Hopf bifurcation control of ecological model via PD controller

Transcription

1 Global Journal of Pure an Applie Mathematics. ISSN Volume 1, Number 6 (016), pp Research Inia Publications Hopf bifurcation control of ecological moel via PD controller Mohamma Darvishi 1 Department of Mathematics, Faculty of Mathematics an Statistics, University of Birjan, Birjan, Iran. Haji Mohamma Mohammaineja Department of Mathematics, Faculty of Mathematics an Statistics, University of Birjan, Birjan, Iran. Abstract In this paper, we investigate the problem of bifurcation control for an ecological mole with timeelay. By choosing the timeelay as the bifurcation parameter, we present a Proportional-Derivative (PD) feeback controller to control Hopf bifurcation. We show that the onset of Hopf bifurcation can be elaye or avance via a PD controller by setting proper controling parameters. Uner consieration moel as operator Equation, apply orthogonal ecomposition, compute the center manifol an normal form we etermine the irection an stability of bifurcating perioic solutions. Therefore the Hopf bifurcation of the moel became controllable to achieve esirable behaviors which are applicable in certain circumstances. AMS subject classification: 34H0, 34D3. Keywors: Hopf bifurcation, bifurcation control, PD controller, ecological mole. 1 Corresponing author.

2 484 Mohamma Darvishi an Haji Mohamma Mohammaineja 1. Introuction The elay ifferential equation (DDE) s x(s) = ax(s τ)e bx(s τ) cx(s) (1.1) which is one of the important ecological systems, escribes the ynamics of Nicholsons blowflies equation. Here x(s) is the size of the population at time s, a is the maximum per capita aily egg prouction rate, 1 is the size at which the population b reprouces at the maximum rate, c is the per capita ( ) aily ault eath rate, an τ is the 1 ( a ) generation time, the positive equilibrium x = ln. Equation (1.1) has been b c extensively stuie in the literature. The majority of the results on (1.1) eal with the global attractiveness of the positive equilibrium an oscillatory behaviors of solutions. The aim of bifurcation control is to elay (avance) the onset of an inherent bifurcation, change the parameter value of an existing bifurcation point, the PD control strategy is use to control the bifurcation [1].. Existence Hopf bifurcation in uncontrol ecological mole We consier the elay ifferential equation (1.1) an let u(s) = x(τs) then s u(s) = s x(τs) = τ[ax(τs τ)e bx(τs τ) cx(τs)] (.) an equation(1.1) can be rewritten as s u(s) = aτu(s 1)e bu(s 1) cτu(s) (.3) We consier ( ) the timeelay as the bifurcation parameter, The critical point in the system 1 ( a ) is u = ln an With the change of variables z(s) = u(s) u we have b c s z(s) = aτ(z(s 1) + u )e b(z(s 1)+u ) cτ(z(s) + u ) (.4) the linearization of equation (,3) at z(s)=0 is whose characteristic equation is s z(s) = cτ[(1 bu )z(s 1) z(s)] (.5) λ = cτ[(1 bu )e λ 1] (.6)

3 Hopf bifurcation control of ecological mole via PD controller 4843 an Assume that λ = iω is a root of the equation (.6) that ω>0 then iω = cτ[(1 bu )e iω 1] (.7) iω = cτ[(1 bu )(cos ω i sin ω) 1] (.8) Separating the real an imaginary parts, we obtain cτ(1 bu ) cos ω = cτ (.9) an cτ(1 bu ) sin ω = ω (.10) then ω =±τc (1 bu ) 1if(1 bu ) 1 > 0. Let λ k (τ) = α k (τ) + iβ k (τ) enote a root of equation (.6) an near τ = τ k such that α k (τ k ) = 0,β k (τ k ) = ω so that λ k (τ k ) = α k (τ k ) + iβ k (τ k ) = iω an iω = cτ k [(1 bu )e iω 1], icτ k (1 bu ) 1 = cτ k [(1 bu )(cos(cτ k (1 bu ) 1) isin(cτ k (1 bu ) 1) 1] Then ecompose the equationin to real an imaginary parts cτ k (1 bu )(cos(cτ k (1 bu ) 1) cτ k = 0 (.11) an cτ k (1 bu ) 1 + cτ k (1 bu )sin(cτ k (1 bu ) 1) = 0 (.1) which lea to 1 (1 τ k = c (1 bu ) 1 [arcsin( bu ) 1 )+kπ] k = 0, 1,,... (.13) (1 bu ) Lemma.1. We have to equation (.6). Proof. Differentiating both sies of equation (.6) with respect to τ, we obtain λ τ = c[(1 bu )e λ 1] 1 + τc[(1 bu )e λ ] (.14)

4 4844 Mohamma Darvishi an Haji Mohamma Mohammaineja therefore λ τ τ=τ k = c(1 bu )cosω c ic(1 bu )sinω 1 + τ k c(1 bu )cosω iτ k c(1 bu )sinω (.15) this implies that then Re(λ) τ=τk = τ c(1 bu )cosω + τ k [c(1 bu )] [1 + τ k c(1 bu )cosω] +[τ k c(1 bu )sinω] (.16) Re(λ) c + τ k [c(1 bu )] τ=τk τ [1 + τ k c(1 bu )cosω] +[τ k c(1 bu )sinω] (.17) so that completing the proof. Theorem.. For the system (1.1), the following statements are true: (i) if c<a<ce,then x = x is asymptotically stable. (ii) if a>ce (bx > ), then x = x is asymptotically stable for τ [0,τ 0 ) an unstable for τ>τ 0 (iii) Equation (1.1) unergoes a Hopf bifurcation at x = x when τ = τ k for k = 0, 1,,...[1]. Figure 1: the numerical solution of uncontrol ecological mole with corresponing to (A) a = 30,b =,c =,τ = 0.4 an (B) a = 30,b =,c =,τ = 0.8.

5 Hopf bifurcation control of ecological mole via PD controller Hopf bifurcation in controlle moel base on PD controller In this section,we focus on esigning a controller to control the Hopf bifurcation in moel base on the PD control strategy. The PD controller uses a single-input an single-output system as follows: the input error signal is efine as e(t) = z(s) z an the output control law u(t) is efine as u(t) = k p z(s) + k ( z(s)) (3.18) s where k p is proportional control parameter an k is erivative control parameter an k p < 1; k < 1. Apply the PD controller to system (.4), we get s z(s) = aτ(z(s 1)+u )e b(z(s 1)+u ) cτ(z(s)+u )+k p z(s)+k z(s) (3.19) s Therefore, the controlle moel can bee expresse as follows s z(s) = 1 1 k [aτ(z(s 1) + u )e b(z(s 1)+u ) cτ(z(s) + u ) + k p z(s)] (3.0) Comparing Equations (3.0) an (.4), we fin that the controlle moel an uncontrolle moel have the same equilibrium point. Expaning the right sie of Equation (3.0) into Taylor series at z(t) = 0, we consier the linear part s z(s) = 1 [cτ((1 bu )z(s 1) z(s)) + k p z(s)] (3.1) 1 k let a 1 = τc(1 bu ) 1 k an a = k p τc 1 k then s z(s) = a 1z(s 1) + a z(s) (3.) Characteristic equation corresponing to Equation (3.) is λ a 1 e λ a = 0 (3.3) Assume Equation (3.3) has a pair of pure imaginary roots λ =±iω with ω > 0. Inserting them into Equation (3.3). Then ecompose the characteristice quationin to real an imaginary parts a + a 1 cos(ω ) = 0 (3.4) ω + a 1 sin(ω ) = 0 (3.5) From Equations (3.4), (3.5) an combine with the formula: sin (ω )+cos (ω ) = 1, we obtain ω = a1 a (3.6)

6 4846 Mohamma Darvishi an Haji Mohamma Mohammaineja an ( ω = arccos a ) a 1 (3.7) Combine Equations (3.6), (3.7) an we can get the equationas follows: ( a 1 a = arccos a ) (3.8) a 1 equation (3.8) is a transcenental equation, we can get its numerical solution by numerical simulation software. [], We nee to get the fixe point τ k of the system such that λ (τ k) = iω, the value of ω can be erive from equation (3.3) then. We still nee to check the following transversal conition in orer to verify the onset of Hopf bifurcation: Lemma 3.1. For equation (3.3) if τ k k p > c k 1 + τ kc then Re(λ) τ=τ τ k > 0. Proof. Let r = c(1 bu ), ifferentiating both sies of equation (3.3) with respect to 1 k τ, we obtain Therefore this implies that λ re λ τ + τre λ λ + λ τ τ=τ k c 1 k τ = 0 c rcosω irsinω = 1 k 1 + τ krcosω iτ krsinω Re(λ) τ=τ τ k = k p(k 1) + τ kc(k p τ kc) + τ k r (1 k ) τ k(1 k ) [(1 + τ krcosω ) + (τ krsinω ) ] so that completing proof. Therefore, the transversal conition for Hopf bifurcation in the PD controlle system (3.0) is satisfie. We conclue above analyzes in following: Theorem 3.. For the controlle system (3.0), there exists a Hopf bifurcation emerging from its equilibrium z = 0 when the positive parameter τ passes through the critical value τ k. We note that the equilibrium point remains unchange an the critical value τ k varies with control parameters k p an k. Theorem (3.19) shows that the PD controller can be applie to the ecological moel for the purpose of bifurcation control.

7 Hopf bifurcation control of ecological mole via PD controller 4847 Figure : the numerical solution of control ecological mole with corresponing to (A) a = 30,b =,c =,τ = 0.4,k = 0.5,k p = 0.1 an (B) a = 30,b =,c =,τ = 0.4,k = 0.3,k p = 0.1. Figure 3: the numerical solution of control ecological mole with corresponing to (A) a = 30,b =,c =,τ = 0.4,k = 0.3,k p = 0.4 an (B) a = 30,b =,c =,τ = 0.8,k = 0.5,k p = Stability an irection of bifurcating perioic solutions In this section we stuy the irection of the Hopf bifurcation an the stability of the bifurcation perioic solutions when τ = τ τ k k uner the conition k p > c k 1 + τ kc an using techniques from normal form an center manifol theory by Hassar et al. [7]. For convenience, let µ = τ τ k, therefore µ = 0 is the value of hopf bifurcation for equation (3.0). In this section we will be iscusse in multiple parts. In the frist part we will formulate the operator form for (3.0), escribe a process of etermining the critical eigenvalues for the problem, formulate the ajoint operator equation an Apply Orthogonal Decomposition. then Compute the Center Manifol Form, Develop the Normal Form on the Center Manifol an at the en examine perioic solutions conitions on center manifol.

8 4848 Mohamma Darvishi an Haji Mohamma Mohammaineja Figure 4: the numerical solution of control ecological mole with corresponing to (A) a = 30,b =,c =,τ = 0.8,k = 0.3,k p = 0.1 an (B) a = 30,b =,c =,τ = 0.8,k = 0.3,k p = 0.4. Step 1: Form the Operator Equation With expaning the right sie of Equation (3.0) into Taylor series at z(s) = 0, then x(s) = U(µ)x(s) + V (µ)x(s 1) + f(x(s),x(s 1), µ) (4.9) s that U(µ) = a, V (µ) = a 1 an f(x(s),x(s 1), µ) = O(x (s), x (s 1)) we consier The linear portion of equation (4.9) that is given by s x(s) = a x(s) + a 1 x(s 1) (4.30) the equations (4.9) an (4.30) can be thought of as maps of entire functions. we will start with the class of continuous functions efine on the interval [-1, 0] with values in R n an refer to this as class C 0. The maps are constructe by efining a family of solution operators for the linear portion of equation (4.9) by (ϒ(s)φ)( ) = (x s (φ))( ) = x(s + ) (4.31) for φ C 0, [ 1, 0],s 0. This is a mapping of a function in C 0 to another function in C 0. so equations (4.9), (4.30) can be thought of as maps from C 0 to C 0 an ϒ(s),s 0 is a semigroup. An infinitesimal generator of a semigroup ϒ(s) is efine by φ = lim s [ϒ(s)φ φ] s for φ C 0. in the linear equation (4.30) the infinitesimal generator can be constructe as { φ (µ)φ = ( ) 1 <0 (4.3) a φ(0) + a 1 φ( 1) = 0

9 Hopf bifurcation control of ecological mole via PD controller 4849 Figure 5: The solution operator maps functions in C 0 to functions in C 0 Then ϒ(s)φ satisfies ϒ(s)φ = (µ)ϒ(s)φ, the operator form for the nonlinear s equation (4.9) is { s x s(φ) = ϒ(µ)x s (φ)+f(x s (φ), µ), that F (φ, µ)( ) = 0 1 <0 f (φ, µ) = 0 (4.33) Step : Define an Ajoint Operator We efine a form of ajoint Operator associate with (4.33). we consier C 0 = C([0, 1],R n ) that is the space of continuous functions from[0,1] to R n. the ajoint equation associate with the equation (4.30) is we efine s y(s,µ) = a y(s,µ) a 1 y(s + 1,µ) (4.34) (ϒ (s)ψ)( ) = (y s (ψ))( ) = y(s + ) (4.35) for [o, 1],s 0,y s C 0 an y s (ψ) be the image of ϒ (s)ψ, so relationship (4.35) efines semigroup with infinitesimal generator ψ ( ) 0 < 1 (µ)ψ = ψ (4.36) (0) = a ψ(0) + a 1 ψ(1) = 0 Note that, although the formal infinitesimal generator for (4.35) is efine as 0 ψ = lim s 0 1 s [ϒ (s)ψ φ]

10 4850 Mohamma Darvishi an Haji Mohamma Mohammaineja for ψ C 0 Hale[5], that 0 = an family of operators (4.35) satisfies s ϒ (s)ψ = (µ)ϒ (s)ψ. Step 3: Define a natural Inner prouct by way of an ajoint operator Let x(s) C 0, y(s) C 0 an efine an inner prouct y(s),x(s) =y T (0)x(0) y T (s + 1)a 1 x(s)s (4.37) Step 4: Get the Critical Eigenvalues an Apply Orthogonal Decomposition As for φ C 0,ψ C 0 we have ψ, (µ)φ = (µ)ψ, φ, obviously (0) an (0) are ajoint operators an ±iω are eigenvalues of (0) an (0). to gain Orthogonal Decomposition we nee to calculate the eigenvector ξ(θ)of (0) an ξ (θ) of (0) associate with the eigenvalues iω an iω respectively. let ξ(θ) = e iω θ A, ξ(θ) = e iω θ Ā that θ [ 1, 0) an ξ (θ) = e iω θ B, ξ (θ) = e iω θ B that θ (0, 1]. let A = 1 we choose B so that ξ,ξ =1 an ξ, ξ =0. from (4.37) we have: ξ,ξ =ξ T (0)ξ(0) ξ T (s + 1)a 1 ξ(s)s = B e iω (s+1) Ba 1 e iω s s (4.38) then Similarly B = a 1 e iω. ξ, ξ =ξ T (0) ξ(0) + so 0 1 ξ T (s + 1)a 1 ξ(s)s = B + ) B (1 a 1e iω a 1 e iω iω = e iω (s+1) Ba 1 e iω s s (4.39) Let x s C 0 the unique family of solutions of (4.33), where the µ notation has been roppe since ( ) we are ( only working with µ = 0. η ξ ),x s Define = η ξ an the orthogonal family of functions ω s = x s,x s

11 Hopf bifurcation control of ecological mole via PD controller 4851 ( ) η (ξ, ξ) so that the equation (4,5) has now been ecompose as η ( ( )) s η(s) = iω η(s) + B T η(s) f ω s + (ξ, ξ) ( ( η(s) )) s η(s) = iω η(s) + B T η(s) f ω s + (ξ, ξ) { η(s) ( ) } ( ω s )( ) Re ξ η,ω s + (ξ, ξ) ξ( ) s ω s( ) = { ( η ) } ( ω s )(0) Re ξ η,ω s + (ξ, ξ) ξ(0) + f η ( ω s + (ξ, ξ) ( )) η(s) η(s) 1 <0 = 0 (4.40) Step 5: Compute the Center Manifol Form For Compute the center manifol form we start with the equations (4.40) an let ( ( )) η(s) χ 1 (η, η, ω s ) = B T f ω s + (ξ, ξ) (4.41) η(s) { ( ) } Re ξ η,ω s + (ξ, ξ) ξ( ) χ (η, η, ω s ) = { ( η ) } Re ξ η,ω s + (ξ, ξ) ξ(0) + f η an from equation (4.9), (3.0) we have ( ω s + (ξ, ξ) ( )) η(s) η(s) 1 <0 = 0 (4.4) f(x(s),x(s 1), µ) = a 3 x (s 1) + O(x 3 (s 1)) (4.43) that a 3 = (u )τcb for to gain an approximate center manifol, can be given as a quaratic form in η an η with coefficients as functions of then ω s (η, η)( ) = ω 0 ( ) η + ω 11( )η η + ω 0 ( ) η (4.44) f(ω s (η, η)( ) + ξ( )η(s) + ξ( ) η(s)) = a 3 (ω s (η, η)( 1) + ξ( 1)η(s) + ξ(1) η(s)) + O(3) (4.45) f(ω s (η, η)( ) + ξ( )η(s) + ξ( ) η(s)) = a 3 γ η + a 3 γ γη η + a 3 γ η that γ = e iω 1. If we efine + a 3 [ω 0 ( 1) γ + ω 11 ( 1)γ ]η η + O(3) (4.46) g 0 = a 3 γ B,g 11 = a 3 γ γ B,g 0 = a 3 γ B

12 485 Mohamma Darvishi an Haji Mohamma Mohammaineja an g 1 = a 3 [ω 0 ( 1) γ + ω 11 ( 1)γ ] B then we can to write f(ω s (η, η)( ) + ξ( )η(s) + ξ( ) η(s)) = g 0η B + g 11η η B + g 0 η B + g 1η η B (4.47) In orer to compute g 1 we nees to compute the center manifol coefficients ω 0,ω 11 From the efinition (4.4) for 1 <0 χ (η, η)( ) = [g 0 ξ( ) + g 0 ξ( )] η an for = 0 χ (η, η)(0) = [g 11 ξ( ) + g 11 ξ( )]η η [g 0 ξ( ) + g 0 ξ( )] η [ g 0 ξ(0) + g 0 ξ(0) g 0 B [ g 11 ξ(0) + g 11 ξ(0) g 11 B ] η ] η η (4.48) [ g 0 ξ(0) + g 0 ξ(0) g ] 0 η B (4.49) since g 0 B =ḡ 0B then coefficients η η,η η, in χ (η, η)( ) respectively as [g χ 0 [ 0 ξ( ) + g 0 ξ( )] 1 <0 ( ) = g 0 ξ(0) + g 0 ξ(0) g ] 0 (4.50) = 0 B [g χ 11 [ 11 ξ( ) + g 11 ξ( )] 1 <0 ( ) = g 11 ξ(0) + g 11 ξ(0) g ] 11 (4.51) = 0 B an χ 0 ( ) = χ 0 ( ). By taking erivatives, the equation for the manifol becomes ( )( ) ( )( ) ωs (η, η)( ) η(s) ωs (η, η)( ) η(s) + η s η s = ω s (η, η)( ) + χ (η, η)( ) (4.5)

13 Hopf bifurcation control of ecological mole via PD controller 4853 so iω ω 0 ( )η (s) + iω ω 11 ( ) η(s)η(s) iω ω 11 ( )η(s) η(s) iω ω 0 ( ) η (s) = ( ω 0 )( ) η + ( ω 11)( )η η + ( ω 0 )( ) η + χ 0 ( ) η + χ 11 ( )η η + χ 0 ( ) η (4.53) we have iω 0 ( ) ( ω 0 )( ) = χ 0 ( ) ( ω 11 )( ) = χ 11 ( ) iω 0 ( ) ( ω 0 )( ) = χ 0 ( ) (4.54) then for 1 <0, so that ω 11 ( ) = g 11 ξ( ) + g 11 ξ( ) ω 11 ( ) = g 11 iω ξ( g 11 iω ξ( ) + Q an from = 0,µ = 0, (4.3), (4.51) Q is will be calculate that Q = then ω 11 ( ) = g 11 iω ξ( ) g 11 iω ξ( ) for Calculation ω 0 ( ) from (4.54), (4.3), (4.50) we have ω 0 ( ) = g 0 iω ξ( ) so that we can efine g 1. g [ 0 3iω ξ( ) g 11 B(a 1 + a ) g 11 B(a 1 + a ). (4.55) ] g 0 B(iω e iω (4.56) a a 1 e iω ) Step 6: Develop the Normal Form on the Center Manifol With normal form theory, following the argument of Wiggins [6], can be use to reuce (4.40) to the form ϑ s = iω ϑ + ϑ ϑ that = i ω ϑ s = iω ϑ + ϑ ϑ {g 11 g 0 g 11 g 0 } 3 + g 1 (4.57)

14 4854 Mohamma Darvishi an Haji Mohamma Mohammaineja We efine ν = Re{ (0)} Re λ(0) δ = Im{ (0)}+νIm λ(0) ω ϱ = Re{ (0)} (4.58) Accoring to the case escribe above, We can summarize the results in the following theorem: Theorem 4.1. For the controlle system (3.0), the Hopf bifurcationis etermine by the parameters ν, δ an β, the conclusions are summarize asfollows: (i) Parameter ν etermines the irection of the Hopf bifurcation. if ν > 0, the Hopf bifurcation is supercritical,the bifurcating perioic solutions exist for τ>τ k,if ν<0 the Hopf bifurcation is subcritical, the bifurcating perioic solutions exist for τ<τ k. (ii) Parameter ϱ etermines the stability of the bifurcating perioic solutions. If ϱ < 0, the bifurcating perioic solutions is stable; if ϱ > 0, the bifurcating perioic solutions is unstable. (iii) Parameter δ etermines the perio of the bifurcating perioic solution. If δ > 0, the perio increases; If δ<0, the perio ecreases. 5. Conclusions In this paper, the problem of Hopf bifurcation control for an ecological moel with time elay was stuie. In orer to control the Hopf bifurcation, a PD controller is applie to the moel. This PD controller can successfully elay or avance the onset of an inherent bifurcation. The en theoren helpe to improve moel. References [1] Yuanyuan Wang, Lisha Wang, Hybri control of the Neimark-Sacker bifurcation in a elaye Nicholson s blowflies equation, Wang an Wang Avances in Difference Equations (015) 015:306. [] DaweiDing, XiaoyunZhang, JineCao, NianWang, DongLiang, Bifurcation control of complex networks moel via PD controller, Neurocomputing (015). [3] Zunshui Cheng, JineCao, Hybri control of Hopf bifurcation in complex networks with elays, Neurocomputing 131(014), [4] Lina MA, Dawei DING, Tianren CAO, Mengxi WANG, Bifurcation control in a small-worl network moel via TDFC, International Conference on Avances in Mechanical Engineering an Inustrial Informatics (AMEII 015).

15 Hopf bifurcation control of ecological mole via PD controller 4855 [5] Jack K. Hale Sjoer M. Veruyn Lunel, Introuction to Functional Differential Equations, springer, Mathematics Subject Classification (1991): 34K0, 34A30, 39A10. [6] Stephen Wiggins, Introuction to Applie Nonlinear Dynamical Systems an Chaos, Springer, Mathematics Subject Classification (1991): 58 Fxx, 34Cxx, 70Kxx [7] Balakumar Balachanran. Tam s Kalmr-Nagy Davi E. Gilsinn, Delay Differential Equations, Springer Science+Business Meia, LLC 009.