hopf bifurcation control of ecological model via pd controller · 2016-12-31 · hopf bifurcation...

Global Journal of Pure and Applied Mathematics.ISSN 0973-1768 Volume 12, Number 6 (2016), pp. 4841–4855© Research India Publicationshttp://www.ripublication.com/gjpam.htm

Hopf bifurcation control of ecological modelvia PD controller

Mohammad Darvishi1

Department of Mathematics,Faculty of Mathematics and Statistics,University of Birjand, Birjand, Iran.

Haji Mohammad Mohammadinejad

Department of Mathematics,Faculty of Mathematics and Statistics,University of Birjand, Birjand, Iran.

Abstract

In this paper, we investigate the problem of bifurcation control for an ecologicalmodle with timedelay. By choosing the timedelay as the bifurcation parameter, wepresent a Proportional-Derivative (PD) feedback controller to control Hopf bifur-cation. We show that the onset of Hopf bifurcation can be delayed or advancedvia a PD controller by setting proper controling parameters. Under considerationmodel as operator Equation, apply orthogonal decomposition, compute the centermanifold and normal form we determined the direction and stability of bifurcatingperiodic solutions. Therefore the Hopf bifurcation of the model became control-lable to achieve desirable behaviors which are applicable in certain circumstances.

AMS subject classification: 34H20, 34D23.Keywords: Hopf bifurcation, bifurcation control, PD controller, ecological modle.

1Corresponding author.

4842 Mohammad Darvishi and Haji Mohammad Mohammadinejad

1. Introduction

The delay differential equation (DDE)

d

dsx(s) = ax(s − τ)e−bx(s−τ) − cx(s) (1.1)

which is one of the important ecological systems, describes the dynamics of Nichol-sons blowflies equation. Here x(s) is the size of the population at time s, a is the

maximum per capita daily egg production rate,1

bis the size at which the population

reproduces at the maximum rate, c is the per capita daily adult death rate, and τ is the

generation time, the positive equilibrium x∗ =(

1

b

)ln

(a

c

). Equation (1.1) has been

extensively studied in the literature. The majority of the results on (1.1) deal with theglobal attractiveness of the positive equilibrium and oscillatory behaviors of solutions.

The aim of bifurcation control is to delay (advance) the onset of an inherent bifurca-tion, change the parameter value of an existing bifurcation point, the PD control strategyis used to control the bifurcation [1].

2. Existence Hopf bifurcation in uncontrol ecological modle

We consider the delay differential equation (1.1) and let u(s) = x(τs) then

d

dsu(s) = d

dsx(τs) = τ [ax(τs − τ)e−bx(τs−τ) − cx(τs)] (2.2)

and equation(1.1) can be rewritten as

d

dsu(s) = aτu(s − 1)e−bu(s−1) − cτu(s) (2.3)

We consider the timedelay as the bifurcation parameter, The critical point in the system

is u∗ =(

1

b

)ln

(a

c

)and With the change of variables z(s) = u(s) − u∗ we have

d

dsz(s) = aτ(z(s − 1) + u∗)e−b(z(s−1)+u∗) − cτ(z(s) + u∗) (2.4)

the linearization of equation (2,3) at z(s)=0 is

d

dsz(s) = cτ [(1 − bu∗)z(s − 1) − z(s)] (2.5)

whose characteristic equation is

λ = cτ [(1 − bu∗)e−λ − 1] (2.6)

Hopf bifurcation control of ecological modle via PD controller 4843

Assume that λ = iω is a root of the equation (2.6) that ω > 0 then

iω = cτ [(1 − bu∗)e−iω − 1] (2.7)

andiω = cτ [(1 − bu∗)(cos ω − i sin ω) − 1] (2.8)

Separating the real and imaginary parts, we obtain

cτ(1 − bu∗) cos ω = cτ (2.9)

andcτ(1 − bu∗) sin ω = −ω (2.10)

then ω = ±τc

√(1 − bu∗)2 − 1 if (1 − bu∗)2 − 1 > 0. Let λk(τ ) = αk(τ ) + iβk(τ )

denote a root of equation (2.6) and near τ = τk such that αk(τk) = 0, βk(τk) = ω so that

λk(τk) = αk(τk) + iβk(τk) = iω

andiω = cτk[(1 − bu∗)e−iω − 1],

icτk

√(1 − bu∗)2 − 1 = cτk[(1 − bu∗)(cos(cτk

√(1 − bu∗)2 − 1)

− isin(cτk

√(1 − bu∗)2 − 1) − 1]

Then decompose the equationin to real and imaginary parts

cτk(1 − bu∗)(cos(cτk

√(1 − bu∗)2 − 1) − cτk = 0 (2.11)

and

cτk

√(1 − bu∗)2 − 1 + cτk(1 − bu∗)sin(cτk

√(1 − bu∗)2 − 1) = 0 (2.12)

which lead to

τk = 1

c√

(1 − bu∗)2 − 1[arcsin(

√(1 − bu∗)2 − 1

(1 − bu∗))+2kπ ] k = 0, 1, 2, ... (2.13)

Lemma 2.1. We have to equation (2.6).

Proof. Differentiating both sides of equation (2.6) with respect to τ , we obtain

dλ

dτ= c[(1 − bu∗)e−λ − 1]

1 + τc[(1 − bu∗)e−λ] (2.14)


therefore

dλ

dτ|τ=τk

= c(1 − bu∗)cosω − c − ic(1 − bu∗)sinω

1 + τkc(1 − bu∗)cosω − iτkc(1 − bu∗)sinω(2.15)

this implies that

dRe(λ)

dτ|τ=τk

= c(1 − bu∗)cosω + τk[c(1 − bu∗)]2

[1 + τkc(1 − bu∗)cosω]2 + [τkc(1 − bu∗)sinω]2 (2.16)

then

dRe(λ)

dτ|τ=τk

c + τk[c(1 − bu∗)]2

[1 + τkc(1 − bu∗)cosω]2 + [τkc(1 − bu∗)sinω]2 (2.17)

so that completing the proof. �

Theorem 2.2. For the system (1.1), the following statements are true:

(i) if c < a < ce2 ,then x = x∗ is asymptotically stable.

(ii) if a > ce2(bx∗ > 2), then x = x∗ is asymptotically stable for τ ∈ [0, τ0) andunstable for τ > τ0

(iii) Equation (1.1) undergoes a Hopf bifurcation at x = x∗ when τ = τk for k =0, 1, 2, . . . [1].

Figure 1: the numerical solution of uncontrol ecological modle with corresponding to(A) a = 30, b = 2, c = 2, τ = 0.4 and (B) a = 30, b = 2, c = 2, τ = 0.8.


3. Hopf bifurcation in controlled model based on PD controller

In this section,we focus on designing a controller to control the Hopf bifurcation in modelbased on the PD control strategy. The PD controller uses a single-input and single-outputsystem as follows: the input error signal is defined as e(t) = z(s) − z∗ and the outputcontrol law u(t) is defined as

u(t) = kpz(s) + kd(d

dsz(s)) (3.18)

where kp is proportional control parameter and kd is derivative control parameter andkp < 1; kd < 1. Apply the PD controller to system (2.4), we get

d

dsz(s) = aτ(z(s−1)+u∗)e−b(z(s−1)+u∗)−cτ(z(s)+u∗)+kpz(s)+kd

d

dsz(s) (3.19)

Therefore, the controlled model can bee expressed as follows

d

dsz(s) = 1

1 − kd

[aτ(z(s − 1) + u∗)e−b(z(s−1)+u∗) − cτ(z(s) + u∗) + kpz(s)] (3.20)

Comparing Equations (3.20) and (2.4), we find that the controlled model and uncon-trolled model have the same equilibrium point. Expanding the right side of Equation(3.20) into Taylor series at z(t) = 0, we consider the linear part

d

dsz(s) = 1

1 − kd

[cτ((1 − bu∗)z(s − 1) − z(s)) + kpz(s)] (3.21)

let a1 = τc(1 − bu∗)1 − kd

and a2 = kp − τc

1 − kd

then

d

dsz(s) = a1z(s − 1) + a2z(s) (3.22)

Characteristic equation corresponding to Equation (3.22) is

λ∗ − a1e−λ∗ − a2 = 0 (3.23)

Assume Equation (3.23) has a pair of pure imaginary roots λ∗ = ±iω∗ with ω∗ > 0.Inserting them into Equation (3.23). Then decompose the characteristice quationin toreal and imaginary parts

a2 + a1cos(ω∗) = 0 (3.24)

ω∗ + a1sin(ω∗) = 0 (3.25)

From Equations (3.24), (3.25) and combined with the formula: sin2(ω∗)+cos2(ω∗) = 1,we obtain

ω∗ =√

a21 − a2

2 (3.26)


and

ω∗ = arccos

(−a2

a1

)(3.27)

Combine Equations (3.26), (3.27) and we can get the equationas follows:√a2

1 − a22 = arccos

(−a2

a1

)(3.28)

equation (3.28) is a transcendental equation, we can get its numerical solution by nu-merical simulation software. [2], We need to get the fixed point τ ∗

k of the system suchthat λ∗(τ ∗

k) = iω∗, the value of ω∗ can be derived from equation (3.23) then.We still need to check the following transversal condition in order to verify the onset

of Hopf bifurcation:

Lemma 3.1. For equation (3.23) if

kp >τ ∗2

kc2

kd − 1 + τ ∗kc

thendRe(λ)

dτ|τ=τ∗

k> 0.

Proof. Let r = c(1 − bu∗)1 − kd

, differentiating both sides of equation (3.23) with respect to

τ , we obtaindλ − re−λdτ + τre−λdλ + c

1 − kd

dτ = 0

Therefore

dλ

dτ|τ=τ∗

k=

rcosω∗ − c

1 − kd

− irsinω∗

1 + τ ∗krcosω∗ − iτ ∗

krsinω∗this implies that

dRe(λ)

dτ|τ=τ∗

k= kp(kd − 1) + τ ∗

kc(kp − τ ∗kc) + τ ∗

k2r2(1 − kd)

2

τ ∗k(1 − kd)

2[(1 + τ ∗krcosω∗)2 + (τ ∗

krsinω∗)2]so that completing proof. �

Therefore, the transversal condition for Hopf bifurcation in the PD controlled system(3.20) is satisfied. We conclude above analyzes in following:

Theorem 3.2. For the controlled system (3.20), there exists a Hopf bifurcation emergingfrom its equilibrium z∗ = 0 when the positive parameter τ passes through the criticalvalue τ ∗

k.

We note that the equilibrium point remains unchanged and the critical value τ ∗k

varies with control parameters kp and kd . Theorem (3.19) shows that the PD controllercan be applied to the ecological model for the purpose of bifurcation control.


Figure 2: the numerical solution of control ecological modle with corresponding to (A)a = 30, b = 2, c = 2, τ = 0.4, kd = 0.5, kp = 0.1 and (B) a = 30, b = 2, c = 2, τ =0.4, kd = 0.3, kp = 0.1.


4. Stability and direction of bifurcating periodic solutions

In this section we study the direction of the Hopf bifurcation and the stability of the

bifurcation periodic solutions when τ = τ ∗k under the condition kp >

τ ∗2kc

2

kd − 1 + τ ∗kc

and using techniques from normal form and center manifold theory by Hassard et al. [7].

For convenience, let µ = τ − τ ∗k, therefore µ = 0 is the value of hopf bifurcation

for equation (3.20). In this section we will be discussed in multiple parts. In the fristpart we will formulate the operator form for (3.20), describe a process of determiningthe critical eigenvalues for the problem, formulate the adjoint operator equation andApply Orthogonal Decomposition. then Compute the Center Manifold Form, Developthe Normal Form on the Center Manifold and at the end examine periodic solutionsconditions on center manifold.



Step 1: Form the Operator EquationWith expanding the right side of Equation (3.20) into Taylor series at z(s) = 0, then

d

dsx(s) = U(µ)x(s) + V (µ)x(s − 1) + f (x(s), x(s − 1), µ) (4.29)

that U(µ) = a2, V (µ) = a1 and f (x(s), x(s − 1), µ) = O(x2(s), x2(s − 1)) weconsider The linear portion of equation (4.29) that is given by

d

dsx(s) = a2x(s) + a1x(s − 1) (4.30)

the equations (4.29) and (4.30) can be thought of as maps of entire functions. we willstart with the class of continuous functions defined on the interval [-1, 0] with valuesin Rn and refer to this as class C0. The maps are constructed by defining a family ofsolution operators for the linear portion of equation (4.29) by

(ϒ(s)φ)() = (xs(φ))() = x(s + ) (4.31)

for φ ∈ C0, ∈ [−1, 0], s ≥ 0. This is a mapping of a function in C0 to anotherfunction in C0. so equations (4.29), (4.30) can be thought of as maps from C0 to C0 andϒ(s), s ≥ 0 is a semigroup.

An infinitesimal generator of a semigroup ϒ(s) is defined by

φ = lims→0+

1

s[ϒ(s)φ − φ]

for φ ∈ C0. in the linear equation (4.30) the infinitesimal generator can be constructedas

(µ)φ ={

dφ

d() −1 ≤ < 0

a2φ(0) + a1φ(−1) = 0(4.32)


Figure 5: The solution operator maps functions in C0 to functions in C0

Then ϒ(s)φ satisfiesd

dsϒ(s)φ = (µ)ϒ(s)φ, the operator form for the nonlinear

equation (4.29) is

d

dsxs(φ) = ϒ(µ)xs(φ)+F(xs(φ), µ), that F (φ, µ)() =

{0 −1 ≤ < 0

f (φ, µ) = 0(4.33)

Step 2: Define an Adjoint OperatorWe define a form of adjoint Operator associated with (4.33). we consider C0

� =C([0, 1], Rn) that is the space of continuous functions from[0,1] to Rn. the adjointequation associated with the equation (4.30) is

d

dsy(s, µ) = −a2y(s, µ) − a1y(s + 1, µ) (4.34)

we define(ϒ�(s)ψ)() = (ys(ψ))() = y(s + ) (4.35)

for ∈ [o, 1], s � 0, ys ∈ C0� and ys(ψ) be the image of ϒ�(s)ψ , so relationship

(4.35) defines semigroup with infinitesimal generator

�(µ)ψ =

−dψ

d() 0 < ≤ 1

−dψ

d(0) = a2ψ(0) + a1ψ(1) = 0

(4.36)

Note that, although the formal infinitesimal generator for (4.35) is defined as

0�ψ = lim

s→0−1

s[ϒ�(s)ψ − φ]


for ψ ∈ C0 Hale[5], that0

� = −�

and family of operators (4.35) satisfies

d

dsϒ�(s)ψ = −�(µ)ϒ�(s)ψ.

Step 3: Define a natural Inner product by way of an adjoint operatorLet x(s) ∈ C0, y(s) ∈ C0

� and define an inner product

〈y(s), x(s)〉 = yT (0)x(0) +∫ 0

−1yT (s + 1)a1x(s)ds (4.37)

Step 4: Get the Critical Eigenvalues and Apply Orthogonal DecompositionAs for φ ∈ C0, ψ ∈ C0

� we have 〈ψ, (µ)φ〉 = 〈�(µ)ψ, φ〉, obviously (0) and�(0) are adjoint operators and ±iω∗ are eigenvalues of (0) and �(0). to gainOrthogonal Decomposition we need to calculate the eigenvector ξ(θ) of (0) and ξ�(θ)

of �(0) associated with the eigenvalues iω∗ and −iω∗ respectively. let ξ(θ) = eiω∗θA,ξ (θ) = e−iω∗θ A that θ ∈ [−1, 0) and ξ�(θ) = eiω∗θB , ξ�(θ) = e−iω∗θ B that θ ∈(0, 1]. let A = 1 we choose B so that 〈ξ�, ξ 〉 = 1 and 〈ξ�, ξ〉 = 0. from (4.37) wehave:

〈ξ�, ξ 〉 = ξ�T(0)ξ(0) +

∫ 0

−1ξ�T

(s + 1)a1ξ(s)ds = B +∫ 0

−1e−iω∗(s+1)Ba1e

iω∗sds

(4.38)

then

B = 1

1 + a1e−iω∗ .

Similarly

〈ξ�, ξ〉 = ξ�T(0)ξ (0) +

∫ 0

−1ξ�T

(s + 1)a1ξ (s)ds = B +∫ 0

−1e−iω∗(s+1)Ba1e

−iω∗sds

(4.39)so

B

(1 − a1e

−iω∗ − a1eiω∗

2iω∗

)= 0.

Let xs ∈ C0 the unique family of solutions of (4.33), where the µ notation has beendropped since we are only working with µ = 0.

Define

(η

η

)=

(〈ξ�, xs〉〈ξ�, xs〉

)and the orthogonal family of functions ωs = xs −


(ξ, ξ )

(η

η

)so that the equation (4,5) has now been decomposed as

d

dsη(s) = iω∗η(s) + BT f

(ωs + (ξ, ξ )

(η(s)

η(s)

))d

dsη(s) = −iω∗η(s) + BT f

(ωs + (ξ, ξ )

(η(s)

η(s)

))

d

dsωs() =

(ωs)() − 2Re

{⟨ξ�, ωs + (ξ, ξ )

(η

η

)⟩ξ()

}−1 ≤ < 0

(ωs)(0) − 2Re

{⟨ξ�, ωs + (ξ, ξ )

(η

η

)⟩ξ(0)

}+ f

(ωs + (ξ, ξ )

(η(s)

η(s)

)) = 0

(4.40)

Step 5: Compute the Center Manifold FormFor Compute the center manifold form we start with the equations (4.40) and let

χ1(η, η, ωs) = BT f

(ωs + (ξ, ξ )

(η(s)

η(s)

))(4.41)

χ2(η, η, ωs) =

−2Re

{⟨ξ�, ωs + (ξ, ξ )

(η

η

)⟩ξ()

}−1 ≤ < 0

−2Re

{⟨ξ�, ωs + (ξ, ξ )

(η

η

)⟩ξ(0)

}+ f

(ωs + (ξ, ξ )

(η(s)

η(s)

)) = 0

(4.42)

and from equation (4.29), (3.20) we have

f (x(s), x(s − 1), µ) = a3x2(s − 1) + O(x3(s − 1)) (4.43)

that a3 = (u∗ − 2)τcb for to gain an approximate center manifold, can be given as aquadratic form in η and η with coefficients as functions of

ωs(η, η)() = ω20()η2

2+ ω11()ηη + ω02()

η2

2(4.44)

then

f (ωs(η, η)() + ξ()η(s) + ξ ()η(s)) = a3(ωs(η, η)(−1) + ξ(−1)η(s)

+ ξ (1)η(s))2 + O(3) (4.45)

f (ωs(η, η)() + ξ()η(s) + ξ ()η(s)) = a3γ2η2 + 2a3γ γ ηη + a3γ

2η2

+ a3[ω20(−1)γ + 2ω11(−1)γ ]η2η + O(3)

(4.46)

that γ = e−iω − 1. If we define

g20 = 2a3γ2B, g11 = 2a3γ γ B, g02 = 2a3γ

2B


and

g21 = 2a3[ω20(−1)γ + 2ω11(−1)γ ]Bthen we can to write

f (ωs(η, η)() + ξ()η(s)+ ξ ()η(s)) = g20η2

2B+ g11ηη

B+ g02η

2

2B+ g21η

2η

2B(4.47)

In order to compute g21 we needs to compute the center manifold coefficients ω20, ω11From the definition (4.42) for −1 ≤ < 0

χ2(η, η)() = −[g20ξ() + ¯g02ξ ()]η2

2

− [g11ξ() + ¯g11ξ ()]ηη − [g02ξ() + ¯g20ξ ()] η2

2(4.48)

and for = 0

χ2(η, η)(0) = −[g20ξ(0) + ¯g02ξ (0) − g20

B

]η2

2

−[g11ξ(0) + ¯g11ξ (0) − g11

B

]ηη −

[g02ξ(0) + ¯g20ξ (0) − g02

B

]η2

2(4.49)

sinceg02

B= g20B

then coefficientsη2

2, ηη,

η2

2in χ2(η, η)() respectively as

χ220() =

−[g20ξ() + ¯g02ξ ()] −1 ≤ < 0

−[g20ξ(0) + ¯g02ξ (0) − g20

B

] = 0

(4.50)

χ211() =

−[g11ξ() + ¯g11ξ ()] −1 ≤ < 0

−[g11ξ(0) + ¯g11ξ (0) − g11

B

] = 0

(4.51)

and χ202() = χ20

2 (). By taking derivatives, the equation for the manifold becomes(dωs(η, η)()

dη

) (dη(s)

ds

)+

(dωs(η, η)()

dη

) (dη(s)

ds

)= ωs(η, η)() + χ2(η, η)() (4.52)


so

iω∗ω20()η2(s) + iω∗ω11()η(s)η(s) − iω∗ω11()η(s)η(s) − iω∗ω02()η2(s) · · ·= (ω20)()

η2

2+ (ω11)()ηη + (ω02)()

η2

2+ χ2

20()η2

2

+ χ211()ηη + χ2

02()η2

2· · · (4.53)

we have

2iω20() − (ω20)() = χ220()

−(ω11)() = χ211()

−2iω02() − (ω02)() = χ202()

(4.54)

then for −1 ≤ < 0,dω11()

= g11ξ() + ¯g11ξ ()

so that

ω11() = g11

iωξ( − ¯g11

iωξ () + Q

and from = 0, µ = 0, (4.32), (4.51) Q is will be calculated that Q = − g11

B(a1 + a2).

then

ω11() = g11

iω∗ ξ() − ¯g11

iω∗ ξ () − g11

B(a1 + a2)(4.55)

for Calculation ω20() from (4.54), (4.32), (4.50) we have

ω20() = − g20

iω∗ ξ() − ¯g02

3iω∗ ξ () −[

g20

B(2iω∗ − a2 − a1e−2iω∗)

]e2iω∗

(4.56)

so that we can define g21.

Step 6: Develop the Normal Form on the Center ManifoldWith normal form theory, following the argument of Wiggins [6], can be used to reduce(4.40) to the form

dϑ

ds= iω∗ϑ + �ϑ2ϑ

dϑ

ds= −iω∗ϑ + �ϑ2ϑ

(4.57)

that

� = i

2ω

{g11g20 − 2|g11|2 − |g02|2

3

}+ g21

2


We define

ν = −Re{�(0)}Reλ(0)

δ = −Im{�(0)} + νImλ(0)

ω∗� = 2Re{�(0)}

(4.58)

According to the case described above, We can summarize the results in the followingtheorem:

Theorem 4.1. For the controlled system (3.20), the Hopf bifurcationis determined bythe parameters ν, δ and β, the conclusions are summarized asfollows:

(i) Parameter ν determines the direction of the Hopf bifurcation. if ν > 0, the Hopfbifurcation is supercritical,the bifurcating periodic solutions exist for τ > τ ∗

k, ifν < 0 the Hopf bifurcation is subcritical, the bifurcating periodic solutions existfor τ < τ ∗

k.

(ii) Parameter � determines the stability of the bifurcating periodic solutions. If � < 0,the bifurcating periodic solutions is stable; if � > 0, the bifurcating periodicsolutions is unstable.

(iii) Parameter δ determines the period of the bifurcating periodic solution. If δ > 0,the period increases; If δ < 0, the period decreases.

5. Conclusions

In this paper, the problem of Hopf bifurcation control for an ecological model with timedelay was studied. In order to control the Hopf bifurcation, a PD controller is applied tothe model. This PD controller can successfully delay or advance the onset of an inherentbifurcation. The end theoren helped to improve model.

References

[1] Yuanyuan Wang, Lisha Wang, Hybrid control of the Neimark-Sacker bifurcation ina delayed Nicholson’s blowflies equation, Wang and Wang Advances in DifferenceEquations (2015) 2015:306.

[2] DaweiDing, XiaoyunZhang, JindeCao, NianWang, DongLiang, Bifurcation controlof complex networks model via PD controller, Neurocomputing (2015).

[3] Zunshui Cheng, JindeCao, Hybrid control of Hopf bifurcation in complex networkswith delays, Neurocomputing 131(2014), 164–170.

[4] Lina MA, Dawei DING, Tianren CAO, Mengxi WANG, Bifurcation control in asmall-world network model via TDFC, International Conference on Advances inMechanical Engineering and Industrial Informatics (AMEII 2015).


[5] Jack K. Hale Sjoerd M.Verduyn Lunel, Introduction to Functional Differential Equa-tions, springer, Mathematics Subject Classification (1991): 34K20, 34A30, 39A10.

[6] Stephen Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos,Springer, Mathematics Subject Classification (1991): 58 Fxx, 34Cxx, 70Kxx

[7] Balakumar Balachandran . Tam’s Kalmr-Nagy David E. Gilsinn, Delay DifferentialEquations, Springer Science+Business Media, LLC 2009.

hopf bifurcation control of ecological model via pd controller · 2016-12-31 · hopf bifurcation...

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