Abstract

In this paper, based on the earlier research, a new fractional-order chaotic Genesio-Tesi model is established. The chaotic phenomenon of the fractional-order chaotic Genesio-Tesi model is controlled by designing two suitable time-delayed feedback controllers. With the aid of Laplace transform, we obtain the characteristic equation of the controlled chaotic Genesio-Tesi model. Then by regarding the time delay as the bifurcation parameter and analyzing the characteristic equation, some new sufficient criteria to guarantee the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model are derived. The research shows that when time delay remains in some interval, the equilibrium point of the controlled chaotic Genesio-Tesi model is stable and a Hopf bifurcation will happen when the time delay crosses a critical value. The effect of the time delay on the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model is shown. At last, computer simulations check the rationalization of the obtained theoretical prediction. The derived key results in this paper play an important role in controlling the chaotic behavior of many other differential chaotic systems.

1. Introduction

As is known to us, chaos control issue has been widely studied in the last decades because of its potential practical value in various areas. How to control the chaotic phenomenon to serve human beings has become a hot issue in today’s world. Chaos control has attracted much attention of researchers from various fields. In recent years, many chaos control techniques (for example, delayed feedback approach [1], Ott–Grebogi–York (OGY) technique [2], variable structure control [3], observer-based control [4], backstepping design technique [5], and active control [6]) have been proposed. Many excellent fruits have been reported. For example, Kocamaz et al. [7] investigated the chaos control by applying the sliding mode control technique; Chen [8] proposed an adaptive feedback control technique for chaos and hyperchaos control; Din [9] studied the chaos control of a discrete-time prey-predator system by using three different types of feedback control strategies; Singh and Gakkhar [10] controlled the chaos of a food chain model with a time-delayed feedback controller; Yan et al. [11] discussed the chaos control of continuous unified chaotic systems applying discrete rippling sliding mode control. For more knowledge on chaos control, readers can refer to [1219, 3444].

In 1992, Genesio and Tesi [20] put up the chaotic Genesio-Tesi system:where denote different voltage of electronic components and are all negative coefficients. Genesio and Tesi [20] investigated the chaotic behavior of model (1) by applying harmonic balance approaches. In 2012, Guan et al. [21] discussed the chaos control of model (1) by designing distributed delay feedback controller. By analyzing the characteristic equation of the controlled Genesio-Tesi model, the sufficient conditions to ensure the stability of the equilibrium point and the existence of Hopf bifurcation for the controlled Genesio-Tesi model are established. In addition, the stability and the direction of bifurcating periodic solution are determined by the centre manifold theorem and normal form theory. In 2009, Sun [22] proposed a tracking control to realize chaos synchronization for the Genesio-Tesi chaotic system (1) based on the time-domain approach. In 2007, Park [23] designed a novel feedback controller to realize exponential synchronization of the Genesio-Tesi chaotic system (1). In 2009, Park [24] further considered the functional projective synchronization problem for the Genesio-Tesi chaotic system (1) and Zhou and Chen [25] dealt with the Hopf bifurcation and Si’lnikov chaos of Genesio model (1).

It is worth mentioning that all the above papers focus only on the integer-order differential models. In recent years, many researchers argue that fractional-order differential equations play a key role in describing the real phenomena and are found to have potential applications in various areas such as physics, economics, physics, heat transfer, and chemical engineering [2628]. Thus, it is important to deal with the fractional-order differential models. Based on discussion above, we revise model (1) in the following fractional-order form:where denote different voltage of electronic components, are all negative coefficients, and denotes the fractional order. Model (2) is the commensurate fractional-order system. Let and , then the chaotic phenomenon will occur in model (2). The fact is shown in the following (see Figures 1 and 2).

This paper mainly focuses on two aspects: (a) designing two appropriate controllers to control the chaotic phenomenon of model (2) and (b) revealing the impact of time delay on the stability and bifurcation behavior of the controlled fractional-order Genesio-Tesi chaotic model. The superiority of this paper can be summarized as follows:(a)A new fractional-order Genesio-Tesi chaotic model is proposed(b)Two controllers are designed to control the chaotic phenomenon of the fractional-order Genesio-Tesi chaotic model(c)The advantages and disadvantages of two controllers are compared

The outline of this paper is organized as follows: In Section 2, some elementary knowledge on fractional-order differential is prepared. In Section 3, two different control techniques are designed to control the chaotic phenomenon of the fractional-order chaotic Genesio-Tesi model. In Section 4, two examples are given to illustrate the theoretical analysis. In Section 5, we end this manuscript with conclusion.

2. Elementary Knowledge

In this section, we list several basic results about fractional-order differential equations.

Definition 1 (see [29]). The Caputo fractional-order derivative is defined as follows:where , , and m is a positive integer such that . If , then

Definition 2 (see [30]). The fractional-order system is given as follows:where and . is said to the equilibrium point of system (5) if .

Lemma 1 (see [31]). Denote the root of the characteristic equation of the autonomous systemwhere Then, system (6) is asymptotically stable and also stable , and those critical eigenvalues that satisfy have geometric multiplicity one.

Lemma 2 (see [32]). For the systemwhere , the characteristic equation takes the following form:

The zero solution of system (7) is asymptotically stable if all the roots of equation (8) are negative real roots.

Remark 1. In Lemma 1 and Lemma 2, we say that system (6) (or system (7)) is asymptotically stable (or stable), implying that the fixed point E of system (6) (or system (7)) is asymptotically stable (or stable).

3. Chaos Control via Time-Delayed Feedback Controllers

It is easy to see that system (2) has two unique equilibrium points:

A long time ago, there were many publications that handle the chaos control issue of integer-order differential models by applying time-delayed feedback controllers. But the studies on the chaos control of fractional-order chaotic models via time-delayed feedback controllers are very rare. To make up the deficiency, in this paper, we will design two time-delayed feedback controllers to eliminate the chaotic behavior of system (2). In this paper, we are concerned only with the equilibrium point . The other equilibrium point is easily analyzed in a similar way.

3.1. Adding Time-Delayed Feedback Controller to the First Equation of Model (2)

In this section, we add a time-delayed feedback controller to the first equation of model (2), and then system (2) takes the following form:where is the gain coefficient and ϱ is the time delay. The linear system of system (10) near the equilibrium point reads

The corresponding characteristic equation of (11) takes the following form:

Hence,where

Suppose that is the root of (13), then we getwhere

In view of (15), we have

Let

Hence,

By (19), we have

In addition,where

By (20), we havewhere

Let

Lemma 3. (i) Assume that , then the root that has zero real parts does not exist in (13).(ii) Assume that and which satisfies , then (13) possesses at least two pairs of purely imaginary roots.

Proof of Lemma 3(i). According to (25), we haveIn view of , we have , . In view of , we know that (26) does not possess positive real roots. Furthermore, is not the root of (13). This proves Lemma 3(i).

Proof of Lemma 3(ii). According to , and , and such that Hence, (26) possesses at least two positive real roots and (13) possesses at least two pairs of purely imaginary roots. This proves Lemma 3 (ii).

Suppose that (23) possesses ten positive real roots . In view of (17), we getwhere Hence, is a pair of purely imaginary roots of (13) when . Let

Next, the hypothesis is given: where

Lemma 4. Suppose that is the root of (13) around such that , then

Proof of Lemma 4. In view of (13), we getSincewe obtainwhereThus,It follows from thatThis proves Lemma 4.

Next, the hypothesis is given:

Lemma 5. If and holds true, then system (13) is locally asymptotically stable.

Proof of Lemma 5. Under the condition , (13) becomesIn view of , where is the root of (38). Thus, Lemma 5 holds true, implying the proof.

The following results are established by above discussion.

Theorem 1 (under the condition of Lemma 3(ii)). Suppose that and hold true, then (i) the equilibrium point of system (10) at the origin is locally asymptotically stable if and (ii) a Hopf bifurcation will happen near the equilibrium point if .

3.2. Adding Time-Delayed Feedback Controller to the Second Equation of Model (2)

In this section, we add a time-delayed feedback controller to the second equation of model (2), and then system (2) becomeswhere is the gain coefficient and ρ is the time delay. The linear equation of equation (39) around is

The characteristic equation of (40) takes the following form:

Hence,where

Assume that is the root of (42), then we havewhere

According to (44), we get

Let

Then,

By (46), we have

Furthermore,where

In view of (49), we havewhere

By (52), we have

Let

Lemma 6. (i) Suppose that , then the root that has zero real parts does not exist in (42).(ii) Suppose that and which satisfies , then (42) possesses at least two pairs of purely imaginary roots.

Proof of Lemma 6(i). In view of (55), we haveIn view of , we have , . In view of , we know that (56) does not possess positive real roots. Furthermore, is not the root of (42). This proves Lemma 6(i).

Proof of Lemma 6(ii). In view of , and , and such that Hence, (55) has at least two positive real roots and (42) has at least two pairs of purely imaginary roots. This proves Lemma 6(ii).

Assume that (54) has eight positive real roots . By (46), we havewhere Hence, is a pair of purely imaginary roots of (42) when . Let

Next, we give the hypothesis as follows: where

Lemma 7. Assume that is the root of (42) around such that , then

Proof of Lemma 7. By (42), we haveSincewe havewhereThus,By , we haveThis proves Lemma 7.

Next, we give the hypothesis as follows:

Lemma 8. If and holds true, then system (42) is locally asymptotically stable.

Proof of Lemma 8. If , then (42) takes the following form:In view of , where is the root of (68). Therefore, Lemma 8 holds true, implying the proof.

Theorem 2 (in addition to the condition of Lemma 6(ii)). Assume that and are fulfilled, then (i) the equilibrium point of model (39) at the origin is locally asymptotically stable if and (ii) a Hopf bifurcation will happen near the equilibrium point if .

Remark 2. Although there are many papers [17, 18, 21, 23] that deal with the chaos control by applying the time-delayed feedback controller, they only focus on the integer-order differential systems. In this paper, we handle the chaos control for the fractional-order differential model. All the derived results and analysis ways for the chaos control of integer-order differential systems cannot be transferred to (2) to control the chaotic phenomenon. From this viewpoint, we can say that the results of this paper are completely innovative and supplement the earlier investigation.

Remark 3. By computer simulations and maximum Lyapunov exponent, we show that model (2) has chaotic behavior. If the time delay is introduced into the controlled chaotic model (10) or (39), then we can control the chaotic phenomenon of the origin system (2) under some suitable conditions. Also, we can examine the chaotic nature of the fractional-order system by maximum Lyapunov exponent and computer simulations.

Remark 4. Adding the time-delayed feedback controller to the first equation and the second equation of model (2) is only a try. Of course, we can add the controller to the third equation. Through the later analysis, we can check whether this control technique is right or not.

Remark 5. Although the fractional-order chaotic Genesio-Tesi model and the integer-order chaotic Genesio-Tesi model have the same form, the dynamical behavior and the research method on stability and Hopf bifurcation are very different because of introduction and the variation of fractional order. According to some previous related papers (see [1719, 21, 38, 39]), we can see that the dynamical behavior and the research method on stability and Hopf bifurcation for fractional-order differential systems are very different from those for integer-order differential systems.

4. Two Examples

In this section, we use a new predictor-corrector method (NPCM) [33] to carry out computer simulations.

Example 1. The following model is given:Obviously, model (69) possesses the zero equilibrium point. Let , the critical frequency ,