Hybrid control of Hopf bifurcation in a fractional order small-world network model Lina MA 1,a, Xiuli ZHANG 1,b, Dawei DING 1,c,*

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1 International Conference on Avances in Mechanical Engineering an Inustrial Informatics (AMEII 2015 Hybri control of Hopf bifurcation in a fractional orer small-worl network moel Lina MA 1,a, Xiuli ZHANG 1,b, Dawei DING 1,c,* 1 School of Electronics an Information Engineering, Anhui University, Hefei, China a @qq.com, b @qq.com, c wing@au.eu.com *Corresponing author Keywors: Fractional orer system, Small worl network, Hybri control, Hopf bifurcation Abstract. In this paper, a hybri control strategy is applie to control the Hopf bifurcation in a fractional orer small-worl network moel. By choosing time-elay as a bifurcation parameter an analyzing the associate characteristic equation, we prove that the moel exhibits Hopf bifurcation when time elay passes through a critical value. In orer to control the unesirable Hopf bifurcation, a hybri control strategy is propose. By ajusting the control parameters of the strategy, it shows that the onset of Hopf bifurcation has been postpone without changing the original equilibrium point of the system. Finally, a numerical simulation is presente to verify the theoretical results. Introuction Bifurcation control refers to the issue of moifying the bifurcation characteristics so that achieving some esirable ynamical behaviors [1-2]. Various methos have been use to control bifurcations in both iscrete an continuous systems [4-5]. For Hopf bifurcation control, much work has been use to avoi this kin of behaviors by aing ifferent kin of controller. In[6], a static state feeback controller was propose by Abe an Fu. In [2,3], the authors propose a ynamic elaye feeback control metho which is utilize for stabilizing unstable fixe points near Hopf bifurcation. In [7], parameters elay feeback control was use in controlling the Hopf bifurcation. Later, in orer to control bifurcation an chaos in iscrete nonlinear ynamical systems, a hybri control strategy using both state feeback an parameter perturbation was put forwar to control the Hopf bifurcation in [8]. Starting with the work of Watts an Strogatz [9] small-sorl networks, a lot of interesting researches on the theory an application of small-worl networks have arisen [9-11]. However, the negligence of nonlinear elements such as time elay etc, the traitional moel can not reflect the realistic network transmission situation. Thus, in [10], a more general nonlinear elaye ifferential equation moel was formulate for small-worl networks by Yang. Li [11] testifie the local stability of one-orer small-worl system, an showe that Hopf bifurcation may occur as the measure parameter passes through a criticalpoint. Bifurcation perioic solutions was calculate by applying the center manifol theorem also. Besies, in orer to evelop the stability of system further, Xiao et al. [12] an Zhao [13] esigne a elaye feeback controller to introuce a new Hopf bifurcation with regar to measure parameter an time-elay parameter respectively as well as make the Hopf bifurcation to have certain characteristics. However, it is not obvious to obtain the change effect of stability region, there is also important to optimize the control technique of Hopf bifurcation aiming at the improvement of the control performance. During the past ecaes, the stuy of fractional ifferential calculus [14] has attracte increasing interest. The applications of fractional orer system have arouse exceeingly an increasingly attention not only by scientists but also by engineers in many fiels, such as physics, chemical an various engineering an so on [15]. Compare with the classical integer-orer moels, fractional-orer moels provie an meaningful instrument for the escription of memory an hereitary properties of various materials an processes [16]. Shi et al. [17] stuie stability an The authors - Publishe by Atlantis Press 1226

2 Hopf bifurcation control of a fractional-orer small worl network moel an it has shown that small nonlinear can also result in instability of system an the onset of Hopf bifurcation. In this paper, we attempt to apply a hybri control strategy to a fractional orer small-worl network moel, stuy the Hopf bifurcation an its stability aiming at elaying the onset of Hopf bifurcation.thanks to the avantage of hybri strategy that combines state feeback with parameter perturbation to realize bifurcation control without changing the equilibrium point, the performance of the original system can be retaine completely. Here we choose time-elay as a bifurcation parameter.with emphasis on the relationship between the Hopf bifurcation an the time-elay, we will investigate the effect of time-elay on bifurcating behaviors in the small-worl network moel. The remainer of this paper is organize as follows. In Section 2, some properties of the uncontrolle fractional orer small-worl network system are summarize. In Section 3, hybri control strategy is applie to original moel, an the existence of the Hopf bifurcation of this system is stuie. Numerical simulations an conclusion are given to verify the theoretic analysis in Section 4 an Section 5, respectively. Hopf bifurcation of uncontrolle System In this section, we consier a time-elaye Hopf bifurcation in a fractional orer small-worl network moel. The uncontrolle system can be moele by the following elay ifferential equation. 2 (1 D V ( t = ξ + V ( t τ µξ V ( t τ wherev is the total influence volume, µ is a measure of nonlinear interactions in the network, is the imension of the network an the orer of ifferential equation is 0 < < 1, D V ( t is fractional-orer erivative anξ is the Newman-Watts length scale. LetV be the nonzero equilibrium point of system (1.It then satisfies the following equation: V = (2 2µξ For convenience,the results of stability an Hopf bifurcation of system (1 are summarize here for comparison an completeness.the etaile analysis for the system can be obtaine in [20]. Theorem 1 For the system (1,combing Eq.(2 we can get the following results in: (1Whenτ < τ 0, the equilibrium point of the system (1 is locally asymptotically stable; (2Whenτ = τ 0,a Hopf bifurcation occurs; (3Whenτ > τ 0,the equilibrium point of the system (1 is asymptotically unstable an a limit cycle exists. π π where the critical valueτ 2 0 =. 1+ Hybri control of bifurcation Now we turn to esign a controller to accomplish the control of the Hopf bifurcation arising from the fractional orer small-worl network system (1.Equation (1 is onate as D V ( t = g( V ( t, µ,τ (3 By escribe the Hybri control strategy an aing it to the moel (3,the controlle system is as follows: D V ( t = αg( V ( t, µ, t + ( 1 α ( V ( t t V 2 = α[ ξ + V ( t t µξ V ( t t ] + ( 1 α ( V ( t t V (4 whereα is a control parameter an 0 < α < 1.The controlle system (4 reuces to the original system 1227

3 (3 ifα = 1.By selecting the appropriate control parameterα,the Hopf bifurcation can be elaye or even eliminate completely without changing the equilibrium point of the system.so,we set u ( t = V ( t V,the right-han sie of Eq.(4 is expane by a Taylor expansion aroun the equilibrium pointv,we have ( α α µξ 2 D V t = v( t τ + αµξ V ( t τ (5 The linearize prat of system (5 is D u( t = (1 α v( t τ (6 With the initial conition at zero,applying fractional orer Laplace transformation [ ( D α ζ t u( t ]( s = s [ ζ ( u( t ]( s we can get characteristic equation of Eq.(6 is λτ λ α + α + µξ e = 0. (7 If the characteristic Eq.(7 has pure imaginary roots λ = ± i ω, ω > 0,then we have π ω cos + 1 α cos( ωτ = 0 2 π ω sin ( = 1 α sin ωτ 0 2 (8 From Eq.(8,we obtain 2nπ + π ( π / 2 ω = 1 α, τ n =, n = 0,1,2,. (9 1 α In orer to ensure that Eq.(7 has roots with positive real parts except forτ = τ 0,we let λ = r cos θ + i sinθ = s + iω, ω >,then ( 0 ( sτ r cos θ 1 α α 1 e cos( ωτ ( + ( + sτ r sin θ 1 α α 1 e sin ωτ From Eq.(11,we get ταn( ωτ ταn( αθ = 0 = 0 =,an it exists a non-negative integer m to make ωτ = 2 mπ + π αθ.ifσ > 0,thenθ is satisfie π / 2 < θ < π / 2,we get (11 2mπ + π ( π / 2 < ωτ < 2mπ + π + ( π / Insertingτ = τ 0,we obtain ωτ < 2nπ + π ( π /,Therefore,we can obtain a conclusion as follows: When n 1,it hasω in certain to make two equations of Eq.(11 come into existence at the same time.eq.(7 has roots with positive real parts,anv is unstable. When n = 0,there is no roots with positive real parts.that is to say,whenτ > τ 0,Eq.(7 at least has one root with positive real parts,an π π τ 0 = 2 (12 1 α In orer to fin a Hopf bifurcation point,the following transversality conition is neee λ R τ = τ 0 τ > 0 (13 Hence,let λ = ± iω be the root of Eq.(7,by calculating,we have λ ω 1 α R τ = τ 0, λ = ± iω = > (14 τ ω 1 α + α µξ τ Therefore,the final conition for the occurrence of a Hopf bifurcation in the nonlinear moel ( (10

4 is inee satisfie.thus we have the following theorem. Theorem 2 For the controlle system (4,we can easily obtain (1When τ ( 0, τ 0, the equilibrium pointv of the controlle system (4 is locally asymptotically stable; (2Whenτ = τ 0,the controlle system (4 exists a Hopf bifurcation at equilibrium pointv. (3When τ ( τ 0,,the equilibrium pointv of the controlle system (4 is asymptotically unstable. Numerical simulations In this section,we present numerical results to verify the analytic preictions obtaine in the previous section,using the hybri control strategy to control the Hopf bifurcation in a fractional orer small-worl network moel (4.For a consistent comparison,we choose the system parameters as use in [12],withξ = 3, µ = We consier the effect of hybri control parameterα on the stability of Eq.(4.We choose α =1,as we know,the system is the uncontrolle moel.for the uncontrolle moel,we know that when = 0. 8,we getτ 0 =1. 355, V = The ynamical behavior of the uncontrolle moel (1 is illustrate in Fig.1 an Fig.2.It is shown that whenτ < τ0,trajectories converge to the equilibrium point,while asτ is increase to passτ 0, V loses stability an a Hopf bifurcation occurs.then,we chooseα = 0. 85the system becomes the controlle moel,the critical valueτ 0 increases from τ 0 = to τ 0 = The ynamical behavior of the controlle moel (4 is illustrate in Fig.3 an Fig.4.We can see that the onset of Hopf bifurcation is elaye, an the stable range in parameter space is extene. Thus,by the hybri control strategy,we can increase the critical value of time elay,exten the stable region of equilibrium point of controlle system an elay the onset of Hopf bifurcation.it is shown that this metho can also be use in fractional orer moel,an easy to be applie in reality. Fig. 1 Waveform graph an phase portrait of uncontrolle system with τ =1. 35 Fig. 2 Waveform graph an phase portrait of uncontrolle system with τ =

5 Fig. 3 Waveform graph an phase portrait of controlle system withτ = 1.35, α = Fig. 4 Waveform graph an phase portrait of controlle system withτ = 1.45, α = Conclusion In this paper,the problem of Hopf bifurcation control for a fractional orer small-worl network moel has been stuie.to control the Hopf bifurcation,a hybri control strategy has been propose.by selecting appropriate control parameters,this metho can effectively elay the onset of Hopf bifurcation an exten the stable region of equilibrium point of controlle system.numurical results have valiate the correctiness of the theoretical analysis. Acknowlegments This work is supporte National Natural Science Founation of China(No , College Stuents' innovation an entrepreneurship training plan of Anhui University (J ,College Stuents' research training plan of Anhui University (J Reference [1] P. Hou, Z. Wang, Stability an Hopf bifurcation of a flui-flow moel for congestion control in wireless networks, Journal of Electronics an Information Technology. 32 ( [2] S. Guo, G. Feng, X. Liao, Q. Liu, Hopf bifurcation control in a congestion control moel via ynamic elaye feeback, Chaos.18 ( [3] B. Rezaie, M. Jahe Motlagh, M. Analoui, S. Khorsani, Stabilizing fixe points of time-elay systems close to the Hopf bifurcation using a ynamic elaye feeback control metho, J. Phys. A: Math. Theor. 42 ( [4] Hopf-transcritical bifurcation in retare functional ifferential equations.t,m&a.73 ( [5] Zhaiybai.T,Equilibrium-torus bifurcation in nonsmooth systems.nonlinear phenomena.7 ( [6] L. Nguyen, K. Hong, Hopf bifurcation control via a ynamic state-feeback control.phys. Lett. A 376 (

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