Hopf bifurcation for a class of fractional differential equations with delay

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1 Nonlinear Dyn (01 69:71 79 DOI /s ORIGINAL PAPER Hopf bifurcation for a class of fractional differential equations with delay Azizollah Babakhani Dumitru Baleanu Reza Khanbabaie Received: 5 March 011 / Accepted: 30 November 011 / Published online: 5 January 01 Springer Science+Business Media B.V. 01 Abstract The main purpose of this manuscript is to prove the existence of solutions for delay fractional order differential equations (FDE at the neighborhood of its equilibrium point. After we convert the delay FDE into linear delay FDE by using its equilibrium point, we define the 1 : resonant double Hopf point set with its characteristic equation. We find the members of this set in different cases. The bifurcation curves for a class of delay FDE are obtained within a differential operator of Caputo type with the lower terminal at. Keywords Fractional calculus Hopf bifurcation A. Babakhani ( R. Khanbabaie Faculty of Basic Science, Babol University of Technology, Babol , Iran babakhani@nit.ac.ir R. Khanbabaie rkhanbabaie@nit.ac.ir D. Baleanu Department of Mathematics Computer Science, Cankaya University, Ankara, Turkey dumitru@cankaya.edu.tr D. Baleanu Institute of Space Sciences, P.O. BOX, MG-3, 76900, Magurele-Bucharest, Romania 1 Introduction Fractional calculus deals with the study of fractional order integral derivative operators over real or complex domains their applications. Fractional differential equations have gained considerable importance due to their valuable applications [1 7] inviscoelasticity, electroanalytical chemistry as well as in various engineering physical problems (see, for example, [1 11], the references therein. The combination of the delay with fractional calculus was used successfully in many areas of science engineering especially when we try to make models to describe the complex systems with memory effect [1 15]. In fact, by using such a combination, we are able to recover the fractional calculus model by making zero the delay also we can recover the classical case with delay by making the order of derivatives being one. This specific behavior leads us to the conclusion that the combined models can reveal new aspects of a given complex model. A bifurcation of a dynamical system is a qualitative change in its dynamics produced by varying parameters [16 19]. Bifurcation theory provides a strategy for investigating the bifurcations that occur within a family. It does so by identifying ubiquitous patterns of bifurcations. Each bifurcation type or singularity has a name, for example, Andronov Hopf bifurcation. No distinction has been made in the literature between bifurcation bifurcation type, both being called bifurcations. The Hopf Hopf bifurcation is a bifur-

2 7 A. Babakhani et al. cation of an equilibrium point in a two-parameter family of autonomous ODEs at which the critical equilibrium has two pairs of purely imaginary eigenvalues. This phenomenon is also called the double-hopf bifurcation [16, 18]. We recall that an equilibrium (or equilibrium point of a dynamical system generated by an autonomous system of ordinary differential equations is a solution that does not change with time [0, 1]. Geometrically, equilibria are points in the system s phase space. More precisely, the ordinary differential equations dx dt = f(xhas an equilibrium solution x(t = x e if f(x e = 0. Recently, the stability of equilibrium points are studied by Ahmed et al. on the fractional differential equations []. Motivated by the previously arguments, in this paper, we consider the fractional order differential equation with a delay τ, namely c D ρ u(t + αc D μ u(t + βu = f ( u(t τ, t (,T], (1 where m 1 <ρ m, n 1 <μ n, f : R R is a given smooth function there exists u such that βu = f(u. c D ρ c D μ are Caputo fractional derivatives with the lower terminal at. The outline of the paper is as follows. In Sect., we present the basic definitions of fractional calculus equilibrium point. The linearization of (1 is discussed in Sect. 3. The existence of parameter values for which the equilibrium solution undergoes two simultaneous Hopf bifurcations with frequencies at 1-to- ratio is discussed in Sect. 4. The Hopf bifurcation curves are shown in Sect. 5 when α = 0 also two examples are given to illustrate our results. Finally, in Sect. 6, we reformulate (1 into a system representation. Basic tools In this section, we present some stard definitions results used throughout this paper [1 3]. Caputo Riemann Liouville fractional derivative/integral their properties are defined below. Definition.1 For a function u defined on an interval [a,b], the Riemann Liouville fractional integral of f of order α>0 is defined by I ρ a + u(t = 1 Ɣ(ρ t a (t s ρ 1 u(s ds, t > a, Riemann Liouville fractional derivative of u of order ρ>0 defined by D ρ a + u(t = dn n ρ I dt n a + u(t }, where n 1 <ρ n. The definition of Caputo fractional derivative of u of order ρ>0is c D ρ a + u(t = I n ρ a + u (n (t }. The relation between Caputo Riemann Liouville fractional derivative is given by the following expression: D ρ a + u(t = c D ρ n 1 u (j (a a + u(t + Ɣ(j ρ + 1 (t aj ρ. j=1 ( Under natural conditions on the function u(t,thecaputo derivative becomes a conventional nth derivative of the function u(t, forρ n. Indeed, let us assume that 0 n 1 <ρ<n that the function u(t has n + 1 continuous bounded derivatives in [a,t ] for every T>a. Then lim α n c D ρ a + u(t = u (n (t, n = 1,,... One difference between the Riemann Liouville definition the Caputo definition is that the Caputo derivative of a constant is always 0, whereas in the case of a finite value of the lower terminal a, the Riemann Liouville fractional derivative of a constant λ is different from zero 0, but c D ρ λ 0 + u(t = Ɣ(1 ρ t ρ, u(t= λ. (3 However, by putting a = in both definitions requiring reasonable behavior of u(t its derivatives for t, we end up with the same formula c D ρ u(t = Dρ u(t 1 t u (n (t = ds. Ɣ(n ρ (t s ρ n+1 Note that Riemann Liouville fractional derivatives of u(t = e υt with the lower terminal at is given by

3 Hopf bifurcation for a class of fractional differential equations with delay 73 the following expression [3, p. 311]: D λ u(t = υλ e υt. (4 We denote c D ρ u(t as c D ρ u(t I ρ a + u(t as I ρ a u(t. Definition. [] A point x 0 R is called an equilibrium point or critical point of D ρ u(t = f(t if f(x 0 = 0. An equilibrium point x 0 is called a hyperbolic equilibrium point of c D ρ u(t = f(t if f (x 0 0, which in turn it is called sink if f (x 0 <0, source if f (x 0 >0 saddle if f (x 0 = 0. 3 The characteristic equations In this section, we study the linearization of (1 near to the equilibrium solution. We also prove that under some conditions (1 has explicit solutions. Proposition 3.1 Let us suppose that f : R R is a smooth function there exists u R such that βu = f(u, β R. Then the linearization of (1 near the equilibrium solution becomes the following fractional differential equation with a delay τ : c D ρ x(t + α c D μ x(t + βx(t = Ax(t τ (5 for which its characteristic equation is υ ρ + αυ μ + β = Ae υτ. (6 Proof If we substitute x = u u into (1, we obtain c D ρ x + α c D μ x + β(x + u = f ( x(t τ+ u. (7 By using the properties of f into (7, we get c D ρ x + α c D μ x + βx = f ( x(t τ+ u f(u. (8 The mean value theorem deduces that there exists a A R such that f ( x(t τ+ u f(u = Ax(t τ, (9 where x(t τ is sufficiently small. Hence A f (u. Substituting (9 into(8 it leads to (5. Using (4, it is easy to show that (6 is a characteristic equation when we substitute x = e νt into (5, where υ is a complex number. For given values of α, τ A, a necessary condition for Hopf bifurcation to occur at the equilibrium solution of (1 is that there exists an ω 0 such that ν = iω is a solution of (6 [6, p. 169]. Proposition 3. υ = iω,ω > 0 for given values of α, β, τ A is a solution to (6 if only if the following pair of real equations hold: ( πρ β + ω ρ cos + αω μ cos ( πρ ω ρ sin + αω μ sin ( πμ ( πμ = A cos ωτ = A sin ωτ. Proof By substituting υ = iω into (6, we get (10a (10b i ρ ω ρ + αi μ ω μ + β = Ae iωτ, (11 where i ρ = exp(ρ ln(i = exp i = cos ( + i sin ρπ i μ = exp(μ ln(i = exp ( μπ i = cos ( μπ ( + i sin μπ exp( iωτ = cos(ωτ i sin(ωτ. (1 Substituting the three equations (1into(11 leads to (10a (10b. Remark 3.3 (i It is easy to show that the characteristic equation (6 is invariant under the reflection ω ω. If we substitute υ = iω in (6, then (1 also yields. Therefore, we assume without losing the generality that ω>0. (ii Squaring both (10a (10b then adding together yields to β + ω ρ + α ω μ + βω ρ cos ( μπ + αβω μ cos μπ + αω ρ+μ cos = A. (13

4 74 A. Babakhani et al. We try to find an explicit solution for (1 under the conditions on the constant values of α, β A. Theorem 3.4 Consider the fractional delay differential equation (1 so that ρ (m 1,m] μ (n 1,n] where m, n are positive integers. Suppose that a smooth function f : R R is given there exists u R such that βu = f(u, β R. (1 If α = 0 β < A, then there exist a real value ω such that x(t = exp(iωt is a solution for the equation c D ρ x(t + βx(t = Ax(t τ. ( If β = 0,A= 0 ρ μ =k 1 or ρ + μ = k 1 where k is a positive integer, then the equation c D ρ x(t+ α c D μ x(t = 0 has trivial solution x(t = C, where C is constant. Proof (1 Using (13, we have ω ρ + β cos ω ρ = βcos > 0. } ω ρ + β A = 0, + β cos + A β Hence, x(t = exp (iωt is an explicit solution for the fractional differential equation D ρ x(t + βx(t = Ax(t τ. ( If β = 0,A = 0 then (10a (10b, respectively, reduce to ω ρ cos ( πρ + αω μ cos ( πμ = 0 ω ρ sin ( πρ + αω μ sin ( πμ (14 = 0. Squaring both (14 then adding them together yields μπ ω ρ + α ω μ + αω ρ+μ cos = 0. (15 Substituting ρ μ =k + 1into(15 yields ω ρ + α ω μ = 0, hence ω = 0. Therefore, x(t = C is a solution of equation c D ρ x(t + α c D μ x(t = 0 where C is a constant. Further, squaring both (14 then subtracting yields ( ( ρπ μπ ω ρ cos + α ω μ cos ( ρπ + μπ + αω ρ+μ cos = 0. Hence, by using ρ + μ = k + 1 into the above equation we get ω ρ cos + α ω μ cos ( μπ = 0. Therefore, ω = 0 x(t = C is a solution of equation D ρ x(t + αd μ x(t = 0 where C is a constant. Example 3.5 Let α = 0,ρ = /3,f(u = (3u u β = such that f(u = β u. Then u = ,A f (u = β < A. Thus, the fractional delay differential equation c D 3 u(t + βu(t = 3u(t τ u (t τ, τ >0 is equivalent to c D 3 x(t + βx(t = Ax(t τ,τ >0, where x(t = u(t u satisfies Theorem 3.4 (1. 4 Bifurcation analysis In this section, we establish the existence of the parameter values for which the equilibrium solution undergoes two simultaneous Hopf bifurcations with frequencies at 1-to- ratio. Definition 4.1 The point (α,τ,a in the parameter space is a 1 : resonant double Hopf point if the following two conditions hold: (a there exists one only one ω>0 such that (10a, (10b two equations ( ( πρ πμ β + (ω ρ cos + α(ω μ cos = A cos ωτ (16a ( ( πρ πμ (ω ρ sin + α(ω μ sin = A sin ωτ (16b are satisfied, (b if (α,τ,a ω are as in (a, then there are no ω ω,ω such that υ = i ω satisfies (6. Note that by squaring both (16a (16b then add together we yield:

5 Hopf bifurcation for a class of fractional differential equations with delay 75 ( ρπ β + (ω ρ + α (ω μ + β(ω ρ cos ( μπ + αβ(ω μ cos ( ρπ μπ + α(ω ρ+μ cos = A. (17 If α = 0, then (10a, (10b, (16a, (16b, respectively, reduce to ( πρ β + ω ρ cos = A cos ωτ (18a ( πρ ω ρ sin = Asin ωτ (18b ( πρ β + (ω ρ cos = A cos ωτ (18c ( πρ (ω ρ sin = Asin ωτ. (18d Squaring both (18a (18b adding yields ( ρπ β + ω ρ + βω ρ cos = A, (19 while doing the same with (18c (18d yields ( ρπ β + (ω ρ + ρ+1 βω ρ cos = A (0 where n is integer. Comparing (19 (0, it gives ω ρ = β ( ρπ ρ + 1 cos =: λ. (1 If we consider β = (ρ + 1 cos (, substituting it into (1 yields ω ρ = 1. Comparing (18b (18dgives[ ρ 1 cos(ωτ] sin = 0, hence ρ 1 ρ = cos(ωτ or = 1,, 3,... (3 Case 1. If ρ 1 = cos(ωτ, then this equation is defined when 0 <ρ 1, hence (1 is defined when β<0. Note that ω 0, therefore, ωτ = arccos ( ρ 1 > 0, 0 <ρ<1. Hence, ωτ = kπ ± λπ, where 0 <λ<1/ k Z +. Using cos ωτ = ρ 1, 0 <ρ<1 substituting ω ρ = 1into(18a (18c leads to A = (ρ 1 cos cos(ωτ cos(ωτ = (ρ 1 cos 4 ρ ρ, 0 <ρ<1. (4 Note that cos(ωτ = cos (ωτ sin (ωτ sin(ωτ = 1 cos (ωτ = 1 4 ρ 1. Since 4 ρ ρ < 0 for all ρ (0, 1, then A<0 for all ρ (0, 1. Case. If ρ = n, n Z+ A 0, then (18b (18d yields ωτ = kπ or ωτ = (k + 1π where k is a positive integer. If ωτ = kπ, then (18a (18c give a contradiction 1 = ρ,ρ 0. Hence, ωτ = (k +1π. Substituting ωτ = (k +1π, ω ρ = 1 into (18a (18c adding these two new equations gives A = ρ 1 ( ρπ cos = ( 1 ρ ρ 1. (5 Proposition 4. If α = 0 ρ be an even positive integer, then the 1 : resonant double Hopf points are precisely those in the set that R 1: = (α, τ, A = ( 0,(k + 1π, ( 1 ρ ρ 1 k = 0, 1,,..., ρ Z+ }., Proof If α = 0, the members of R 1: will be precisely those which are discussed in case. Now we would like to show that there are no ω ω,ω such that υ = i ω satisfies (6. By (18a (18b, suppose that there exists an ω 1, such that ( 1 ρ ρ ( 1 ρ ω ρ = ( 1 ρ ρ 1 cos ( ω(k + 1π (6a ( 1 ρ ρ 1 sin ( ω(k + 1π = 0 (6b Equation (6b yields to ω = m Z, hence (6agives ( ρ + 1/ + ω = ( 1 m ( ρ 1/. Therefore, when m is an even or an odd positive integers ω = 1or ω =, respectively.

6 76 A. Babakhani et al. Proposition 4.3 If α = 0 0 <ρ<1, then the 1 : resonant double Hopf points are precisely those in the set that ( R 1: = (α, τ, A = 0, arccos ( ρ 1, sin }, 0 <ρ< ρ 1 Proof If α = 0 0 <μ<1, then the prove of the members of R 1: is discussed in case 1. Now let us suppose that there exists an ω 1, such that υ = i ω satisfies (6. Hence, using (18b values A τ, we have ( ρπ ω ρ sin = sin sin( ω arccos ( ρ 1 or 1 4 ρ 1 (7 ω ρ 1 4 ρ 1 = sin ( ω arccos ( ρ 1. Equation (7 holds if ω = 1 or which is a contradiction. Thus, the 1 : resonant double Hopf points are precisely those given by R 1:. Note that sin(arccos θ= 1 θ sin θ = sinθ cos θ. According to Theorem 3.4 (1, (1 has nontrivial solutions under the condition β < A mentioned in that theorem. The following corollary indicate a new condition which in (1 has a nontrivial solution. Corollary 4.4 Suppose that f : R R is a given smooth function there exists u R such that βu = f(u, β R. Let 0 <ρ<1 λ = ρ. If 4 λ 8 ( λ 1 λ cos ( π ln λ ln, 1 <λ<, (8 then (1 has two solutions x j (t = exp(iωt, j = 1,. Proof Based on Theorem 3.1(i, ω ρ is a real value if ( ρπ β cos + A β = (A β sin ωτ(a + β sin ωτ 0, (9 where sin ωτ > 0. In view of ( (4, the real numbers A β are negative hence A+β sin ωτ < 0. Thus, (9 is valid if A β sin ωτ. Substituting ( (4 intoa β sin ωτ using λ = ρ, we can conclude the result. Example 4.5 Consider a fourth-order ordinary delay differential equation, u (4 (t + βu(t = 9u (t τ 18u(t τ, τ>0. (30 Here, α = 0,ρ = 4 f(u= 9(u u.using( (5, we have that β = 9 A = 15. Set u = 3, hence 9 3 = f(3. Thus, the delay ordinary differential equation (8 is equivalent to x (4 (t 9 x(t 15 x(t τ, τ >0. Based on Proposition 4., the1: resonant double Hopf points are precisely those in the set that ( R 1: = (α, τ, A = 0,(k 1π, 15 },k Z +. Example 4.6 Consider the following delay fractional differential equation c D 1 u(t + βu(t = u (t τ 18u(t τ, τ>0. (31 Here, α = 0,ρ = 1/ f(u= (u u. Using ( (4, we have β = ( + / A = 1 f (u. Set u = ( /, hence β u = f(u. Thus, the delay factional differential equation (9 is equivalent to c D 1 + x(t x(t x(t τ, τ >0. Using Proposition 4., the1: resonant double Hopf points are precisely those in the set that R 1: = (α, τ, A = (0, kπ ± λ,1 : ( 0 <λ= arccos < π },k Z+. 5 Bifurcation curves While α = 0, the linearized form of (1 about the equilibrium solution becomes c D ρ x + βx = Ax(t τ. (3

7 Hopf bifurcation for a class of fractional differential equations with delay 77 The corresponding characteristic equation is given by υ ρ + β = Ae υτ. (33 When ρ/ Z + A = 0, i.e., along the τ axis, we have υ =±i ρ β. This is just the usual result of neutral stability for the forced simple harmonic oscillator. When ρ/ Z + A 0, by using (18b (18d, we have ωτ = π the two priory Hopf bifurcation curves are given by ρ π ρ A = ( 1 ρ τ ρ ρ + 1 ρ cos, Z+ = ( 1 ρ ρ π ρ } τ ρ ρ + 1 (π bifurcation curve (34a A = ρ + 1 cos ( 1 ρ π ρ τ ρ, = ( 1 ρ ρ + 1 π ρ } τ ρ ρ Z+ (π bifurcation curve. (34b If ρ (0, 1 then like in the case (1 wehaveω = (1/τ arccos( ρ 1 from (18a (18c, the two priory Hopf bifurcation curves are given by A = 1 ( arccos( ρ 1 ρ 1 β + τ ρ cos }, 0 <ρ<1 (35a A = 1 ( arccos( ρ 1 ρ 1 β + τ ρ cos }, 0 <ρ<1. (35b Example 5.1 If ρ =,α = 0, then two priory Hopf bifurcation curves are shown in Fig. given by A = 5 4π τ A = π τ 5 (π bifurcation curve (π bifurcation curve. Fig. 1 Hopf bifurcation curves, if ρ = 1,α= 0 If ρ = 1,α = 0, then priory Hopf bifurcation curvesareshowninfig.1 given by A = 1 } π +, τ A = 1 } π +. τ 6 System representation To study the behavior of (1 near the resonant double Hopf point, we have to underst the effects of the nonlinear terms in the equation. To do this, we must first reformulate the equation by rewriting (1 as a system of two fractional order equations exping those about the equilibrium solution. Although currently the most important results have not been found in this section but this idea may lead to new results.

8 78 A. Babakhani et al. We assume that X is small (we are near the equilibrium solution so that the Taylor s expansion of f is valid. By neglecting the higher order terms, this becomes in vector form c D μ X(t = BX(t + DX(t τ+ D X (t τ,(37 where X(t =[X(t,Y(t] T B,D,D are matrices. Let us define the function space C = C([ τ,0], R the function X t (s = X(t + s, τ s 0. Assuming that X t C, (35 induces an equation on C: c D μ X t (s = (LX t (s + (N X t (s, where L N represent operators on C corresponding to the linear nonlinear parts of (37, respectively. 7 Conclusion Fig. Hopf bifurcation curves, if ρ =,α= 0 If μ = ρ (0, 1 X = u u,y = c D μ u then c D μ X(t = Y(t c D μ Y(t= c D μ( c D μ u(t t 1 μ = c D μ u(t I (1 μ u(t } t=0 Ɣ( μ = c D ρ u(t = βx(t αy(t + AX(t τ + A X (t τ+ O ( X 3 (t τ, where A = f (u A = 1 f (u. Hence, rewriting (4 as a system of two fractional order equations exping those around the equilibrium solution gives c D μ X(t = Y(t c D μ Y(t= βx(t αy(t + A 1 X(t τ (36 + A X (t τ+ O ( X 3 (t τ. We considered a class of FDE with delay involving Caputo fractional derivative possessing a lower terminal at in order to study the Hopf bifurcation analysis. We investigated the existence of solutions for a class of delay fractional differential equations at the neighborhood of its equilibrium. In order to study the 1 : double Hopf resonant set, we have shown that there are points in the parameter space by converting the delay FDE into linear delay FDE. The bifurcation curves for a class of delay FDE were reported. In order to illustrate our results, several illustrative examples were analyzed in detail. Acknowledgements The authors would like to thank to the anonymous referees for their valuable comments suggestions. References 1. Oldham, K.B., Spanier, I.: The Fractional Calculus. Academic Press, New York (1974. Miller, K.B., Ross, B.: An Introduction to the Fractional Calculus Fractional Differential Equations. Wiley, New York ( Podlubny, I.: Fractional Differential Equations. Academic Press, New York ( Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory Applications of Fractional Differential Equations. North- Holl Mathematics Studies, vol. 04. Elsevier, Amsterdam (006

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