamical behaviors of such a system are discussed numerically by means of a bifurcation diagram, the largest Lyapunov exponents, a phase portrait, and a 0–1 test. In addition, a one-dimensional controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Next, a synchronization control scheme for two different fractional order discrete-time systems with hidden attractors is reported, where the master system is a two-dimensional system that has been reported in the literature. Numerical results are presented to confirm the results.

I. INTRODUCTION

Recently, interest has grown in the subject of discrete fractional calculus and its application in science and engineering.^1,2 This gave rise to many fractional order chaotic discrete-time systems.^3–9 From what has been reported, the dynamical behaviors of fractional order discrete-time systems are heavily dependent on the fractional order, which introduces new degrees of freedom and makes them more suitable for secure communications and encryption.¹⁰ Research on control and synchronization of such systems has also been widely investigated.^11–15

Until now, great consideration has been given to the study of no equilibrium fractional order chaotic systems due to their practical application in engineering systems. The attractors associated with these systems are called “hidden attractors.” Such attractors are observed in nonlinear systems with no equilibrium point or with only stable points or in nonlinear systems with line equilibrium points. The theory and application of these systems are shown to be useful in the mathematical modeling of real world phenomena.^16,17 There are only a few investigations of hidden attractors in discrete-time chaotic systems.^18–21

Based on this consideration, a new fractional order discrete-time system with no equilibrium is developed. By the help of numerical simulation, we try to find out whether it is possible to have new hidden chaotic attractors in the proposed system. The rest of this paper is organized as follows: In Sec. II, preliminary definitions are presented, and the fractional order discrete-time system with no equilibrium is described. In Sec. III, the complex dynamics of the proposed system are investigated numerically by means of a bifurcation diagram, the largest Lyapunov exponent, and a 0–1 test. A one-dimensional control law is proposed in Sec. IV to stabilize the states of the system. Synchronization of different-dimensional systems with hidden attractors is studied in Sec. V. Section VI contains the conclusion.

II. THE FRACTIONAL MAP

In this section, we describe the construction of a new fractional order discrete-time system with hidden attractors. To do so, we first recall some necessary definitions and essential results from discrete fractional calculus theory. We consider N_a = {a, a + 1, a + 2, …} where a ∈ R. The ν-th Caputo like difference operator ${}^{C}Δ_{a}^{ν} X (t)$ of a function $X (t) : N_{a} \to R$ is defined as²²

\begin{matrix} {}^{C}Δ_{a}^{ν} X (t) & = & Δ_{a}^{- (n - ν)} Δ^{n} X (t) \\ = & \frac{1}{Γ (n - ν)} \sum_{s = a}^{t - (n - ν)} {(t - s - 1)}^{(n - ν - 1)} Δ_{s}^{n} X (s), \end{matrix}

(1)

where $ν \notin N$ is the fractional order, $t \in N_{a + n - υ}$ ⁠, and $n = ⌈ν⌉ + 1$ ⁠. The term $t^{(ν)}$ denotes the falling function defined in terms of the Gamma function Γ as

t^{(ν)} = \frac{Γ (t + 1)}{Γ (t + 1 - ν)} .

(2)

The definition of the ν − th fractional sum can be given according to Ref. 23 as

Δ_{a}^{- ν} X (t) = \frac{1}{Γ (ν)} \sum_{s = a}^{t - ν} {(t - s - 1)}^{(ν - 1)} X (s),

(3)

where $t \in N_{a + ν}$ ⁠, and ν > 0.

Now, we propose the following new fractional order discrete-time system:

\{\begin{matrix} {}^{C}Δ_{a}^{ν} x (t) = y (t - 1 + ν) - x (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} y (t) = z (t - 1 + ν) - y (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} z (t) = a_{1} x (t - 1 + ν) + a_{2} z (t - 1 + ν) \\ + a_{3} x (t - 1 + ν) y (t - 1 + ν) \\ + a_{4} z^{2} (t - 1 + ν) + a_{5} - z (t - 1 + ν), \end{matrix})

(4)

where 0 < ν < 1, a is the starting point, and a₁, a₂, a₃, a₄, and a₅ are some bifurcation parameters. System (4) has two quadratic nonlinear terms. The equilibrium points of the fractional order discrete-time system (4) are found by solving the following system of equations:

\{\begin{matrix} y_{f} - x_{f} = 0, \\ z_{f} - y_{f} = 0, \\ a_{1} x_{f} + a_{2} z_{f} + a_{3} x_{f} y_{f} + a_{4} z_{f}^{2} + a_{5} - z_{f} = 0 . \end{matrix})

(5)

By using simple calculation, Eq. (5) can be transformed into

(a_{3} + a_{4}) x_{f}^{2} + (a_{1} + a_{2} - 1) x_{f} + a_{5} = 0 .

(6)

Here, we focus on the existence of hidden attractors; for that, we only need to determine the cases where the second order equation, Eq. (6), has no solution. In particular, two cases are considered:

Case (A): When a₃ + a₄ = 0, a₁ + a₂ − 1 = 0, and a₅ ≠ 0, Eq. (6) has no solution. Therefore, there is no equilibrium in the fractional order discrete-time system (4).
Case (B): When ${(a_{2} + a_{1} - 1)}^{2} - 4 a_{5} (a_{3} + a_{4}) < 0$ and a₃ + a₄ ≠ 0, Eq. (6) has no solution. Similarly, there is no equilibrium in the fractional order discrete-time system (4).

These results show that the fractional order discrete-time system (4) can generate hidden attractors in both cases.

III. DYNAMIC ANALYSIS

We first recall Theorem III.1, which will allow us to derive the numerical formula of the fractional order discrete-time system (4). The proof of Theorem III.1 is given in Ref. 24.

Theorem III.1.

For the fractional difference equation

\{\begin{matrix} {}^{C}Δ_{a}^{ν} u (t) = f (t + ν - 1, u (t + ν - 1)), \\ Δ^{k} u (a) = u_{k}, n = ⌈ν⌉ + 1, k = 0,1, \dots, n - 1, \end{matrix})

(7)

and the equivalent discrete integral equation can be obtained as

\begin{matrix} u (t) & = & u_{0} (t) + \frac{1}{Γ (ν)} \sum_{s = a + n - ν}^{t - ν} {(t - σ (s))}^{(ν - 1)} \\ \times f (s + ν - 1, u (s + ν - 1)), t \in N_{α + n}, \end{matrix}

(8)

where

u_{0} (t) = \sum_{k = 0}^{n - 1} \frac{{(t - a)}^{(k)}}{Γ (k + 1)} Δ^{k} u (a) .

(9)

Remark III.1.

Taking into account that

\frac{{(t - σ (s))}^{(ν - 1)}}{Γ (ν)}

is a discrete kernel function that may be chosen as

\frac{Γ (t - s)}{Γ (ν) Γ (t - s - ν + 1)}

⁠. Assuming a = 0 and s + ν = j and limiting ourselves to 0 < ν < 1, the numerical formula can be obtained as

u (n) = u_{0} + \frac{1}{Γ (ν)} \sum_{j = 1}^{n} \frac{Γ (n - j + ν)}{Γ (n - j + 1)} f (j - 1, u (j - 1)) .

(10)

By using Theorem III.1, the numerical solution of the fractional order discrete-time system can be obtained as

\{\begin{matrix} x (n) = x (0) + \frac{1}{Γ (ν)} \sum_{j = 1}^{n} \frac{Γ (n - j + ν)}{Γ (n - j + 1)} (y (j - 1) - x (j - 1)), \\ y (n) = y (0) + \frac{1}{Γ (ν)} \sum_{j = 1}^{n} \frac{Γ (n - j + ν)}{Γ (n - j + 1)} (z (j - 1) - y (j - 1)), \\ z (n) = z (0) + \frac{1}{Γ (ν)} \sum_{j = 1}^{n} \frac{Γ (n - j + ν)}{Γ (n - j + 1)} (a_{1} x (j - 1) + a_{2} z (j - 1)) \\ (+ a_{3} x (j - 1) y (j - 1) + a_{4} z^{2} (j - 1) + a_{5} - z (j - 1)), \end{matrix})

(11)

where x(0), y(0), and z(0) are the initial states. These numerical formulas will allow us to examine the sensitivity of the fractional order discrete-time system throughout the remainder of this paper. A detailed investigation is presented in this section.

A. Bifurcation and Lyapunov exponents

To fully understand the dynamic characteristic of fractional order discrete-time system (4), standard nonlinear analysis techniques, such as bifurcation diagrams, largest Lyapunov exponents, and phase portraits are performed.

Case (A): Here, the fractional order discrete-time system generates chaotic attractors when a₁ = 1.7, a₂ = −0.7, a₃ = 1, a₄ = −1, and a₅ = −0.3 with initial conditions x₀ = 0.84, y₀ = 1.25, and z₀ = 1.46 and the fractional order ν = 0.9762. Figure 1 displays the hidden chaotic attractor in different phase space projections. We investigate the dynamics of the system as the order ν is varied. The bifurcation diagram and the corresponding largest Lyapunov exponent in the x–ν plane are shown in Fig. 2. Note that the largest Lyapunov exponents are approximated in a similar manner to the states in reference.²⁵ Figure 2 shows that the fractional order discrete-time system (4) displays a chaotic behavior for much of the range [0.9732, 0.985] ∪ [0.9908, 1] with small periodic windows at ν = 0.9832. To obtain a global view, we plot the phase portraits for ν = 1, ν = 0.9928, ν = 0.9832, and ν = 0.9762 in the x − y plane, and the results are shown in Fig. 3. This illustrates that when ν = 1, ν = 0.9928, and ν = 0.9762, the fractional order discrete-time system (4) generates hidden chaotic attractors, which is consistent with the corresponding bifurcation diagram and largest Lyapunov exponents shown in Fig. 2.

Now, we consider the effect of a₅ on the dynamic behavior of the fractional order discrete-time system (4) for two fractional order values. Figures 4(a) and 5(a) present the bifurcation diagrams of the system for ν = 0.9832 and ν = 0.9762, respectively. We find that the states of the fractional order discrete-time system (4) changes qualitatively with the variation in a₅ and ν. As the fractional order ν decreases to 0.9762, the chaotic motion decreases, and new periodic windows are observed in the interval [−0.2, −0.1]. This implies that the fractional order stabilizes the states of the fractional order discrete-time system. Although bifurcation plots are a useful tool in determining the existence of chaos and quantifying it, the most agreed upon tool is the largest Lyapunov exponent. Figures 4(b) and 5(b) show the estimated largest Lyapunov exponent for ν = 0.9832 and ν = 0.9762, respectively, to further confirm the results. Taking ν = 0.9762 as an example, the corresponding largest Lyapunov exponent is positive, which confirms that the fractional order discrete-time system has hidden chaotic attractors in this case.

Case (B): When the system parameters are fixed as a₁ = 1, a₂ = 0, a₃ = 0.3, a₄ = −0.2, and a₅ = 1 and for ν = 1, the fractional order discrete-time system behaves chaotically. To investigate the sensitivity of the novel system with respect to the fractional order, we fix a₁ = 1, a₂ = 0, a₃ = 0.3, a₄ = −0.2, and a₅ = 1 and vary ν in the range 0.6 ≤ ν ≤ 1. The bifurcation diagram and the largest Lyapunov exponent are shown in Fig. 6. As ν passes through 0.9904, system (4) becomes totally periodic. The periodic motion is confirmed by the largest Lyapunov exponent, as shown in Fig. 6(b). When ν ∈ (0.6344, 0.8731) the system is stabilized to periodic points, as shown in Fig. 7(e). In this case, the lowest order ν for which the fractional order discrete-time system generates chaos is 0.9904.

FIG. 1.

$FIG. 1. Hidden chaotic attractor of the fractional order discrete-time system (4) for ν = 0.9762 in the (a) x − y plane, (b) x − z plane, (c) y − z plane, and (d) x–y–z space.$

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Hidden chaotic attractor of the fractional order discrete-time system (4) for ν = 0.9762 in the (a) x − y plane, (b) x − z plane, (c) y − z plane, and (d) x–y–z space.

FIG. 2.

$FIG. 2. (a) Bifurcation diagram of the fractional order discrete-time system (4) vs ν for a1 = 1.7, a2 = −0.7, a3 = 1, a4 = −1, and a5 = −0.3 and {x0, y0, z0} = {0.84, 1.25, 1.46} and (b) largest Lyapunov exponent vs ν.$

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(a) Bifurcation diagram of the fractional order discrete-time system (4) vs ν for a₁ = 1.7, a₂ = −0.7, a₃ = 1, a₄ = −1, and a₅ = −0.3 and {x₀, y₀, z₀} = {0.84, 1.25, 1.46} and (b) largest Lyapunov exponent vs ν.

FIG. 3.

$FIG. 3. Phase diagrams of fractional order discrete-time system (4) using the parameter values a1 = 1.7, a2 = −0.7, a3 = 1, a4 = −1, and a5 = −0.3 and the initial condition {x0, y0, z0} = {0.84, 1.25, 1.46} for (a) ν = 1, (b) ν = 0.9928, (c) ν = 0.9832, and (d) ν = 0.9762.$

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Phase diagrams of fractional order discrete-time system (4) using the parameter values a₁ = 1.7, a₂ = −0.7, a₃ = 1, a₄ = −1, and a₅ = −0.3 and the initial condition {x₀, y₀, z₀} = {0.84, 1.25, 1.46} for (a) ν = 1, (b) ν = 0.9928, (c) ν = 0.9832, and (d) ν = 0.9762.

FIG. 4.

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Fractional order discrete-time system (4) with a varying parameter a₅ and setting the parameter values as a₁ = 1.7, a₂ = −0.7, a₃ = 1, and a₄ = −1 for order ν = 0.9832 shows (a) the bifurcation diagram and (b) estimated Lyapunov exponents when a₅ = −0.3.

FIG. 5.

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Fractional order discrete-time system (4) with a varying parameter a₅ and setting the parameter values as a₁ = 1.7, a₂ = −0.7, a₃ = 1, and a₄ = −1 for order ν = 0.9762 shows (a) the bifurcation diagram and (b) estimated Lyapunov exponents when a₅ = −0.3.

FIG. 6.

$FIG. 6. (a) Bifurcation diagram of the fractional order discrete-time system (4) vs ν for a1 = 1, a2 = 0, a3 = 0.3, a4 = −0.2, and a5 = 1 and {x0, y0, z0} = {1.1, 1.28, −1.78} and (b) largest Lyapunov exponent vs ν.$

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(a) Bifurcation diagram of the fractional order discrete-time system (4) vs ν for a₁ = 1, a₂ = 0, a₃ = 0.3, a₄ = −0.2, and a₅ = 1 and {x₀, y₀, z₀} = {1.1, 1.28, −1.78} and (b) largest Lyapunov exponent vs ν.

FIG. 7.

$FIG. 7. Phase diagrams of fractional order discrete-time system (4) using the parameter values a1 = 1, a2 = 0, a3 = 0.3, a4 = −0.2, a5 = 1, and the initial condition {x0, y0, z0} = {1.1, 1.28, −1.78} for (a) ν = 1, (b) ν = 0.9871, (c) ν = 0.9084, and (d) ν = 0.894 and (e) evolution of the state x(n) for ν = 0.6546.$

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Phase diagrams of fractional order discrete-time system (4) using the parameter values a₁ = 1, a₂ = 0, a₃ = 0.3, a₄ = −0.2, a₅ = 1, and the initial condition {x₀, y₀, z₀} = {1.1, 1.28, −1.78} for (a) ν = 1, (b) ν = 0.9871, (c) ν = 0.9084, and (d) ν = 0.894 and (e) evolution of the state x(n) for ν = 0.6546.

B. 0–1 test

Another method to validate the existence of chaos in a fractional order discrete-time system is the 0–1 test method. The 0–1 test method for chaos is a binary test that reveals chaotic behavior in nonlinear systems, where the input is the series of data and the output is 0 or 1 depending on whether the dynamic is chaotic or non-chaotic.²⁶ When the system is chaotic, the output K approaches 1, and when the system is periodic, the output K approaches 0. In addition, the dynamics of the components $p_{c} (n) = \sum_{j = 1}^{n} x (j) \cos j c$ and $q_{c} (n) = \sum_{j = 1}^{n} x (j) \sin j c$ provide a visual test. Basically, if the dynamic is regular, then the behavior of trajectories in the (p–q) plane is bounded, whereas if the dynamic is chaotic, then the (p − q) trajectories depict a Brownian-like behavior.

The test has been applied directly to the series data obtained from numerical formula (11) with system parameters a₁ = 1.7, a₂ = −0.7, a₃ = 1, a₄ = −1, and a₅ = −0.3. The translation component (p − q) of the fractional order discrete-time system (4) is calculated and illustrated in Fig. 8. As it can be seen, the trajectories in the p − q plane depicts a Brownian behavior for ν = 0.9928 and ν = 0.9762, which confirms the chaotic behavior of the fractional order discrete-time system. On the other hand, Fig. 9 depicts the plot of K vs n for the fractional order ν = 0.998 and system parameters a₁ = 0, a₂ = 0, a₃ = −0.57, a₄ = 0.35, and a₅ = −1.27. It shows that the asymptotic growth rate K approaches 1 as n increases, indicating that the new system is chaotic. The corresponding hidden chaotic attractor is shown in Fig. 10.

FIG. 8.

$FIG. 8. Dynamics of the translation components (p, q) of the fractional order discrete-time system for (a) ν = 1, (b) ν = 0.9928, (c) ν = 0.9832, and (d) ν = 0.9762.$

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Dynamics of the translation components (p, q) of the fractional order discrete-time system for (a) ν = 1, (b) ν = 0.9928, (c) ν = 0.9832, and (d) ν = 0.9762.

FIG. 9.

$FIG. 9. The 0–1 test of the fractional order discrete-time system: the asymptotic growth rate K vs n for ν = 0.998 with bifurcation parameters a1 = 0, a2 = 0, a3 = −0.57, a4 = 0.35, and a5 = −1.27 and the initial condition {x0, y0, z0} = {−0.14, −0.7, 0.06}.$

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The 0–1 test of the fractional order discrete-time system: the asymptotic growth rate K vs n for ν = 0.998 with bifurcation parameters a₁ = 0, a₂ = 0, a₃ = −0.57, a₄ = 0.35, and a₅ = −1.27 and the initial condition {x₀, y₀, z₀} = {−0.14, −0.7, 0.06}.

FIG. 10.

$FIG. 10. Hidden chaotic attractor of the fractional order discrete-time system for ν = 0.998 in the (a) x − y plane, (b) x − z plane, (c) y − z plane, and (d) x − y − z space.$

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Hidden chaotic attractor of the fractional order discrete-time system for ν = 0.998 in the (a) x − y plane, (b) x − z plane, (c) y − z plane, and (d) x − y − z space.

Successively, the 0–1 test is applied to system (4) when a₁ = 1, a₂ = 0, a₃ = 0.3, a₄ = −0.2, a₅ = 1, and {x₀, y₀, z₀} = {1.1, 1.28, −1.78}. The translation component (p − q) depicts Brownian-like trajectories for ν = 0.9871 and bound trajectories for ν = 0.9084 and ν = 0.894 (see Fig. 11). These pictures, along with the phase diagrams reported in Fig. 7, indicate that the fractional order discrete-time system (4) has hidden chaotic attractors for ν = 0.9871 and shows a regular behavior for ν = 0.894 and ν = 0.9084, which conforms very well with the results given in Fig. 6.

FIG. 11.

$FIG. 11. Dynamics of the translation components (p, q) of the fractional order discrete-time system for a1 = 1, a2 = 0, a3 = 0.3, a4 = −0.2, and a5 = 1 with (a) ν = 1, (b) ν = 0.9871, (c) ν = 0.9084, and (d) ν = 0.894.$

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Dynamics of the translation components (p, q) of the fractional order discrete-time system for a₁ = 1, a₂ = 0, a₃ = 0.3, a₄ = −0.2, and a₅ = 1 with (a) ν = 1, (b) ν = 0.9871, (c) ν = 0.9084, and (d) ν = 0.894.

IV. CONTROL LAW

When we refer to stabilization, what we are talking about is adding a new time-varying parameter $u (t)$ to one of the system’s states and finding a closed form adaptive formula for these parameters to force the system states to zero in sufficient time. Theorem IV.1 provides the basis for the stability analysis.

Theorem IV.1

(Ref. 27). The zero equilibrium of the linear fractional order discrete-time system

{}^{C}Δ_{a}^{ν} X (t) = M X (t + ν - 1),

(12)

where

X (t) = {(x_{1} (t), \dots, x_{n} (t))}^{T}

, 0 < ν ≤ 1, M ∈ R^n×n, and

\forall t \in N_{a + 1 - ν}

are asymptotically stable if

λ \in \{z \in C : |z| < {(2 \cos \frac{|\arg z| - π}{2 - ν})}^{ν} and |\arg z| > \frac{ν π}{2}\},

(13)

for all the eigenvalues λ of M.

Now, we can give Theorem IV.2 to stabilize the proposed new fractional order discrete-time system. The asymptotic stability of solutions can be established by using condition (13).

Theorem IV.2.

The three-dimensional fractional order discrete-time system can be stabilized under the one-dimensional control law

u (t) = - a_{1} x (t) - a_{2} z (t) - a_{3} x (t) y (t) - a_{4} z^{2} (t) - a_{5} .

(14)

Proof.

The controlled fractional order discrete-time system involves the time-varying control parameter

u (t)

and is given by

\{\begin{matrix} {}^{C}Δ_{a}^{ν} x (t) = y (t - 1 + ν) - x (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} y (t) = z (t - 1 + ν) - y (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} z (t) = a_{1} x (t - 1 + ν) + (a_{2} - 1) z (t - 1 + ν) \\ + a_{3} x (t - 1 + ν) y (t - 1 + ν) \\ + a_{4} z^{2} (t - 1 + ν) + a_{5} + u (t - 1 + ν) . \end{matrix})

(15)

Substituting the proposed control law into yields the following simplified dynamics:

\{\begin{matrix} {}^{C}Δ_{a}^{ν} x (t) = y (t - 1 + ν) - x (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} y (t) = z (t - 1 + ν) - y (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} z (t) = - z (t - 1 + ν) . \end{matrix})

(16)

System can be written in a compact form,

\begin{matrix} {}^{C}Δ_{a}^{ν} {(x (t), y (t), z (t))}^{T} \\ = M \times {(x (t - 1 + ν), y (t - 1 + υ), z (t - 1 + ν))}^{T}, \end{matrix}

(17)

where

M = (\begin{matrix} - 1 & 1 & 0 \\ 0 & - 1 & 1 \\ 0 & 0 & - 1 \end{matrix}) .

(18)

The aim is to show that the zero equilibrium of is asymptotically stable, which means that the system states converge toward zero as time progresses. Asymptotic stability can be established using the stability theory of the linear fractional discrete-time system, as described in Theorem IV.2. Now, it is easy to see that the eigenvalues of matrix are λ₁ = λ₂ = λ₃ = −1 and satisfy

|\arg λ_{i}| = π > \frac{ν π}{2}

and

|λ_{i}| < {(2 \cos \frac{|\arg λ_{i}| - π}{2 - ν})}^{ν}

⁠, for i = 1, 2, 3. From Theorem IV.1, it is evident that the zero solution of is asymptotically stable, and, therefore, the fractional discrete-time system is stabilized.

□

A numerical simulation was carried out to illustrate the result of Theorem IV.2. We choose a = 0, initial conditions $(x (0), y (0), z (0)) = (1.5, 2.84, - 1.2)$ ⁠, and fractional order ν = 0.98. The evolution of states toward zeros is depicted in Fig. 12, which confirms the theoretical control proposed in Theorem IV.2.

FIG. 12.

$FIG. 12. Stabilization of states and attractors of the fractional order discrete-time system with ν = 0.98: (a) evolution of the state x(n), (b) evolution of the state y(n), (c) evolution of the state z(n), and (d) controlled attractors in the x − y plane.$

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Stabilization of states and attractors of the fractional order discrete-time system with ν = 0.98: (a) evolution of the state x(n), (b) evolution of the state y(n), (c) evolution of the state z(n), and (d) controlled attractors in the x − y plane.

V. SYNCHRONIZATION SCHEME

Another interesting aspect is the synchronization of one chaotic system with another. Synchronization refers to the addition of a set of control parameters to the controlled chaotic system and adaptively updating the controls such that the states become synchronized. Let us consider the master system described for ${t \in N}_{a + 1 - ν}$ by

\{\begin{matrix} {}^{C}Δ_{a}^{ν} x_{m} (t) = y_{m} (t - 1 + ν) - x_{m} (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} y_{m} (t) = x_{m} (t - 1 + ν) + α x_{m}^{2} (t - 1 + ν) + β y_{m}^{2} (t - 1 + ν) \\ - γ x_{m} (t - 1 + ν) y_{m} (t - 1 + ν) - y_{m} (t - 1 + ν) - δ . \end{matrix})

(19)

Note that the subscript m in the states refers to the master. It has been shown in Ref. 28 that the fractional order discrete-time system (19) exhibits chaotic behaviors with no fixed points. The two-dimensional fractional order discrete-time system (19) is the first example of a fractional order discrete-time system without equilibrium, i.e., system (19) has hidden attractors. For instance, when $(α, β, γ, δ) = (0.2, 0.71, 0.91, 1.14)$ ⁠, $(x (0), y (0)) = (0.93, - 0.44)$ ], ν = 0.985, and a = 0, Fig. 13 shows the chaotic behavior of the fractional order discrete-time system (19).

FIG. 13.

$FIG. 13. Chaotic behavior of the fractional discrete-time system (19): (a) the chaotic attractor, (b) bifurcation diagram vs β, and (c) the largest Lyapunov exponent corresponding to (b).$

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Chaotic behavior of the fractional discrete-time system (19): (a) the chaotic attractor, (b) bifurcation diagram vs β, and (c) the largest Lyapunov exponent corresponding to (b).

Similarly, the subscripts s are used to denote the states of the three-dimensional fractional order discrete-time system (4). The slave, thus, is given by

\{\begin{matrix} {}^{C}Δ_{a}^{ν} x_{s} (t) = y_{s} (t - 1 + ν) - x_{s} (t - 1 + ν) + u_{1} (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} y_{s} (t) = z_{s} (t - 1 + ν) - y_{s} (t - 1 + ν) + u_{2} (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} z_{s} (t) = a_{1} x_{s} (t - 1 + ν) + (a_{2} - 1) z_{s} (t - 1 + ν) \\ + a_{3} x_{s} (t - 1 + ν) y_{s} (t - 1 + ν) \\ + a_{4} z_{s}^{2} (t - 1 + ν) + a_{5} + u_{3} (t - 1 + ν), \end{matrix})

(20)

where the functions $u_{i} (t)$ for i = 1, 2, 3, 4 denote the synchronization controllers. To synchronize the two-dimensional master system (19) and the three-dimensional slave system (20), we propose a new scheme of synchronization where the error system is defined as follows:

\{\begin{matrix} e_{1} (t) = x_{s} (t) - x_{m} (t), \\ e_{2} (t) = y_{s} (t) - y_{m} (t), \\ e_{3} (t) = z_{s} (t) - x_{m} (t) - y_{m} (t) . \end{matrix})

(21)

The master system (19) and the slave system (20) are said to be synchronized if there exist controllers u₁(t), u₂(t), u₃(t) such that $lim_{t \to \infty} |e_{i} (t)| = 0, i = 1,2,3$ ⁠, with $t \in N_{a + 1 - ν}$ ⁠.

Theorem V.1 presents the proposed control law for this type of synchronization.

Theorem V.1.

Subject to

\{\begin{matrix} u_{1} (t) = 0, \\ u_{2} (t) = - z_{s} (t) + x_{m} (t) + α_{1} x_{m}^{2} (t) + β_{2} y_{m}^{2} (t) - γ_{3} x_{m} (t) y_{m} (t) - δ_{4}, \\ u_{3} (t) = x_{m} (t) - y_{m} (t) + a_{1} x_{s} (t) - a_{2} z_{s} (t) - a_{3} x_{s} (t) y_{s} (t) - a_{4} z_{s}^{2} (t) \\ - a_{5} + α_{1} x_{m}^{2} (t) + β_{2} y_{m}^{2} (t) - γ_{3} x_{m} (t) y_{m} (t) - δ_{4}, \end{matrix})

(22)

the master system and the slave system are synchronized.

Proof.

The error system has fractional order Caputo differences

\{\begin{matrix} {}^{C}Δ_{a}^{ν} e_{1} (t) = y_{s} (t - 1 + ν) - x_{s} (t - 1 + ν) + u_{1} (t - 1 + ν) \\ - y_{m} (t - 1 + ν) + x_{m} (t - 1 + ν), \\ {}^{C}Δ_{a}^{ν} e_{2} (t) = z_{s} (t - 1 + ν) - y_{s} (t - 1 + ν) + u_{2} (t - 1 + ν) \\ - x_{m} (t - 1 + ν) - α_{1}^{2} x_{m} (t - 1 + ν) - β_{2} y_{m}^{2} (t - 1 + ν) \\ + γ_{3} x_{m} (t - 1 + ν) y_{m} (t - 1 + ν) + y_{m} (t - 1 + ν) + δ_{4}, \\ {}^{C}Δ_{a}^{ν} e_{2} (t) = a_{1} x_{s} (t - 1 + ν) + (a_{2} - 1) z_{s} (t - 1 + ν) \\ + a_{3} x_{s} (t - 1 + ν) y_{s} (t - 1 + ν) + a_{4} z_{s}^{2} (t - 1 + ν) + a_{5} \\ + u_{3} {(t - 1 + ν)}_{1}^{2} x_{m} (t - 1 + ν) - α - β_{2} y_{m}^{2} (t - 1 + ν) \\ + γ_{3} x_{m} (t - 1 + ν) y_{m} (t - 1 + ν) + δ_{4} . \end{matrix})

(23)

Substituting the control law into yields reduced dynamics

\begin{matrix} {}^{C}Δ_{a}^{ν} {(e_{1} (t), e_{2} (t), e_{3} (t))}^{T} \\ = M \times {(e_{1} (t - 1 + ν), e_{2} (t - 1 + ν), e_{3} (t - 1 + ν))}^{T}, \end{matrix}

(24)

where

M = (\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}) .

(25)

It is easy to see that the eigenvalues of the matrix M satisfy the stability condition , and by means of Theorem IV.1, we know that the zero solution of is globally asymptotically stable and, consequently, the systems and are synchronized. □

The control laws stated in Theorem V.1 are confirmed through numerical simulations. Figure 14 depicts the time evolution of fractional errors (24) subject to the control law (22). The errors clearly converge toward zero, indicating that the described combined synchronization is successful.

FIG. 14.

View large Download slide

Time evolution of the synchronization errors for ν = 0.98.

VI. CONCLUSION

This paper has introduced a new fractional order discrete-time system with hidden chaotic attractors. Using standard nonlinear analysis techniques such as bifurcation diagrams, largest Lyapunov exponents, phase portraits, and the 0–1 test, we have shown that the proposed system can exhibit different hidden attractors for different fractional orders and system parameters. In addition, a one-dimensional feedback stabilization controller is proposed to force the states of the fractional order discrete-time system to zero asymptotically.

Finally, we have proposed a two-dimensional control law to synchronize the new three-dimensional fractional order discrete-time system by a two-dimensional fractional order discrete-time system with hidden chaotic attractors. The asymptotic convergence of the proposed controller is established by means of the linearization method. Numerical results were presented to support the proposed theoretical synchronization results. Such a system will have potential application in future works.

ACKNOWLEDGMENTS

Adel Ouannas was supported by the Directorate General for Scientific Research and Technological Development of Algeria. Shaher Momani was supported by Ajman University in UAE.

The data that support the findings of this study are available within the article.

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Author(s)

Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization

I. INTRODUCTION

II. THE FRACTIONAL MAP

III. DYNAMIC ANALYSIS

A. Bifurcation and Lyapunov exponents

B. 0–1 test

IV. CONTROL LAW

V. SYNCHRONIZATION SCHEME

VI. CONCLUSION

ACKNOWLEDGMENTS

REFERENCES

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Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization

I. INTRODUCTION

II. THE FRACTIONAL MAP

III. DYNAMIC ANALYSIS

A. Bifurcation and Lyapunov exponents

B. 0–1 test

IV. CONTROL LAW

V. SYNCHRONIZATION SCHEME

VI. CONCLUSION

ACKNOWLEDGMENTS

REFERENCES

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