Hopf Bifurcation Control in a Lorenz Type System

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ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2018.68146 Aug. 28, 2018 1704 Journal of Applied Mathematics and Physics

Hopf Bifurcation Control in a Lorenz Type

System

P. E. Calderon-Saavedra

1

, E. Munoz-Aguirre

2

, J. Alvarez-Mena

2

, S. Gomez-Perez

2 1_{Universidad del Tolima, Ibagu}_é_{, Colombia}

2_{Universidad Veracruzana, Veracruz, Mexico}

Abstract

In this paper Hopf bifurcation control is implemented in order to change the bifurcation from supercritical to subcritical in a differential equations system of Lorenz type. To achieve this purpose: first, a region of parameters is identi-fied where the system has a supercritical Hopf bifurcation; second, a class of non-linear feedback control laws is proposed; finally, it is shown that there are control laws which the disturbed system undergoes subcritical Hopf bi-furcation.

Keywords

Lorenz Type Systems, Bifurcation Control, Hopf Bifurcation Control, Subcritical and Supercritical Hopf Bifurcation

1. Introduction

In [1] the Bifurcation Control is described as the task of designing a control law that modifies the bifurcation characteristics in order to achieve a desirable dynamic behaviour. To specify the above, set a given nonlinear system,

(

, ,

)

f α

=

x x (1) with state vectors _x_∈_n_{and parameters vector}_{α ∈}_m_{, which presents a}

bifurcation. A parametric class of functions u

( )

⋅,α _{is proposed, called control}

laws, with which (1) is perturbed, resulting the nonlinear system,

(

,

) (

, .

)

f α u α

= +

x x x (2)

The bifurcation control problem consists in to find, within the parametric class, a control law that achieves a desirable behaviour or an established objective in the perturbed system.

Objectives of the bifurcation control are diverse, some of them are: to move

How to cite this paper: Calderon-Saavedra, P.E., Munoz-Aguirre, E., Alvarez-Mena, J. and Gomez-Perez, S. (2018) Hopf Bifurca-tion Control in a Lorenz Type System. Journal of Applied Mathematics and Phys-ics, 6, 1704-1719.

https://doi.org/10.4236/jamp.2018.68146

Received: April 12, 2018 Accepted: August 25, 2018 Published: August 28, 2018

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/jamp.2018.68146 1705 Journal of Applied Mathematics and Physics the critical value or the bifurcation points [2] [3], introducing a new bifurcation at a preferable parameter value [4], to change the amplitude or frequency of some limit cycles emerging from bifurcation [1] [5], to stabilize the periodic orbit that emerges from the Hopf bifurcation [3] [6], to modify the shape or type of a bifurcations chain [7] [8], to change and/or create a bifurcation [4] [9].

A kind of bifurcation is the Hopf bifurcation, this corresponds to the following situation: when the parameter crosses a critical value, the Jacobian of the system evaluated in the equilibrium has a pair of conjugate complex eigenvalues, which moves from the left to the right half-plane, while the other eigenvalues remain stable; at the crossing moment the real parts of the two eigenvalues become zero, the stability of the equilibrium changes from stable to unstable and a limit cycle arises. When the limit cycle is stable the Hopf bifurcation is said to be supercritical and when the limit cycle is unstable it is said that the Hopf bifurcation is subcritical [10].

When the system (1) presents Hopf bifurcation in a equilibrium point

(

_x∗,α

)

_{, the}_{Hopf Bifurcation Control}_{consists in to determine a control law}

( )

_, 0

u ⋅α of a parametric class u

( )

⋅,α , in such a way that the equilibrium point

(

_x∗,α

)

_{of (1) moves to an equilibrium point}

(

_x0_,_α0

)

_{of the perturbed system} (2), the Hopf bifurcation is preserved and one of the following situations occurs:

1) Stability of the limit cycle that emerges from the Hopf bifurcation changes from unstable to stable or vice versa.

2) Change the orientation of the limit cycle. In this paper is consider the Lorenz type system

(

)

,

, , x a y x y dy xz

z bz gxy

= −

= − = − +

  

(3)

with state vector

(

_{x y z}_{, ,}

)

T_∈_3_{, parameters}_a_>₀_, _g_>₀_and _{b d}_, _∈__{. The} system (3) is a particular case of the system studied by Xianyi Li y Qianjun Ou in

[11], where is proved, under standar hypotheses, the system has a Hopf bifurcation. Recently, in [12] is proved the Hopf bifurcation in (3) is non-degenerated and supercritical.

The objective is apply Hopf Bifurcation Control to the system (3) in order to change the Hopf bifurcation from supercritical to subcritical, by using non-lineal control laws in state feedback. The used method for Hopf bifurcation analisys include the Hopf Theorem, the Central Manifold Theorem, Normal Forms, and a formule for the first Lyapunov coefficient given in [13].

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DOI: 10.4236/jamp.2018.68146 1706 Journal of Applied Mathematics and Physics On the other hand, in [3] [24] [25] [26], Hopf bifurcation control is done with the purpose of to change the Hopf bifurcation from supercritical to subcritical, for systems that are not Lorenz type.

Remainder of this paper is ordered as follows. The Section 2 describes some preliminary results concerning the existence of the Hopf bifurcation in the system (3). In Section 3 it is proved that the Hopf bifurcation is non-degenerate and supercritical. In Sections 4 and 5 the Hopf bifurcation control of the system (3) is made. Finally, in Section 6 the conclusions of this paper are presented.

2. Dynamics System

In this section, in order to make a self-contained writing, the results of [11] are specialize to the system (3), which correspond to the Hopf bifurcation.

System that Li and Ou considers is as follows,

(

)

2 , ,

, x a y x

y dy xz

z bz fx gxy

= −

= − = − + +

  

(4)

where a>0 _, f ≥0 _, g≥0 _, f g+ >0 _, b d, ∈_ _{, and states vector}

(

_{x y z}_{, ,}

)

T_∈_3_{. It is observed that when} _f ₌₀_{the system (3) is obtained.} Li and Ou [11] found that when

(

a b d f g, , , ,

) (

= 10,3,6,1,0

)

and

(

, ,

)

23 33 7, , 20 10 2 x y z _{= } _

 , the system that results presents the Lyapunov exponents:

1 0.4265

LE

λ = , λLE₂=0.000001, λLE3 = −7.4264, it is observed that two of these exponents are positive, which characterizes the emergence of a chaotic orbit. The

Figure 1 shows the chaotic attractor of the system (4) when the initial conditions and parameters are as indicated at the beginning of the paragraph.

Equilibrium points of the system (4) are classified according to the next cases: • _Case bd<0: There is only one equilibrium point P0=

(

0,0,0

)

.

• _Case bd=0: If b=0, all point of form

(

0,0,z

)

is an equilibrium and for 0

d= the equilibrium is P0.

• _Case bd>0: There are three equilibrium points P0 =

(

0,0,0

)

,

1 bd , bd ,

P d

f g f g

 

= _ _

+ +

  and 2 , ,

bd bd

P d

f g f g

 

= −_ − _

+ +

 .

From now, the analysis focuses on the system (3) that results from taking 0

f = _{in (4), and is restricted to the case} bd>0.

Li and Ou in [11] proved that in P0 the system (3) has a no-bifurcation dynamics and that in the equilibrium points P1 y P2 the system presents Hopf bifurcation, if b>0 the bifurcation critical value is:

. 3 a b d−

+ =

(4)

[image:4.595.251.496.71.253.2]

DOI: 10.4236/jamp.2018.68146 1707 Journal of Applied Mathematics and Physics Figure 1. Chaotic attractor of Li-Ou.

Theorem 1 When a>0 _, b>0 , d>0 and g>0 , the following statements are true to the system (3):

1) For 0

3 a b

d +

< < , the equilibriums P1 and P2 are asymptotically stable.

2) When a b d₃+ < _,_{the equilibriums P}₁_{and P}₂_{are unstables}_.

3) When d=a b₃+ , in each equilibrium P1 and P2 a periodic orbit emerges

with period T 2π

ab

= _.

Hence the system (3) presents Hopf bifurcation at the equilibrium points P1

and P2.

3. Hopf Bifurcation Analysis

In this section it will be showed that the Hopf bifurcation of the system (3) is non-degenerated and supercritical. To achieve this it is used the central manifold theory, it can be consulted in [10] Chapter 5.

Theorem 2 can be found in [12] and the proof presented here is in essence the same except that the Lyapunov coefficient is obtained through a different way.

Theorem 2 If the parameters satisfy b>0, a b= ,

3 a b

d= + _,_and g>0,

then the system (3) exhibits non-degenerated and supercritical Hopf Bifurcation in the equilibrium points P1 and P2.

Proof. Since the conditions in the Theorem 1 are met, the system (3) exhibits Hopf bifurcation in the equilibrium points P1 and P2. It will be demonstrated that Hopf bifurcation is supercritical, at verifying that the first Lyapunov coefficient is negative in the equilibrium point P1. Since the symmetry of the system, the conclusion is the same for the equilibrium point P2.

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DOI: 10.4236/jamp.2018.68146 1708 Journal of Applied Mathematics and Physics 2

0

2 2 _{, with} 2 _,

3 3 3

a a

a a a

A h h

g

gh gh a

−

 

 

= − − =

 

 ₋ 

 

(5)

and corresponding characteristic polynomial

( )

3 4 2 2 4 .3

3 3

a a

P λ =λ + λ +a λ+

Let P be with a couple of pure imaginary roots and a not-zero real root denoted by λ ω1= i, λ2 = −ωi and λ3, respectively, where

ω

>0. Due to the

relationship between the roots and the P coefficients, it follows that

1 2 3 4 ,₃a

λ λ+ +λ = − ₍₆₎

2 1 2 2 3 1 3 a ,

λ λ +λ λ λ λ+ = (7)

3

1 2 3 4 .₃a

λ λ λ

= − ₍₈₎

By replacing λ λ1+ 2=0 into (6) is obtained the root λ3= −4₃a, which is

negative since a is positive. Then, by replacing in (8) λ ω1= i, λ2 = −ωi and 3 4 3a

λ = − _{, is obtained} ω =a.

Resolving the system Aq aiq= , it has that the complex eigenvector is,

2 2

. 1 3 ah

q ah ahi

a a i

 

 

=_ + _

 

 − 

 

By resolving the system _{A p}T _{= −}_aip_{, it has the complex eigenvector,}

2 2 2

2 2

5 1

3 3

.

a gh a i

p a a i

ah

 ₋ ₊ 

  

 

 

= − +

 

 

In order to eigenvectors p and q satisfy the condition p q, =1_{, the} eigenvector p is rewritten in the form:

2 2

3

5 5

9 3

9 7 1 _.

3 3

50

4 3

a a i

p a a i

ha

ah ahi

 ₊ 

 

= _− − _

 

 ₊ 

 

 

The equilibrium point P1= h h, ,2₃a

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DOI: 10.4236/jamp.2018.68146 1709 Journal of Applied Mathematics and Physics transformation x h y= + 1 , y h y= + 2 , z=2₃a+y3 , thus the system (3) beecomes

1

0

2 2 _,

3 3

a a

Y h Y y BY

gh gh a

−       = − − +    ₋   

 ₍₉₎

where

(

)

T

1, ,2 3

Y = y y y and

0 0 0

0 0 1

0 0 B g     =_ − _     .

Observe that the system (9) has the form Y AY F Y= +

( )

, where A is the Jacobian matrix of the system evaluated in the equilibrium point P1 and the

non-linear part

( )

2

F Y =O Y , is a smooth function with Taylor expansion in a neighbourhood of Y =0_{with expression:}

( )

1

(

_,

)

1

(

_{, ,}

)

( )

4 _,

2 6

F Y = B Y Y + C Y Y Y +O Y

where B Y Y

(

,

)

and C Y Y Y

(

, ,

)

, are multi-linear vector functions of Y. To the system (9), the multi-linear functions B and C are:

( )

(

)

(

)

2 2 0 ₀ 1

, 2 , , 2 ,

3

B q q ah a i q q ah

ahg h ag g i

ω ω ω

ω   _ _   _ _   = − + = _− _   _ _  ₊     

(

)

(

)

(

)

2 0 ₀ 1

, 2 , , , 0 ,

3

0

B q q ah a i q q q

h ag g i

ω ω ω   _{ }   _{ }   = − − _{=  }   _{ }  ₋      finally, obtaining

( )

1 3

1 2 0

.

F Y y y

gy y     = −_ _    

First Lyapunov coefficient 1 is calculated using the formula taken from

[10],

( )

(

)

(

)

(

)

(

)

( )

(

)

1 1 1 3 1

0 , , ,

2

2 , , ,

, , 2 , ,

Re p C q q q

p B q A B q q

p B q i I A B q q

ω ω − −  = _ −  + − _  (10)

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DOI: 10.4236/jamp.2018.68146 1710 Journal of Applied Mathematics and Physics

2 2

1

3 2 1

6 2 2

1 2 3 _{1 .}

6 2 2

2 ₁ ₀

3

gh a a

ahg hg g

a h g a

A

h ahg hg g

− − + −       ₊ ₋    =      −     

The matrix 2i Iω −3 A and its inverse are,

3

2 0

2 2

2 2 ,

3 3

2

i a a

i I A a i a h

hg gh i a

ω ω ω ω + −       − = −    ₋ ₋ ₊   

(

)

(

)

(

)

(

)

2 2 2

2 2

1 3

2

9 2 2 2

6 2 2

4 4 2

1 2 3 6

2 ,

6 2 2

2 ₂ 1

3 3

a a i a a i ah

a a i _{h a} _i

a h g i

i I A

r

agh gh i h ag g i a i

ω ω ω

ω ω

ω ω ω ω

− − − + + −      ₋ ₋ ₋ ₋ ₊ ₋ ₊    − =      − + + − +     

where _r_{= −}₂_a3₋₃_ω3_i_.

By replacing this elements in (10), result the first Lyapunov coefficient

( )

2

1 142 0 , 65 a = − 

which is negative, thus the periodic orbit that arises in the equilibrium point

1

P of the system (3) is stable. Hence in the equilibrium points P1 and P2 of

the system (3) the supercritical and non-degenerated Hopf bifurcation is presented.

Example 1 For each case of the Theorem 1 a particular system and the graph of the orbit with initial condition

(

0 0 0

)

5 5 89

, , , ,

4 4 100 x y z _{= } _

  are presented.

System with parameters

(

, , ,

)

9 9 9, , ,2 10 10 20

a b d g _{= } _

  satisfies the conditions

0

a> _, b>0, g>0 and

3 a b

d< + _._{Then by the part}₁_{of the Theorem}_1,_the

equilibrium points P1 and P2 are asymptotically stables. The Figure 2(a)

illustrates the behaviour of the orbit around P1.

When the parameters are

(

, , ,

)

9 9 7, , ,2 10 10 10

a b d g _{= } _

 , the conditions a>0,

0

b> _, g>0_and

3 a b

d> + _{are satisfied}_,_{and the part}₂_{of the Theorem}₁

guaranties that the equilibrium points P1 and P2 are unstable, as is illustrated in

the Figure 2(b).

To the parameters

(

, , ,

)

9 9 3, , ,2 10 10 5

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DOI: 10.4236/jamp.2018.68146 1711 Journal of Applied Mathematics and Physics 0

g> and d=a b+₃ are verified, and the part 3 of the Theorem 1 guaranties

the existence of a periodic orbit. As consequence of the Theorem 2 this orbit is stable. TheFigure 2(c)shows a orbit that converge at limit cycle around P1.

(a)

(b)

[image:8.595.267.477.144.684.2]

(c)

Figure 2. Dynamics of the system (3) when

(

, ,

)

9 9, , 2 10 10

a b g = _ _.

(a) d=₂₀9 _{; (b)} 7 10

d= _{; (c)} 3 5

(9)

4. Hopf Bifurcation Control

In this section the main results about Hopf bifurcation control in the system (3) are presented, the proofs are postponed to the next section.

In order to resolve the Hopf bifurcation control problem in the system (3), the control laws class in closed loop are used,

(

)

(

)

T

, , 0, ,0 ,

u X d k = −kx z d− ₍₁₁₎

where

(

)

T

, ,

X = x y z is the states vector, the parameter k is a positive real number called control gain, and d is a parameter of the system (3). The control laws class (11) was used by Pei in [23] to the Hopf bifurcation control in the Lü system.

Perturbed system with the control laws class (11) is the next,

(

)

(

)

,

, .

x a y x

y dy xz kx z d

z bz gxy

= −

= − − −

= − +

  

(12)

with parameters a>0, g>0, k>0 and b d, ∈.

Problem consists in finding a control law of the class (11) such that the perturbed system preserves the equilibrium points of the system (3), it presents Hopf bifurcation and this is subcritical. When a control law achieve the planted objectives, it is said that it changes the Hopf bifurcation of the system (3) from supercritical to subcritical.

Observe that the perturb system preserves the equilibrium points P1, P2 and P3 of the system (3), since u P d k

(

1, ,

)

=u P d k

(

2, ,

)

=u P d k

(

0, ,

)

=0.

Next theorem states that the perturbed system (12) presents Hopf bifurcation in the equilibrium points P1 and P2.

Theorem 3 To the system (12) with parameters a>0_, a b= _, d>0_, g>0

and k>0, the critical value of the bifurcation is 3 8 9

2

H k

d a

k

− + +

= _{and the}

next sentences are true:

1) When 0< <d dH, the equilibrium points P1 and P2 are stable. 2) For the case dH <d , the equilibrium points P1 and P2 are unstable. 3) When d d= H, in each equilibrium point P1 and P2 arises a periodic orbit

with period T 2π

ω

= where 2 2

(

1

)

2

H

a d k

a d

ω= +

− .

Thus, the system presents Hopf bifurcation in the equilibrium points P1 and

P2.

Next theorem states that there are control laws for which the Hopf bifurcation of the perturbed system is subcritical.

Theorem 4 Let the perturbed system (12) be, with parameters a>0_, a b= _, 0

g> _and 3 8 9

2 k

d a

k

− + +

= _,_{then there is at least a value of k such that the}

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DOI: 10.4236/jamp.2018.68146 1713 Journal of Applied Mathematics and Physics Next corollary summarizes the obtained results.

Corollary 1 The control law in feedback of states

(

)

(

)

T

0, 9x z d ,0 ,

= − −

u

changes the Hopf bifurcation of the system (3) from supercritical to subcritical.

Example 2 In the Figure 3, a numerical simulation is presented to illustrate the Theorem 4 results. Taking the parameter values

(

, , , , ,

)

3 3 19, , ,0,3,10 5 5 100

a b d f g k = _ _ the conditions a>0, a b= , g>0 and

H

d d= _{are met}, and the Theorem 4 ensures the presence of an unstable periodic orbit around each equilibrium P1 and P2. On the graph, the orbit with

initial condition

(

x y z0, ,0 0

)

= _{1250 1250 5}433 433 2, , 

  converges to equilibrium point

and moves off of cycle, while the orbit with initial condition

1501 249 1041_, _, 2000 1250 10000

 

 

 , moves around of each equilibrium point, like a Lorenz

attractor.

5. Proof of Theorems 3 and 4

To proof the Theorems 3 and 4 the following lemmas are used.

Lemma 1 Let _T_:_3_→_3_{be defined as} _{T x y z}

(

_{, ,}

) (

_{= − −}_{x y z}_{, ,}

)

_._{The system}

(12) is invariant under T.

The proof follows from the fact that vectorial field f associated to the system (12) satisfies f T X

(

( )

)

=T f X

(

( )

)

, for all _X_∈_3_.

A consequence of the previous lemma is that the dynamic system, determined by the perturbed system, is symmetric with respect to the z-axis.

Lemma 2 Polynomial with real coefficients

( )

3 2

1 2 3

[image:10.595.247.503.505.686.2]

P λ =λ +lλ +lλ+l has a couple of conjugate imaginary pure roots, if and only if, l3 =l l1 2 with l2 >0.

Figure 3. Unstable limit cycle for

(

, , , , ,

)

3 3 19, , ,0,3,10 5 5 100

(11)

Lemma 3 Let

( )

3

( )

2

( )

1 2 3

=

P

λ

+l

α λ

+l

α λ

+l

α

_{be a polynomial which}

coefficients are differentiable real functions with respect to the parameter α in the neighbourhood V

(

α ε0;

)

. Suppose that for α α= 0, P has a couple of conjugate roots λ α1

( )

0 =µ α

( )

0 +iβ α

( )

0 , λ α2

( )

0 =µ α

( )

0 −iβ α

( )

0 with

( )

0 0

µ α = _and β α

( )

₀ = >ω 0_,_{and a negative real root}λ α₃

( )

₀ _._Then_,_{for all} α in a neighbourhood of α0, the polynomial P has two conjugate roots

( )

1 i

λ α =µ α + β α _, λ α₂

( )

=µ α

( )

−iβ α

( )

_with β α

( )

>0_{and a negative}

real root λ α3

( )

, and the derivative of the real part of the λ1 in α0 is:

( )

(

)

0 0

1 ₃ _{2 1} _{2 1}

2

1 2

.

2 2

D Re _{l l l l l}

D _{α α} l l _{α α}

λ

α ₌

=

′− ′ − ′ =

+ (13)

Proof of Theorem 3. Under the established conditions in the parameters the analysis of stability of the system in the equilibrium point P1 is done. As the system is symmetric, the conclusion is the same to the equilibrium point P2.

The Jacobian matrix A of the system (12) in the point P1 takes the form

(

10

)

, with ,

a a

ad

A d d k h h

g

gh gh a

−

 

 

= −_ − + _ =

 ₋ 

 

(14)

and characteristic polynomial

( )

3

(

₂

)

2

(

2

)

_{2 1}

(

)

2 _.

p λ =λ + a d− λ + a +akd λ+ +k a d (15)

With help of the Lemma 2, the necessary and sufficient conditions so that

( )

p

λ

to have two imaginary pure roots are 1)

(

₂_{a d a}₋

)

(

2₊_akd

)

₌_{2 1}

(

₊_{k a d}

)

2 2) _a2₊_akd_>₀

From 1)

1 3 ₂8k 9 and 2 3 ₂8k 9 .

d a d a

k k

− + + − − +

= =

From 2) and by the hypothesis of the theorem (a>0_and d>0),

3 8 _{9 .}

2

H k

d a

k

− + +

= ₍₁₆₎

Case 1. Let 0< <d dH be, that is,

2dk+3a a< 9 8 .+ k As the terms of inequality are positive,

2 ₃ ₂ 2 _0.

kd + ad− a <

Thus

(

₂_{a d a}₋

)

(

2₊_akd

)

_>_{2 1}

(

₊_{k a d}

)

2 _{. As} _k_>₀_and _d_>₀_{, it follows that}

(

)

2

2 1+k a d>0, and

(

₂_{a d a}₋

)

(

2₊_akd

)

_>₀_{. Hence,} ₂_{a d}_{− >}₀_.

By Routh-Hurwitz criterion, [27] page 460, a necessary and sufficient condition so that eigenvalues λ1, λ2 and λ3 of the matrix A to have negative

real part is that the expressions 2a d− _, _{2 1}

(

₊_{k a d}

)

2 _and

(12)

DOI: 10.4236/jamp.2018.68146 1715 Journal of Applied Mathematics and Physics point P1 of the perturbed system is stable.

Case 2. Let dH <d be, that is,

9 8 2 3 .

a + k < kd+ a Since the terms of inequality are positive,

2 2

0<kd +3ad−2 .a

Then

(

₂_{a d a}₋

)

(

2₊_akd

)

₋_{2 1}

(

₊_{k a d}

)

2 _<₀_{. From the previous inequality and} from _{2 1}

(

₊_{k a d}

)

2 _>₀ _{, by the Routh-Hurwitz criterion there is at least a}

{

1,2,3

}

j∈ _{such that}Re

( )

λj ≥0 . The case λ =j 0 , implies that

(

)

2

2 1+k a d=0, which is not possible. In the case that λj is pure imaginary,

by the Lemma 2, d d= H, which is not possible, since that dH <d. Hence,

when dH <d there is at least a j∈

{

1,2,3

}

such that Re

( )

λj >0. Therefore

the equilibrium point P1 of the system (12) is unstable.

Case 3. Let λ1, λ2 and λ3 be the roots of p

( )

λ

. When d 3 ₂8_kk 9a

− + +

= _,

it follows that two of these are pure imaginary, of the form λ ω1= i, λ2= −ωi

with

ω

>0_{. To find the value of}ω, it is observed that the coefficients of

( )

p

λ

_satisfy,

(

)

1 2 3 2a dH ,

λ λ λ

+ + = − − ₍₁₇₎

2

1 2 2 3 1 3 a akdH,

λ λ λ λ λ λ

+ + = + ₍₁₈₎

(

)

2 1 2 3 2 1 k a dH.

λ λ λ

= − + (19)

By replacing λ λ1+ 2=0 in (17), the real root

λ

3= −

(

2a d− H

)

is obtained.

By replacing λ1, λ2 and λ3 in (19),

(

)

2

2 1

0 2

H

a d k

a d

ω= + >

− is obtained.

Hence, as d d= H, the polynomial (15) has two conjugate eigenvalues λ ω1= i,

2 i

λ = −ω _with 2 2

(

1

)

0 2

H

a d k

a d

ω= + >

− and

λ

3= −

(

2a d− H

)

<0. Since to the Lemma 3 the transversality value is,

( )

(

)

(

)

(

)

2 2

1

2 2

2 1 2 2 _.

2 2 H

H

d d H

DRe k a k a d

Dd _{a d}

λ ω

ω

=

+ + + −

=

 ₋ ₊ 

 

Due to 2a d− H >0, 2 1

(

+k a

)

2>0 and

ω

>0 , it follows that the transversality value is positive. And by the Hopf Theorem, in the equilibrium point P1 of the system (12) arises a periodic orbit with period

_{( )}

1 2π

T Im λ

= ,

where

( )

1 2

(

)

2 1

2 H _H

a d k

Im

a d

λ = +

− .

Proof of Theorem 4. Let the parameters be as in the Theorem 4. According to Theorem 3, the system (12) exhibit Hopf bifurcation in the equilibriums P1 and

(13)

DOI: 10.4236/jamp.2018.68146 1716 Journal of Applied Mathematics and Physics When solving the system

(

A−

ω

iI q

)

=0_{, with}_A_{the Jacobian matrix of the} system (12) evaluated in P1, the complex eigenvector

(

)

(

)

(

)

(

)

2

1

1 1 ,

H

ah k

q ah k k hi

d a i

ω ω ω +     =_ + + + _  ₊ ₋   

is obtained, with _h adH

g

= _{. When solving the system}_{A p}T _{= −}_{i p}

_ω

_{, the}

eigenvector,

(

) (

)

(

)

(

)

(

)

(

)

(

)

(

)(

)

2 2 2 2 2 2

1 ₃ ₂ _,

1 2 1

H H H H H

H H

H

d a d a a d a d a a d i

p a a d a a a d i

r

ah k ah a d k i

ω ω ω ω ω  _ − − − _{ }+ − + − _     _ _  = − − +_ − − _    + + − +   

is obtained, where ₂

(

₁

)

3

(

₂

)

2

H

r₌ ah ₊k ω ₊ a d₋ ω

  . Also p q, =1 is

satisfied.

Equilibrium point P1 is moved to the origin via the transformation 1

X P X= + _{, with}

₍

₎

T , ,

X = x y z , resulting the system (12) in X AX F X= +

( )

_,

where A is given by (14), the non-linear part

( )

₀

( )

2

F X = X is a smooth function with Taylor expansion in a neighbourhood of X =0_,

( )

1

(

)

1

(

)

( )

4

, , , 0 ,

2 6

F X = B X X + C X X X + X

where B X X

(

,

)

y C X X X

(

, ,

)

are vectorial multi-linear functions with

3 X∈_ _.

To the system (12) in the origin, the vectorial multi-linear functions B and C

are:

( )

(

)

(

)

(

)

2 2

0

, 2 1 H ,

B q q ah k d a i

h ag g i

ω ω ω     = + _− − − _  ₊   

(

)

(

)

2 2

0

, 2 1 ,

B q q ah k

ahg ω     = + _− _    

(

)

(

)

(

)

(

)

(

)

2 2 0 0

, 2 1 , , , 0 ,

0

H

B q q ah k d a i C q q q

h ag g i

ω ω ω         = + _− + − _ =_{ }    ₋  _{ }   resulting,

( )

0 .

F X xz kxz

gxy     = − −_ _    

First Lyapunov coefficient 1

( )

0 is founded using the formula (10), below

(14)

DOI: 10.4236/jamp.2018.68146 1717 Journal of Applied Mathematics and Physics Inverse matrix _A−1_{of the matrix}_A_{definite in (14) is,}

(

)

(

)

(

)

(

)

(

)

(

)

(

)

2 2 1 1 1

2 1 2 1 2

1

1 _{1 .}

2 1 2 1 2

1 ₀

1 1

H

gh k ad a

ahg k hg k g

ad h g k a

A

h ahg k hg k g

d

a k k

− − + + −    + +    ₊ ₊ ₋    =  + +    −    ₊ ₊   

Matrix 2i Iω −3 A and her inverse are:

(

)

3

2 0

2 2 1 ,

2

H H

i a a

i I A d i d h k

hg gh i a

ω ω ω ω + −     − =_ − + _  ₋ ₋ ₊   

(

)

(

)

(

)

(

)

(

)(

)

[

]

[

]

(

)

1 3 2 2

2 ₂ ₂

2

3 2 2 1

2 2 2

1 2 1 2

1 4 4 _,

2 2 2

2

H H

H

H H

i I A

ad a d i a a i ah k

ad h g k i _a _{a i} _h _{k a} _i

s

h d g g i h ag g i a d i

ω

ω ω ω

ω ω ω ω

− − − − + − + − +      ₋_ ₊ ₊ ₊ _ ₋ ₊ ₋ ₊ ₊      =    ₋ ₊ ₊ _₋ ₊ ₋ _        

where ₃ 2

(

₁

)

3

H

s= − _a d +k +ω i_.

Finally, these are replaced in (10), resulting the Lyapunov coefficient

( )

2

(

2 3

1

4 5 6 7

8 9 10 11

12 13 2

3 4

0 2 2 196830 1277208 3519855 4921830

2091738 5196138 12026872 13293502

9317612 4357822 1338423 245120

18252 808 65610 396576 1003509

1230894 233486 1749622

a k k k

k k k k

k k t kt k t

k t k t k

= − − − − − + + + + + + + + − + + + + + − 

5_t₋_3243200_{k t}6

) (

)(

)

7 8 9 10

11 12 13 2

3 4 5 6 2

3 4 5 2 3

2 3

3154458 1952508 792330 202421

27452 796 72 2025 6930 11144

10514 5423 1164 24 675 2010 2888

2390 941 96 27 63 56 18

9 17 12 6 3

k t k t k t k t

k t k t k t k k

k k k k t kt k t

k t k t k t k k k

t kt k t k t k t

− − − −  − − + _ − − −  − − − − + + + + + + − − − −

+ + + +

(

− + +

) (

1

)(

3

)

,

3

k t

kt

k t kt



+ − +



− + + _

(20)

with t= 8k+9_.

When k=9_{, the expression (20) takes the form,}

( )

2

1

14175

0 5 0.

58 a

= >



(15)

DOI: 10.4236/jamp.2018.68146 1718 Journal of Applied Mathematics and Physics Hopf bifurcation. By Lemma 1, the conclusion is the same to the equilibrium P2.

6. Conclusions

Via the Theorem 2, the existence of the supercritical and non-degenerate Hopf Bifurcation to the system (3) in the equilibrium points P1 and P2 was proved.

It was proved that at perturbing the system (3) in the system (12), the

control law

(

)

T

0, 9 ,0

u= − x z d− changes the Hopf bifurcation of the system (3) from supercritical to subcritical, in the parameters region a>0_, a b= _,

3 8 9

2 k

d a

k

− + +

= _and g>0.

Acknowledgements

To CONACYT by the scholarship granted to the PhD in Mathematics student Pablo Emilio Calderon Saavedra of Universidad Veracruzana

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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