Stability and Stabilizing of Fractional Complex Lorenz Systems

Abstract

We study the stability and stabilization of complex fractional Lorenz system. The fractional calculus are taken in sense of the Caputo derivatives. The technique is based on stability theory of fractional-order systems. Numerical solutions are imposed.

1. Introduction

Chaos theory is a new branch of mathematics and physics. It provided a new method to understand the behavior of the processes in the world. Chaotic behaviors virtue in different areas of science and engineering such as mechanics, physics, electronics, medicine, biology, ecology, economy. In 1963, Lorenz discovered the chaotic system [1]. A dynamical system is called chaotic if it is deterministic, has long-term aperiodic behavior, and exhibits sensitive dependence on the initial conditions. If the system has one positive Lyapunov exponent, then it is called chaotic. The Lorenz system is a system of ordinary differential equations that imposing chaotic solutions for certain parameter values and initial conditions. Researches along this line ultimately led to the ruling of different classes of relevant chaotic systems. The Lorenz system has advantages, it presents a symmetry, and the equations are invariant under (𝑥,𝑦)→(−𝑥,−𝑦). Also the system is dissipative with a value of the divergence and therefore the volume in phase space always contracts under the flow. Moreover the Lorenz system is bounded.

This system was widely considered over the course of the 1970s, 1980s, and 1990s (see [2–5]) as a paradigmatic chaotic problem. In 1994, Sprott defined the diffusionless Lorenz system (a one-parameter version of Lorenz system) [6]. In 1999, Chen and Ueta established an interesting system [7], which is the dual system to the Lorenz system [1], in the sense defined by Vaněček and Čelikovský [8, 9]. Later, Lü and Chen [10] furthermore introduced a chaotic system, which refers to the transition between the Lorenz and the Chen systems. In 2005, Čelikovský and Chen described another generalized Lorenz canonical form [11]. More systems are posed in 2006-2007; a unified Lorenz-type system is recognized by Yang et al. [12, 13]. In 2009, Huang and Yang found a new chaotic system, which contains special cases as the modified Lorenz system and conjugate Chen system [14].

Recently, the Lorenz system has got many studies and attentions from researchers. Zhang et al. considered the synchronization of two identical hyperchaotic Lorenz systems with time delay only by a single linear controller [15], while Shi and Wang investigated a class of new synchronization (hybrid synchronization) coexisting in the same system [16]. Camargo et al. considered a system of two identical linearly coupled Lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength. They verified the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction, such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa [17]. Finally, Li et al. concerned with model-free control of the Lorenz chaotic system, where only the online input and output are available, while the mathematic model of the system is unknown [18].

Fractional-order chaotic systems have been studied widely in current years in many fields such as chaotic phenomena, chaotic control, chaotic synchronization, and other related studies. It has been provided that many fractional-order dynamical systems, as some well-known integer-order systems, can be displayed complex bifurcation and chaotic phenomena. For example, the fractional-order Lorenz system, the fractional-order Chen system, the fractional-order Lü system, and the fractional-order unified system also exhibit chaotic behavior.

Čelikovský and Chen (2002) generalized a new Lorenz canonical form of one parameter [19, 20]. The dynamics of the fractional-order Lorenz model was also investigated by I. Grigorenko and E. Grigorenko (2003) [21]. Li and Peng (2004) studied the time-domain algorithm to compute the chaotic attractor of the fractional differential system, which is completely different from the frequency-domain one [22]. Yan and Li (2007) [23] considered chaos synchronization of fractional-order Lorenz, Rossler, and Chen systems taking one as master and second one as slave. Zhou and Cheng (2008) [24] discussed a synchronization between different fractional-order chaotic systems, namely, Rossler and Chen systems and Chua and Chen systems. Based on the qualitative theory, the existence and uniqueness of solutions for a class of fractional-order Lorenz chaotic systems have been investigated theoretically by Yu et al. (2009) [25]. Fractional-order diffusion less Lorenz system has been investigated numerically by Sun and Sprott (2009) [26], and results have shown that the system has complex dynamics with interesting characteristics. Recently, Agrawal et al., synchronized fractional-order chaotic systems using active control method [27]. Xu et al., investigated the phenomenon of diffusionless Lorenz system with fractional-order [28]. Zhou and Ding studied the stability and stabilizations of equilibrium points of the fractional-order Lorenz chaotic system [29]. Finally, Si et al. employed an uncertain complex network with four fractional-order Lorenz systems to verify the effectiveness of the proposed approach [30].

In current years, researchers introduced and studied several types of chaotic nonlinear systems with complex variables. These systems which involving complex variables are employed to describe the physics of a detuned laser, rotating fluids, disk dynamos, electronic circuits, and particle beam dynamics in high-energy accelerators. As special model, the chaotic complex Lorenz system which is used to describe and simulate the physics of detuned lasers and thermal convection of liquid flows. The complex Lorenz model is, in fact, the outcome of the study of Baroclinic models. This model corresponds to the equilibrium state of the atmosphere in which surfaces of constant density are not parallel to the surface of constant gravitational potential. In 1982, Fowler et al. defined and studied a complex Lorenz as a generalization of the real Lorenz system [31, 32]. The complex variables of Lorenz system are related, respectively, to electric field, the atomic polarization amplitudes, and the population inversion in a ring laser system of two-level atoms; for more details, see [33, 34]. Later Rauh et al. (1996) [35] introduced a set of equations to model such a system. This has the same structure as that of Lorenz equations, but it occurs in the complex domain. Panchev and Vitanov (2005) [36] studied asymptotic properties of some complex Lorenz-like systems, while other properties and chaotic synchronization of the complex Lorenz model are studied by Mahmoud et al. (2007) [37]. Further studies are introduced due to Mahmoud et al.; they constructed the new hyperchaotic complex Lorenz systems by extending the ideas of adding state feedback controls and introducing the complex periodic forcing [38, 39], while G. M. Mahmoud and E. E. Mahmoud (2010) [40] investigated complete synchronization of n-dimensional chaotic complex systems with uncertain parameters. Li et al. (2012) [41] considered the distributed robust control problems of uncertain linear multiagent systems with undirected communication topologies. It was assumed that the agents have identical nominal dynamics while subject to different norm-bounded parameter uncertainties, leading to weakly heterogeneous multiagent systems. Mahmoud (2012) [42] introduced a new hyperchaotic complex Lorenz system. This hyperchaotic complex system was constructed by adding a linear controller to the second equation of the chaotic complex Lorenz system. Finally, El-Sayed et al. [43] studied the dynamic properties of the continuous dynamical system of the fractional equations of complex variables. The five-dimensional complex Lorenz model is often used to explain and simulate the physics of tuned lasers. These studies have their origin in the shaping expression by Haken [44] of the isomorphism between the three equations of the real Lorenz model and the three real equations for a single-mode laser operating with its resonant cavity tuned to resonance with the material transition. Furthermore, analytic studies of model systems can be found in [45–47].

In this paper, we study the stability and stabilization of fractional complex Lorenz model. The fractional calculus are taken in sense of the Caputo derivatives. Caputo initial value problem holds for both homogeneous and nonhomogeneous conditions. For this reason choice Caputo derivative is better than other fractional derivatives. The technique based on stability theory of fractional-order systems. Numerical solutions are imposed.

2. Fractional Calculus

The idea of the fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) was planted over 300 years ago. Abel in 1823 investigated the generalized tautochrone problem and for the first time applied fractional calculus techniques in a physical problem. Later Liouville applied fractional calculus to problems in potential theory. Since that time the fractional calculus has haggard the attention of many researchers in all area of sciences. Recently, the theory of fractional calculus has found interesting applications in the theory of analytic functions. The classical definitions of fractional operators and their generalizations have fruitfully been applied in obtaining, for example, the characterization properties, coefficient estimates, distortion inequalities, and convolution structures for various subclasses of analytic functions and the works in the research monographs [48–50]. This section concerns with some preliminaries and notations regarding the fractional calculus. Let us start with the Riemann-Liouville fractional operators.

Definition 1. The fractional (arbitrary) order integral of the function 𝑓 of order 𝛼>0 is defined by 𝐼^𝛼_𝑎𝑓(𝑡)=^𝑡_𝑎(𝑡−𝜏)^𝛼−1Γ(𝛼)𝑓(𝜏)𝑑𝜏. (1) When 𝑎=0, we write 𝐼^𝛼_𝑎𝑓(𝑡)=𝑓(𝑡)∗𝜙_𝛼(𝑡), where (∗) denoted the convolution product, 𝜙_𝛼(𝑡)=𝑡^𝛼−1/Γ(𝛼),𝑡>0, and 𝜙_𝛼(𝑡)=0,𝑡≤0 and 𝜙_𝛼→𝛿(𝑡), as 𝛼→0 where 𝛿(𝑡) is the delta function.

Definition 2. The fractional (arbitrary) order derivative of the function 𝑓 of order 0≤𝛼<1 is defined by 𝐷^𝛼_𝑎𝑓(𝑡)=𝑑𝑑𝑡^𝑡_𝑎(𝑡−𝜏)^−𝛼Γ()𝑓(𝜏)𝑑1−𝛼𝑑𝜏=𝐼𝑑𝑡^𝑎1−𝛼𝑓(𝑡). (2)

Remark 3. From Definitions 1 and 2, 𝑎=0, we have 𝐷^𝛼𝑡^𝜇=Γ𝜇+1Γ𝑡𝜇−𝛼+1^𝜇−𝛼𝐼,𝜇>−1;0<𝛼<1,_𝛼𝑡^𝜇=Γ𝜇+1Γ𝑡𝜇+𝛼+1^𝜇+𝛼,𝜇>−1;𝛼>0. (3)

Definition 4. The Caputo fractional derivative of order 𝜇>0 is defined, for a smooth function 𝑓(𝑡) by ^𝑐𝐷^𝜇𝑓(𝑡)1∶=Γ𝑛−𝜇^𝑡₀𝑓^(𝑛)(𝜁)(𝑡−𝜁)^{𝜇−𝑛+1}𝑑𝜁, (4) where 𝑛=[𝜇]+1, (the notation [𝜇] stands for the largest integer not greater than 𝜇).

Remark 5. The following relations hold the following:(i)Representation ^𝑐𝐷^𝜇𝑓(𝑡)=𝐼^𝑛−𝜇𝐷^𝑛𝑓(𝑡),𝑛−1<𝜇<𝑛. (5)(ii)The Caputo fractional derivative of the power function ^𝑐𝐷^𝜇𝑡^𝜇=Γ𝜇+1Γ𝑡𝜇−𝛼+1^𝜇−𝛼=𝐷^𝛼𝑡^𝜇. (6)(iii)𝐼^𝜇𝑐𝐷^𝜇𝑓(𝑡)=𝑓(𝑡),𝑓(0)=0,𝜇∈(0,1). (7)(iv)Linearity ^𝑐𝐷^𝜇𝜆𝑓(𝑧)+𝑔(𝑧)=𝜆^𝑐𝐷^𝜇_𝑧𝑓(𝑡)+^𝑐𝐷^𝜇𝑔(𝑡). (8)(v)Noncommutation ^𝑐𝐷^𝜇𝐷^𝛼𝑓(𝑡)≠𝐷^𝛼𝑐𝐷^𝜇𝑓(𝑡). (9)

3. Stability of Fractional Complex System

A complex Lorenz-like system has been found in Laser Physics and while analyzing baroclinic instability of the geophysical flows in the atmosphere which takes the form 𝜉^.₁=−𝜎𝜉₁+𝜎𝜉₂,𝜉^.₂=𝜌𝜉₁−𝑎𝜉₂−𝜉₁𝜉₃,𝜉^.₃=−𝑏𝜉₃+12𝜉^∗₁𝜉₂+𝜉₁𝜉^∗₂, (10)(11)(12) where 𝜉₁ and 𝜉₂ are complex variables; that is, 𝜉₁=𝑢₁+𝑖𝑢₂,𝜉^∗₁=𝑢₁−𝑖𝑢₂ and 𝜉₂=𝑢₃+𝑖𝑢₄,𝜉^∗₂=𝑢₃−𝑖𝑢₄, but 𝜉₃∶=𝑢₅ is real variable, and 𝑎=1−𝑖ℓ, such that 𝜌,ℓ,𝜎, and 𝑏 are real parameters. It assumed that 𝑢_𝑘,𝑘=1,…,5 is a function in 𝑡.

Here, we consider the following fractional complex Lorenz system, in sense of the Caputo derivative: ^𝑐𝐷^𝜇1𝜉₁=−𝜎𝜉₁+𝜎𝜉₂,_𝑐𝐷^𝜇2𝜉₂=𝜌𝜉₁−𝑎𝜉₂−𝜉₁𝜉₃,_𝑐𝐷^𝜇3𝜉₃=−𝑏𝜉₃+12𝜉^∗₁𝜉₂+𝜉₁𝜉^∗₂𝜉₃, (13) where 0<𝜇_𝑗≤1,𝑗=1,2,3; in this note we shall take 𝜇_𝑗≥0.95. For real variables, system (13) reduces to the system of the form ^𝑐𝐷^𝜇1𝑢₁=−𝜎𝑢₁+𝜎𝑢₃,_𝑐𝐷^𝜇1𝑢₂=−𝜎𝑢₂+𝜎𝑢₄,_𝑐𝐷^𝜇2𝑢₃=𝜌𝑢₁−𝑢₃−ℓ𝑢₄−𝑢₁𝑢₅,_𝑐𝐷^𝜇2𝑢₄=𝜌𝑢₂+ℓ𝑢₃−𝑢₄−𝑢₂𝑢₅,_𝑐𝐷^𝜇3𝑢₅=−𝑏𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄𝑢₅. (14)

In this section, we shall discuss the stability of the system (14). For this purpose we need the following preliminaries which can be found in [51].

Definition 6. The zero solution of the equation ^𝑐𝐷^𝜇𝑢=𝑓(𝑡,𝑢(𝑡)),𝜇∈(0,1] is said to be stable if, for any initial values 𝑢₀, there exists 𝜖>0 such that ‖𝑢(𝑡)‖≤𝜖,forall𝑡>𝑡₀. The zero is said to be asymptotically stable if it is stable, and ‖𝑢(𝑡)‖→0,𝑡→∞.

Lemma 7. Assume the system of the form ^𝑐𝐷^𝜇𝑢=𝐴𝑢(𝑡)+ℎ(𝑢(𝑡)),𝜇∈(0,1], (15) where 𝑢(𝑡)=(𝑢₁,…,𝑢_𝑛(𝑡))^𝑇∈ℝ^𝑛,𝐴∈ℝ^𝑛×𝑛. If |arg(spec(𝐴))|>𝜇𝜋/2,𝜇‖𝐴‖>1,  spec(𝐴) denotes the eigenvalues of 𝐴, and ‖⋅‖ denotes the 𝑙₂ norm; and lim_𝑢→0(‖ℎ(𝑢(𝑡))‖/‖𝑢(𝑡)‖)=0, then the system (15) is asymptotically stable.

Lemma 8. Assume the controller system with the linear feedback control input ^𝑐𝐷^𝜇𝑢=(𝐴+𝐵𝐾)𝑢(𝑡)+ℎ(𝑢(𝑡)),𝜇∈(0,1], (16) where 𝐾∈ℝ^1×𝑛 is a feedback, 𝐵∈ℝ^𝑛×1, 𝑢(𝑡)=(𝑢₁,…,𝑢_𝑛(𝑡))^𝑇∈ℝ^𝑛,and𝐴∈ℝ^𝑛×𝑛. If |arg(spec(𝐴))|>𝜇𝜋/2,𝜇‖𝐴+𝐵𝐾‖>1, and lim_𝑢→0(‖ℎ(𝑢(𝑡))‖/‖𝑢(𝑡)‖)=0, then the system (16) is asymptotically stable.

Theorem 9. If 𝑏>0,𝜎>0, then the system (14) is asymptotically stable at the equilibrium point 𝑝₀=(0,0,0,0,0).

Proof. System (14) can be written in the form (15), where ℎ(𝑢)=00𝐴=−𝜎0𝜎000−𝜎0𝜎0𝜌0−1−ℓ00𝜌ℓ−100000−𝑏,−𝑢₁𝑢₅−𝑢₂𝑢₅𝑢₁𝑢₃+𝑢₂𝑢₄𝑢₅. (17)(18) It is obvious that ℎ(𝑢) satisfies lim_𝑢→0‖ℎ(𝑢(𝑡))‖‖𝑢(𝑡)‖=lim_𝑢→0−𝑢₁𝑢₅²+−𝑢₂𝑢₅²+𝑢₅²𝑢₁𝑢₃+𝑢₂𝑢₄²𝑢₁²+𝑢₂²+𝑢₃²+𝑢₄²+𝑢₅²≤lim_𝑢→0−𝑢₁𝑢₅²+−𝑢₂𝑢₅²+𝑢₅²𝑢₁𝑢₃+𝑢₂𝑢₄²𝑢²₅=lim_𝑢→0𝑢²₁+𝑢²₂+𝑢₁𝑢₃+𝑢₂𝑢₄²=0. (19) Moreover the characteristic equation of the system is (𝜆+𝑏)(𝜎+𝜆)(1+𝜆)−𝜌𝜎²+(𝜎+𝜆)²ℓ²=0 (20) and when ℓ=0, the eigenvalues are 𝜆₁=−𝑏,𝜆_2,3=−(𝜎+1)±(𝜎+1)²−4𝜎1−𝜌2, (21) where 𝜆_2,3 are of multiplicity 2. Since −(𝜎+1)<0, therefore arg𝜆_𝑖>𝜋2>𝜋2𝜇,𝜇=max𝜇_𝑖,𝑖=1,2,3 (22) and for suitable choice of the parameters 𝑏,𝜎,ℓ, and 𝜇_𝑖 we have ‖𝐴‖𝜇>1,𝜇∶=min(𝜇_𝑖). According to Lemma 7, it implies that the equilibrium point 𝑝₀ of system (14) is asymptotically stable. Now for ℓ≠0 and 𝜌=0, the eigenvalues are 𝜆₁=−𝑏,𝜆_2,3=−𝜎,𝜆_4,5=−1±𝑖ℓ, (23) hence relation (16) holds, and consequently the system (14) is asymptotically stable. Finally, the case ℓ≠0 and 𝜌≠0(𝜌>1) such that 𝜎=1, and ℓ²>2(𝜌−1) implies the eigenvalues of multiplicity 2 𝜆₁=−𝑏,𝜆_2,3=−2−ℓ²−2𝜌±ℓ²−2𝜌²−4𝜌²2 (24) which yields that the system is asymptotically stable at 𝑝₀. Note that the other cases impose complex 𝜎. This completes the proof.

The system (14) can be assumed in the form (16), we derive the following result.

Theorem 10. If 𝑏>0,𝜎>0, then the controlled system (14) is asymptotically stable at the equilibrium point 𝑝₀=(0,0,0,0,0).

Proof. Applying control input 𝑣(𝑡)=𝐵𝐾𝑢(𝑡) on (14), where 𝐵=(−𝜎,0,0,0,0)^𝑇 and 𝐾=(0,0,1,0,0) are taken in which |arg(spec(𝐴+𝐵𝐾))|>𝜇𝜋/2, where the characteristic equation of the system is (𝜆+𝑏)(𝜎+𝜆)−(𝜎+𝜆)(𝜆+1)²+𝜎𝜌(1+𝜆)−ℓ²(𝜎+𝜆)=0. (25) Obviously, for ℓ=0 the eigenvalues of the system are 𝜆₁=−𝑏,𝜆₂=−𝜎,𝜆₃=−1, (26) where 𝜆₂ and 𝜆₃ of multiplicity 2. Also when 𝜌=0, the eigenvalues are 𝜆₁=−𝑏,𝜆₂=−𝜎,𝜆₃=−1±𝑖ℓ. (27) Finally for ℓ≠0,𝜌≠0, and𝜎=1, the system admits the following eigenvalues: 𝜆₁=−𝑏,𝜆_2,3=−1,𝜆_4,5=−1±𝜌−ℓ². (28) Furthermore, 𝜇‖𝐴+𝐵𝐾‖>1; hence in view of Lemma 8, controlled system (14) is asymptotically stable at 𝑝₀.

4. Stabilizing 𝑝₀

In this section, we design an evaluation controller for fractional-order Lorenz chaotic system (14) via fractional-order derivative. For this puropose we need the following result which can be found in [52].

Lemma 11. The fixed points of the following nonlinear commensurate fractional-order autonomous system, ^𝑐𝐷^𝜇𝑢=𝑓(𝑢),𝜇∈(0,1), (29) are asymptotically stable if all eigenvalues (𝜆) of the Jacobian matrix evaluated at the fixed points satisfy |arg𝜆|>0.5𝜋𝜇, where 0<𝜇<1,𝑢∈𝑅^𝑛,𝑓∶𝑅^𝑛→𝑅^𝑛 are continuous nonlinear vector functions, and the fixed points of this nonlinear commensurate fractional-order system are calculated by solving equation 𝑓(𝑢)=0.

We can obtain the following results.

Theorem 12. Assume the controlled fractional-order Lorenz chaotic system ^𝑐𝐷^𝜇1𝑢₁=−𝜎𝑢₁+𝜎𝑢₃,_𝑐𝐷^𝜇1𝑢₂=−𝜎𝑢₂+𝜎𝑢₄,_𝑐𝐷^𝜇2𝑢₃=𝜌𝑢₁−𝑢₃−ℓ𝑢₄−𝑢₁𝑢₅+𝑓₁𝑢₁,_𝑐𝐷^𝜇2𝑢₄=𝜌𝑢₂+ℓ𝑢₃−𝑢₄−𝑢₂𝑢₅,_𝑐𝐷^𝜇3𝑢₅=−𝑏𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄𝑢₅, (30) where 𝑓₁(𝑢₁)=−𝑘^𝑐₁₁𝐷^𝜇1𝑢₁−𝑘₁₂𝑢₁ is the fractional-order controller, and 𝑘_1𝑖,𝑖=1,2 is the feedback coefficient. For the case ℓ=0, if 𝑏>0,𝜎>0, 1+𝑘₁₁𝜎>0,𝑘₁₂=𝜌+𝑘₁₁𝜎, (31) then system (30) will asymptotically converge to the unstable equilibrium point 𝑝₀. Furthermore, for the case ℓ≠0,𝜌≠0, and 𝜎=1, if 𝑏>0 and 𝑘₁₁>−3, then system (30) will asymptotically converge to the unstable equilibrium point 𝑝₀.

Proof. The Jacobi matrix of the controlled fractional-order Lorenz chaotic system (30) at 𝑝₀ is 𝐽=−𝜎0𝜎000−𝜎0𝜎0𝜌+𝑘₁₁𝜎−𝑘₁₂0−1−𝑘₁₁𝜎−ℓ00𝜌ℓ−100000−𝑏. (32) Since 𝑘₁₂=𝜌+𝑘₁₁𝜎, we have 𝐽=−𝜎0𝜎000−𝜎0𝜎000−1−𝑘₁₁𝜎−ℓ00𝜌ℓ−100000−𝑏. (33) Thus the characteristic equation takes the form (𝑏+𝜆)(𝜎+𝜆)²(1+𝜆)1+𝑘₁₁−()𝜎+𝜆𝜎+𝜆𝜌𝜎1+𝑘₁₁𝜎+𝜆+(𝜎+𝜆)²ℓ²=0 (34) which leads to the eigenvalues 𝜆₁=−𝑏,𝜆₂=−𝜎,𝜆₃=−1−𝑘₁₁𝜆𝜎,_4,5=−(𝜎+1)±(𝜎+1)²−4𝜎1−𝜌2 (35) when ℓ=0. And for 𝜌=0, we have 𝜆₁=−𝑏,𝜆_2,3𝜆=−𝜎,_4,5=−2+𝑘₁₁𝜎±2+𝑘₁₁𝜎²−4ℓ²+𝑘₁₁𝜎+12. (36) Since 𝑏>0,𝜎>0, and 1+𝑘₁₁𝜎>0, therefore, arg𝜆_𝑖>0.5𝜋𝜇,𝑖=1,…,5, (37) where 𝜇∶=max𝜇_𝑗,and𝑗=1,2,3. According to Lemma 11, it yields that the equilibrium point 𝑝₀ of system (30) is asymptotically stable; that is, the unstable equilibrium point 𝑝₀ in fractional-order Lorenz system (30) can be stabilized via fractional-order derivative. For the case ℓ≠0,𝜌≠0, and 𝜎=1, since 𝑘₁₁>−3, then the characteristic equation 𝜆³+3+𝑘₁₁𝜆²+3+2𝑘₁₁−𝜌+ℓ²𝜆+1+𝑘₁₁−𝜌−𝜌𝑘₁₁+ℓ²=0 (38) has negative eigenvalues, and hence the system is asymptotically stable. The proof is completed.

Theorem 13. Assume the controlled fractional-order Lorenz chaotic system ^𝑐𝐷^𝜇1𝑢₁=−𝜎𝑢₁+𝜎𝑢₃,_𝑐𝐷^𝜇1𝑢₂=−𝜎𝑢₂+𝜎𝑢₄,_𝑐𝐷^𝜇2𝑢₃=𝜌𝑢₁−𝑢₃−ℓ𝑢₄−𝑢₁𝑢₅,_𝑐𝐷^𝜇2𝑢₄=𝜌𝑢₂+ℓ𝑢₃−𝑢₄−𝑢₂𝑢₅+𝑓₂𝑢₂,_𝑐𝐷^𝜇3𝑢₅=−𝑏𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄𝑢₅, (39) where 𝑓₂(𝑢₂)=−𝑘^𝑐₂₁𝐷^𝜇1𝑢₂−𝑘₂₂𝑢₂ is the fractional-order controller, and 𝑘_2𝑖,𝑖=1,2 is the feedback coefficient. For the case ℓ=0, if 𝑏>0,𝜎>0, 1+𝑘₂₁𝜎>0,𝑘₂₂=𝜌+𝑘₂₁𝜎, (40) then system (39) will asymptotically converge to the unstable equilibrium point 𝑝₀. Furthermore, for the case ℓ≠0,𝜌≠0, and 𝜎=1, if 𝑏>0 and 𝑘₂₁>−3, then system (39) will asymptotically converge to the unstable equilibrium point 𝑝₀.

Proof. The Jacobi matrix of the controlled fractional-order Lorenz chaotic system (39) at 𝑝₀ is 𝐽=−𝜎0𝜎000−𝜎0𝜎0𝜌0−1−ℓ00𝜌+𝑘₂₁𝜎−𝑘₂₂ℓ−1+𝜎𝑘₂₁00000−𝑏. (41)
Since 𝑘₂₂=𝜌+𝑘₂₁𝜎, we have 𝐽=−𝜎0𝜎000−𝜎0𝜎0𝜌0−1−ℓ000ℓ−1+𝜎𝑘₂₁00000−𝑏. (42)
Therefore, the characteristic equation is (𝑏+𝜆)(𝜎+𝜆)(𝜎+𝜆)(1+𝜆)1+𝑘₂₁𝜎+𝜆+𝜌𝜎1+𝑘₂₁𝜎+𝜆−(𝜎+𝜆)ℓ²=0 (43) which yields the eigenvalues 𝜆₁=−𝑏,𝜆₂=−𝜎,𝜆₃=−1−𝑘₂₁𝜆𝜎,_4,5=−(𝜎+1)±(𝜎+1)²−4𝜎1+𝜌2 (44) when ℓ=0. while for 𝜌=0, we impose 𝜆₁=−𝑏,𝜆_2,3𝜆=−𝜎,_4,5=−2+𝑘₂₁𝜎±2+𝑘₂₁𝜎²−41−ℓ²+𝑘₂₁𝜎2 (45) Since 𝑏>0,𝜎>0, and 1+𝑘₂₁𝜎>0, therefore, arg𝜆_𝑖>0.5𝜋𝜇,𝑖=1,…,5, (46) where 𝜇∶=max𝜇_𝑗,𝑗=1,2,3. According to Lemma 11, it yields that the equilibrium point 𝑝₀ of system (39) is asymptotically stable; that is, the unstable equilibrium point 𝑝₀ in fractional-order Lorenz system (39) can be stabilized via fractional-order derivative. For the case ℓ≠0,𝜌≠0,and𝜎=1, since 𝑘₂₁>−3, then the characteristic equation 𝜆³+3+𝑘₂₁𝜆²+3+2𝑘₂₁+𝜌−ℓ²𝜆+1+𝑘₂₁1+𝜌−ℓ²=0 (47) has negative eigenvalues, and hence the system is asymptotically stable. This completes the proof.

Theorem 14. Assume the controlled fractional-order Lorenz chaotic system ^𝑐𝐷^𝜇1𝑢₁=−𝜎𝑢₁+𝜎𝑢₃+𝑓₃𝑢₃,_𝑐𝐷^𝜇1𝑢₂=−𝜎𝑢₂+𝜎𝑢₄,_𝑐𝐷^𝜇2𝑢₃=𝜌𝑢₁−𝑢₃−ℓ𝑢₄−𝑢₁𝑢₅,_𝑐𝐷^𝜇2𝑢₄=𝜌𝑢₂+ℓ𝑢₃−𝑢₄−𝑢₂𝑢₅,_𝑐𝐷^𝜇3𝑢₅=−𝑏𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄𝑢₅, (48) where 𝑓₃(𝑢₃)=−𝑘^𝑐₃₁𝐷^𝜇2𝑢₃−𝑘₃₂𝑢₃ is the fractional-order controller, and 𝑘_2𝑖,𝑖=1,2 is the feedback coefficient. If ℓ=0,𝑏>0,𝜎>0,𝑘₃₂=𝜎+𝑘₃₁, and 𝜎+𝜌𝑘₃₁>0, then system (48) will asymptotically converge to the unstable equilibrium point 𝑝₀.

Proof. The Jacobi matrix of the controlled fractional-order Lorenz chaotic system (48) at 𝑝₀ is 𝐽=−𝜎−𝜌𝑘₃₁0𝜎+𝑘₃₁−𝑘₃₂ℓ𝑘₃₁00−𝜎0𝜎0𝜌0−1−ℓ00𝜌ℓ−100000−𝑏. (49) Because ℓ=0,𝑘₃₂=𝜎+𝑘₃₁𝜎, so we have 𝐽=−𝜎−𝜌𝑘₃₁00000−𝜎0𝜎0𝜌0−1000𝜌0−100000−𝑏. (50) The characteristic equation is (𝑏+𝜆)𝜎+𝜌𝑘₃₁+𝜆(1+𝜆)(𝜎+𝜆)(1+𝜆)−𝜌𝜎=0 (51) which implies the eigenvalues 𝜆₁=−𝑏,𝜆₂=−𝜎,𝜆₃𝜆=−1,_4,5=−(𝜎+1)±(𝜎+1)²−4𝜎1−𝜌2. (52) By the assumptions of the theorem and in virtue of Lemma 11, we obtain that system (48) will asymptotically converge to the unstable equilibrium point 𝑝₀.

In the same manner of Theorem 14, we have the following result.

Theorem 15. Assume the controlled fractional-order Lorenz chaotic system ^𝑐𝐷^𝜇1𝑢₁=−𝜎𝑢₁+𝜎𝑢₃,_𝑐𝐷^𝜇1𝑢₂=−𝜎𝑢₂+𝜎𝑢₄+𝑓₄𝑢₄,_𝑐𝐷^𝜇2𝑢₃=𝜌𝑢₁−𝑢₃−ℓ𝑢₄−𝑢₁𝑢₅,_𝑐𝐷^𝜇2𝑢₄=𝜌𝑢₂+ℓ𝑢₃−𝑢₄−𝑢₂𝑢₅,_𝑐𝐷^𝜇3𝑢₅=−𝑏𝑢₅+𝑢₁𝑢₃+𝑢₂𝑢₄𝑢₅, (53) where 𝑓₄(𝑢₄)=−𝑘^𝑐₄₁𝐷^𝜇2𝑢₄−𝑘₄₂𝑢₄ is the fractional-order controller, and 𝑘_2𝑖,𝑖=1,2 is the feedback coefficient. If ℓ=0,𝑏>0,𝜎>0,𝑘₄₂=𝜎+𝑘₄₁, and 𝜎+𝜌𝑘₄₁>0, then system (53) will asymptotically converge to the unstable equilibrium point 𝑝₀.

5. Numerical Solution

In this section, we discuss the numerical solution of fractional differential equations. All the numerical simulations of fractional-order system in this paper are based on [22]. Set ℎ=𝑇/𝑁,𝑡_𝑛=𝑛ℎ,and𝑛=1,…,𝑁 with the initial condition (𝑢₁(0),…,𝑢₅(0)), therefore, the system (14) can be described as follows: 𝑢₁(𝑛+1)=𝑢₁(0)+ℎ^𝜇1Γ𝜇₁𝜎+2−𝑢^𝑝₁(𝑛+1)+𝑢^𝑝₃(),+𝑛+1_𝑛𝑗=0𝐴_{1,𝑗,𝑛+1}×𝜎−𝑢₁𝑗+𝑢₃𝑗𝑢₂(𝑛+1)=𝑢₂(0)+ℎ^𝜇1Γ𝜇₁𝜎+2−𝑢^𝑝₂(𝑛+1)+𝑢^𝑝₄(),+𝑛+1_𝑛𝑗=0𝐴_{2,𝑗,𝑛+1}×𝜎−𝑢₂𝑗+𝑢₄𝑗𝑢₃(𝑛+1)=𝑢₃(0)+ℎ^𝜇2Γ𝜇₂×+2𝜌𝑢^𝑝₁(𝑛+1)−𝑢^𝑝₃(𝑛+1)−ℓ𝑢^𝑝₄(𝑛+1)−𝑢^𝑝₁(𝑛+1)𝑢^𝑝₅()+𝑛+1_𝑛𝑗=0𝐴_{3,𝑗,𝑛+1}𝜌𝑢₁𝑗−𝑢₃𝑗−ℓ𝑢₄𝑗−𝑢₁𝑗𝑢₅𝑗,𝑢₄(𝑛+1)=𝑢₄(0)+ℎ^𝜇2Γ𝜇₂×+2𝜌𝑢^𝑝₂(𝑛+1)−𝑢^𝑝₄(𝑛+1)+ℓ𝑢^𝑝₃(𝑛+1)−𝑢^𝑝₂(𝑛+1)𝑢^𝑝₅()+𝑛+1_𝑛𝑗=0𝐴_{4,𝑗,𝑛+1}𝜌𝑢₂𝑗−𝑢₄𝑗+ℓ𝑢₃𝑗−𝑢₂𝑗𝑢₅𝑗,𝑢₅(𝑛+1)=𝑢₅(0)+ℎ^𝜇3Γ𝜇₃×+2−𝑏𝑢^𝑝₅()+𝑢𝑛+1_𝑝1(𝑛+1)𝑢^𝑝₃(𝑛+1)+𝑢^𝑝₂(𝑛+1)𝑢^𝑝₄(𝑛+1)×𝑢^𝑝₅(𝑛+1)+^𝑛_𝑗=0𝐴_{5,𝑗,𝑛+1}×−𝑏𝑢₅𝑗+𝑢₁𝑗𝑢₃𝑗+𝑢₂𝑗𝑢₄𝑗𝑢₅𝑗, (54) where 𝑢^𝑝₁(𝑛+1)=𝑢₁(0)+^𝑛_𝑗=0𝐵_{1,𝑗,𝑛+1}×𝜎−𝑢₁𝑗+𝑢₃𝑗,𝑢^𝑝₂(𝑛+1)=𝑢₂(0)+^𝑛_𝑗=0𝐵_{2,𝑗,𝑛+1}×𝜎−𝑢₂𝑗+𝑢₄𝑗,𝑢^𝑝₃(𝑛+1)=𝑢₃(0)+^𝑛_𝑗=0𝐵_{3,𝑗,𝑛+1}×𝜌𝑢₁𝑗−𝑢₃𝑗−ℓ𝑢₄𝑗−𝑢₁𝑗𝑢₅𝑗,𝑢^𝑝₄(𝑛+1)=𝑢₄(0)+^𝑛_𝑗=0𝐵_{4,𝑗,𝑛+1}×𝜌𝑢₂𝑗−𝑢₄𝑗+ℓ𝑢₃𝑗−𝑢₂𝑗𝑢₅𝑗,𝑢^𝑝₅(𝑛+1)=𝑢₅(0)+^𝑛_𝑗=0𝐵_{5,𝑗,𝑛+1}×−𝑏𝑢₅𝑗+𝑢₁𝑗𝑢₃𝑗+𝑢₂𝑗𝑢₄𝑗𝑢₅𝑗, (55) and for 𝑘=1,…,5𝐴_{𝑘,𝑗,𝑛+1}=𝑛^𝜇+1−𝑛−𝜇(𝑛+1)^𝜇,𝑗=0,𝑛−𝑗+2^𝜇+1+𝑛−𝑗^𝜇+1−2𝑛−𝑗+2^𝜇+1𝐵,1≤𝑗≤𝑛,1,𝑗=𝑛+1,_{𝑘,𝑗,𝑛+1}=ℎ^𝜇𝜇𝑛−𝑗+1^𝜇−𝑛−𝑗^𝜇,0≤𝑗≤𝑛. (56) The error of this approximation can be computed as follows: 𝑢_𝑖𝑡_𝑛−𝑢_𝑖(𝑛)=𝑜ℎ^𝜇,𝑝=min2,1+max𝜇_1,2,3. (57)

6. Synchronizing 𝑝₀

we consider a feedback controller for the fractional-order Lorenz chaotic system (14) via fractional-order derivative and obtain the controlled response system ^𝑐𝐷^𝜇1𝑣₁=−𝜎𝑣₁+𝜎𝑣₃,_𝑐𝐷^𝜇1𝑣₂=−𝜎𝑣₂+𝜎𝑣₄,_𝑐𝐷^𝜇2𝑣₃=𝜌𝑣₁−𝑣₃−ℓ𝑣₄−𝑣₁𝑣₅+𝑉,_𝑐𝐷^𝜇2𝑣₄=𝜌𝑣₂+ℓ𝑣₃−𝑣₄−𝑣₂𝑣₅,_𝑐𝐷^𝜇3𝑣₅=−𝑏𝑣₅+𝑣₁𝑣₃+𝑣₂𝑣₄𝑣₅, (58) where 𝑉=𝑘₁^𝑐𝐷^𝜇1𝑣₁−𝐷^𝜇1𝑢₁+𝑘₂𝑣₁−𝑢₁+𝑣₁𝑣₅−𝜌𝑢₁+𝑢₃+ℓ𝑢₄ (59) is the fractional-order controller, and 𝑘_𝑖,𝑖=1,2 is the feedback coefficient. We need the following result which can be found in [53].

Lemma 16. The following linear commensurate fractional-order autonomous system, ^𝑐𝐷^𝜇𝑢=𝐴𝑢,𝜇∈(0,1), (60) is asymptotically stable if and only if |arg𝜆|>0.5𝜋𝜇 is satisfied for all eigenvalues (𝜆) of matrix 𝐴. Also, this system is stable if and only if |arg𝜆|≥0.5𝜋𝜇 is satisfied for all eigenvalues (𝜆) of matrix 𝐴, and those critical eigenvalues which satisfy |arg𝜆|=0.5𝜋𝜇 have geometric multiplicity one, where 0<𝜇<1,𝑢∈𝑅^𝑛, and 𝐴∈𝑅^𝑛×𝑅^𝑛.

We have the following theorem.

Theorem 17. If 𝑏>0,𝜎>0,𝑘₂=−𝜌+𝑘₁𝜎, and 𝜎(𝑘₁+𝜌)<1, then the fractional-order Lorenz chaotic system (14) and the controlled fractional-order Lorenz chaotic system (58) achieved synchronization via fractional-order derivative.

Proof. Define the synchronization error variables as follows: 𝑒_𝑗=𝑣_𝑗−𝑢_𝑗,𝑗=1,…,5. (61) Therefore, we obtain the system ^𝑐𝐷^𝜇1𝑒_1𝑐𝐷^𝜇1𝑒_2𝑐𝐷^𝜇2𝑒_3𝑐𝐷^𝜇2𝑒_4𝑐𝐷^𝜇3𝑒₅𝑒=𝐴¹𝑒₂𝑒₃𝑒₄𝑒₅, (62) where 𝐴=−𝜎0𝜎000−𝜎0𝜎0𝜌−𝑘₁𝜎+𝑘₂0−1+𝑘₁𝜎−ℓ00𝜌−𝑢₅ℓ−1−𝑣₂𝑢₅𝑢₃𝑢₅𝑢₄𝑢₅𝑣₁𝑢₅𝑣₂−𝑏+𝑣₁𝑣₃+𝑣₂𝑣₄. (63) Consider 𝑢_𝑖 and 𝑣_𝑖 with 𝑘₂=−𝜌+𝑘₁𝜎, such that 𝐴=−𝜎0𝜎000−𝜎0𝜎000−1+𝑘₁𝜎−ℓ00𝜌ℓ−100000−𝑏. (64) Thus, the characteristic equation is of the form (𝑏+𝜆)(𝜎+𝜆)(𝜎+𝜆)(1+𝜆)𝑘₁()𝑘𝜎−𝜆−1−𝜌𝜎1+𝜆₁𝜎−𝜆−1−ℓ²(𝜎+𝜆)=0. (65) Because 𝜎(𝑘₁+𝜌)<1 yields that |arg𝜆_𝑖(𝐴)|>𝜋/2>(𝜋/2)𝜇,𝑖=1,…,5. According to Lemma 16, it implies that the equilibrium point 𝑝₀ of error system (60) is asymptotically stable,and hence the fractional-order Lorenz chaotic system (14) and the controlled fractional-order Lorenz chaotic system (58) achieved synchronization via fractional-order derivative. The proof is completed.

7. Conclusion and Discussion

The five-dimensional complex models are habitually used to clarify and simulate the physics of tuned lasers. By using fractional-order derivative (in sense of the Caputo derivatives), we stabilized the unstable equilibrium points of the complex fractional-order Lorenz chaotic system and included chaos synchronization for the fractional-order Lorenz chaotic system. Numerical solutions are computed for these systems. Conditions are imposed to study the stability, and several cases are discussed for these systems. Figure 1 shows the asymptotically stable at 𝑝₀, such that 𝜎>0,𝑏>0, and the values of 𝜌 are taken positive in (a) with the initial values (4,5,6), negative in (b) with the initial values (1,1,1), and for 𝜌=0 in (c), while in (d) the parameters are valued as 𝜎=1,𝜌=2,and𝑏=3 with the initial values (4,5,6). Figures 2 and 3 confirm the time evolution at 𝑝₀. Figures 4–6 show the affectedness of ℓ=0,1,10, where (𝜎,𝜌,𝑏,ℓ)=(10,28,8/3,0) in Figure 4, (10,28,8/3,1) in Figure 5, and (10,28,8/3,10) in Figure 6. Figure 7 shows the phase plane of 𝑝₀. Here we discuss the system for 𝜇_𝑖=0.998,𝑖=1,2,3. Note that 𝑢₂ and 𝑢₄ are congruent on 𝑢₁ and 𝑢₃, respectively. Finally, Figure 8 represents to the chaotic attractors of the proposed system with the parameters (𝜎,𝜌,𝑏,ℓ)=(10,28,8/3,0) and (𝜎,𝜌,𝑏,ℓ)=(10,8,8/3,0) with the initial values (1,2,3), respectively.

(a)

(b)

(c)

(d)

Figure 1 

Asymptotically stable at 𝑝0.

(a)

(b)

(c)

Figure 2 

Time evolution for (𝑢1,𝑢3,and𝑢5) at 𝑝0.

(a)

(b)

(c)

Figure 3 

Time evolution for (𝑢2,𝑢4,and𝑢5) at 𝑝0.

Figure 4 

(10,   28,  8/3,   0).

Figure 5 

(10, 28, 8/3, 1).

Figure 6 

(10, 28, 8/3, 10).

Figure 7 

The phase plane of 𝑝0.

(a)

(b)

(c)

(d)

Figure 8 

Chaotic attractors of the proposed system.