Abstract

Fractional analysis provides useful tools to describe natural phenomena, and therefore, it is more convenient to describe models of satellites. This work illustrates rich chaotic behaviors that exist in a fractional-order model for satellite with and without time-delay. The proof for existence and uniqueness of the satellite model’s solution with and without time-delay is shown. Chaos control is achieved in this system via a simple linear feedback control criterion. Chaotic attractors and chaos control are also found in a time-delay version of the proposed fractional-order satellite system. Various tools based on numerical simulations such as 2D and 3D attractors and bifurcation diagrams are used to illustrate the variety of rich chaotic dynamics in the satellite models.

1. Introduction

Fractional analysis has become a basic topic for research since it presents appropriate mathematical tools to explain a wide variety of engineering, physical and biological phenomena, and some interdisciplinary topics such as neural networks [1–9]. During the past century, some fractional differential operators were successfully proposed to describe fractional derivatives such as Caputo’s type [10] and Riemann–Liouville’s type [11]. The aforementioned operators are defined by integration so they are sorted as nonlocal operators with singular kernels. However, the so called Caputo–Fabrizio operator [12] is nonlocal operator with the nonsingular kernel. As a matter of fact, nonlocal operators are a better candidate to model a variety of rich complex dynamics arising in natural phenomena and provide higher adequacies in describing them.

In the past five decades, applications of chaos theory to natural models have received growing interest. Therefore, chaotic dynamics were reported from natural phenomena arising from ecological, economical, physical, and engineering models [13–16]. Moreover, chaotic dynamics in engineering models involving fractional derivatives were reported in [17–20].

Obviously, these kinds of the aforementioned studies help us to understand, quantify, and predict the complex dynamics arising from real world phenomena.

Lyapunov analysis for the stability of the nonlinear system has also become a focal topic for research in integer-order dynamical systems [21–23]. Recently, researchers have developed fractional versions of the Lyapunov stability theory (LST) [24]. Furthermore, the LST has been developed for the case of time-delayed fractional-order systems [25–28].

Studying models of satellites has received increasing attention due to their potential applications to scientific, military, and civil activities and space technology [29–33]. On the contrary, chaotic dynamics arising from satellite models have been reported by many researchers [34–38]. In [39], Khan and Kumar reported that chaotic dynamics from a mathematical model of satellite consists of the 3D integer-order system. In addition, Khan and Kumar investigated the chaotic integer-order satellite (IOS) system based on T-S fuzzy modeling [40]. Moreover, in [41], Khan and Kumar reported some conditions for chaos synchronization in the IOS system with time delay. Furthermore, a fractional-order version of the IOS system, namely, the fractional-order satellite (FOS) model, was studied by Kumar et al. [18].

Motivated by the aforementioned statements, we further investigate the chaotic dynamics in the FOS model with and without time delay. We also prove existence and uniqueness of solutions in this model with and without time delay. Then, we intend to suppress chaotic dynamics to the FOS’s origin steady state using a simple linear control criterion which depends on a specific selection of the feedback control gains (FCGs). Chaotic attractors and other interesting dynamics are found in a time-delay version of the FOS model. Finally, the abovementioned linear feedback controller is also implemented into a controlled time-delay version of the FOS model. The obtained results show that the time-delay version of the FOS model is also stabilized to its origin equilibrium state when using the same FCGs, whose selection is based on the LST of fractional-order systems with time delay. The numerical results point out that the state variables of the controlled FOS equations are controlled to the origin state faster than its time-delay counterpart does when using the same linear controllers.

2. Fractional Calculus

Throughout this work, the following Caputo’s type [10] represents the imposed fractional differential operator:where is the fractional order, namely, the fractional parameter, represents the -order derivative of , and refers to the minimum integer that is not less than the fractional parameter, and is described bywhere stands for the Gamma function.

Then, we take into account the following initial value problem (IVP):which is equivalent to the nonlinear Volterra integral equation of the second kind, provided that the function is continuous; then, IVP (3) is solved by

According to Diethelm et al. [42, 43], the predictor-correctors (PECE) method can be used to discretize equation (4) as follows:where for some integer N defined using the uniform grid refers to an approximation to , and the coefficient defined in the last term of RHS of equation (5) is described asand the predictor is obtained from

A modification of the PECE scheme was presented in [44] to integrate fractional-order equations with time delay. It can also be described as follows; for the delayed time fractional equation,where can be denoted by an element of the Banach space of all continuous functions over , that is, . Then, assume thatwhere for some integer N defined using the uniform grid with the following notation:

So, can be calculated via

Now, according to [44], the formula for the corrector variable is obtained viaand the predictor variable can be evaluated via

Also, we introduce the following basic lemmas.

Lemma 1 (see [45]). Assume that the vector-value function is continuous and differentiable. Then, and the following inequality holds:

Lemma 2 (see [46]). Assume that a nonnegative function is given that it is continuously differentiable and satisfies the following conditions (as ):where is a nonnegative real number and . If and then

Remark 1. A function is called a Lipschitz continuous if such that the constant satisfies the following inequality:for any , where refers to a matrix norm.
If the delayed time fractional system has the formwhere represents and and are constant matrices, then we obtain the following results:

Remark 2 (see [47]). A necessary and sufficient condition for a continuously differentiable function to be a solution of (18) is

Theorem 1 (see [47]). If the nonlinear function is Lipschitz continuous, then there exists a unique and continuous solution for general system (18).

Finally, we end this section with the following lemmas:

Lemma 3 (see [48]). Consider that the generalized fractional system , such that there exists a Lyapunov function for the equilibrium solution in case of then, is at least locally asymptotically stable (LAS) as .

Lemma 4 (see [48]). Let be a Hermitian matrix whose all principal minor determinants ; then, the quantity and belong to a domain that contains a neighborhood of .

3. The Chaotic FOS Model

The FOS is described by Kumar et al. [18]:where . FOS model (20) exhibits a variety of rich dynamics including chaotic attractors, as outlined in Figure 1, using , and . The corresponding bifurcation diagrams are depicted in Figure 2.

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Figure 1 

3D plots of FOS model (20) with showing that (a) chaos exists at ; (b) chaos exists at ; (c) chaos exists at ; (d) chaos exists at ; (e) quasi-periodic attractor exists at ; (f) asymptotic attractor exists at .

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Figure 2 

Bifurcation diagram of FOS model (20) as varying (a) a with ; (b) b with setting ; (c) c with setting ; (d) with setting .

Calculations of the Lyapunov spectrum show that the FOS system is still chaotic for a wide range of parameters beyond since the maximal Lyapunov exponents are positive. This interesting observation is depicted in Figure 3, and it verifies the existence of rich chaotic dynamics in the FOS system.

Figure 3 

Lyapunov spectrum of FOS model (20) as the parameter with setting .

4. Solution’s Existence and Uniqueness in the FOS Model

The FOS system is represented in the following form:where

Let be a class of continuous functions. Hence, the supremum norm is defined as

Furthermore, for a matrix , the norm is described as

We next discuss the existence and uniqueness of the solution of the FOS model in the region where

According to Remark 2, with setting the solution of chaotic FOS system (20) is obtained as

So,

Consequently, we obtainwhich is reduced towhere

If then is a contraction mapping, and the following theorem gives a sufficient condition for existence and uniqueness of the solution of FOS model (20).

Theorem 2. A sufficient condition for solution’s existence and uniqueness of FOS system (20) in the region with initial conditions and is

5. Chaos Control of the Chaotic FOS Model via a Linear Feedback Criterion

Suppose that denotes an equilibrium point of chaotic FOS model (20) and k₁, k₂, and k₃ are the positive feedback control gains (FCGs). Then, a controlled form of FOS model (20) is introduced by

In case of the node equilibrium point of chaotic FOS model (20), we have the following theorem:

Theorem 3. The chaotic behaviors in FOS model (20) are suppressed to its zero steady state if the following conditions hold:where .

Proof. Chaotic FOS system (32) matches the following Lyapunov function:Hence, we obtainwhereThe Hermitian matrix is strictly positive if all conditions (33) hold. Therefore, according to Lemma 4, it follows that the integer-order derivative of is negative when is not origin, while it is vanished at the origin. Then, based on Lemma 3, the origin equilibrium of chaotic FOS model (20) is at least LAS.
Controlled FOS model (32) is numerically integrated with , and . Now, according to inequalities (33), the FCGs can be chosen as The results are given in Figure 4.

Figure 4 

Trajectories of chaotic FOS system (32) converge to the origin steady state using and the FCGs .

6. Chaotic Dynamics in the FOS Delay Model

A time-delay version of FOS model (20) is represented here bywhere is the delay constant. Chaotic attractors in FOS delay model (37) are depicted in Figure 5 using different values of and fixing other model’s parameters at , and . Moreover, chaotic behaviors are depicted in plane, as shown in Figure 6. Also, the bifurcation plot corresponding to the aforementioned parameter set is illustrated in Figure 7.

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Figure 5 

3D plots of FOS model (37) with , and , showing that (a) chaos exists at ; (b) chaos exists at ; (c) quasi-periodic attractor exists at ; (d) asymptotic attractor exists at .