Hopf Bifurcation Control in a Lorenz Type System

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 identified 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 bifurcation.


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, ( ) with state vectors n ∈ x  and parameters vector m α ∈  , 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, ( ) ( ) 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 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 ( ) ,α x 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 with state vector ( ) 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].
On one side, the system (3) is a particular case of the system studied in [11], another side it is an extension of the Lu system [14], namely, when the parameter g takes the value 1, the system (3) is reduced to the Lu system [14].
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.

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, ( ) where , , x y z ∈  .It is observed that when 0 f = the system (3) is obtained.
Li and Ou [11] found that when ( ) ( ) , , , , From now, the analysis focuses on the system (3) that results from taking 0 f = in (4), and is restricted to the case 0 bd > .
Li and Ou in [11] proved that in P 0 the system (3) has a no-bifurcation dynamics and that in the equilibrium points P 1 y P 2 the system presents Hopf bifurcation, if 0 b > the bifurcation critical value is: . 3 Next theorem guarantees the existence of the Hopf bifurcation of the system (3) in the equilibrium points P 1 and P 2 ., the equilibriums P 1 and P 2 are asymptotically stable.
3) When , in each equilibrium P 1 and P 2 a periodic orbit emerges with period 2π T ab = .
Hence the system (3) presents Hopf bifurcation at the equilibrium points P 1 and P 2 .

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 , and 0 g > , then the system (3) exhibits non-degenerated and supercritical Hopf Bifurcation in the equilibrium points P 1 and P 2 .
Proof.Since the conditions in the Theorem 1 are met, the system (3) exhibits Hopf bifurcation in the equilibrium points P 1 and P 2 .It will be demonstrated that Hopf bifurcation is supercritical, at verifying that the first Lyapunov coefficient is negative in the equilibrium point P 1 .Since the symmetry of the system, the conclusion is the same for the equilibrium point P 2 .
The Jacobian matrix A of the system (3) in the point P 1 is given by, and corresponding characteristic polynomial ( ) Let P be with a couple of pure imaginary roots and a not-zero real root denoted by 1 i λ , respectively, where 0 ω > .Due to the relationship between the roots and the P coefficients, it follows that , is obtained a ω = .
Resolving the system Aq aiq = , it has that the complex eigenvector is, , it has the complex eigenvector, a gh a i p a a i ah In order to eigenvectors p and q satisfy the condition , 1 p q = , the eigenvector p is rewritten in the form:  is moved to the origin via the transformation where ( ) , , Y y y y = and Observe that the system (9) has the form ( ) , where A is the Jacobian matrix of the system evaluated in the equilibrium point 1 P and the non-linear part , is a smooth function with Taylor expansion in a neighbourhood of 0 Y = with expression: where ( ) C Y Y Y , are multi-linear vector functions of Y.To the system (9), the multi-linear functions B and C are: q q ah a i q q ah ahg h ag g i , , , 0 , 3 0 B q q ah a i q q q h ag g i finally, obtaining ( ) Re p C q q q p B q A B q q p B q i I A B q q where , ⋅ ⋅ denotes the usual inner product in the field of complex numbers and ( ) Re ⋅ denotes the real part function of a complex number.Some elements of the formula are written below.The inverse matrix of A defined in (5) The matrix 3 2i I A ω − and its inverse are, where By replacing this elements in (10), result the first Lyapunov coefficient ( ) 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 1 P and 2 P 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 ( ) System with parameters ( ) .Then by the part 1 of the Theorem 1, the equilibrium points P 1 and P 2 are asymptotically stables.The Figure 2(a) illustrates the behaviour of the orbit around P 1 .

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, 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, with parameters 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 P 1 , P 2 and P 3 of the system (3), since ( ) ( ) ( ) , , , , , , 0 u P d k u P d k u P d k = = = .
Next theorem states that the perturbed system (12) presents Hopf bifurcation in the equilibrium points P 1 and P 2 .
Theorem 3 To the system (12) with parameters Thus, the system presents Hopf bifurcation in the equilibrium points P 1 and P 2 .
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 , then there is at least a value of k such that the system presents subcritical Hopf bifurcation in the equilibrium points P 1 y P 2 .
Next corollary summarizes the obtained results.

Proof of Theorems 3 and 4
To proof the Theorems 3 and 4 the following lemmas are used.
Lemma 1 Let , ,  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 , for all 3 X ∈  .A consequence of the previous lemma is that the dynamic system, determined by the perturbed system, is symmetric with respect to the z-axis.λ α .Then, for all α in a neighbourhood of 0 α , the polynomial P has two conjugate roots ( ) ( ) ( ) with ( ) 0 β α > and a negative real root ( ) 3 λ α , and the derivative of the real part of the 1 λ in 0 α is: ( ) ( ) Proof of Theorem 3.Under the established conditions in the parameters the analysis of stability of the system in the equilibrium point P 1 is done.As the system is symmetric, the conclusion is the same to the equilibrium point P 2 .
The Jacobian matrix A of the system (12) in the point P 1 takes the form With help of the Lemma 2, the necessary and sufficient conditions so that ( ) p λ to have two imaginary pure roots are From 2) and by the hypothesis of the theorem ( 0 a > and As the terms of inequality are positive, By Routh-Hurwitz criterion, [27] page 460, a necessary and sufficient condition so that eigenvalues 1 λ , 2 λ and 3 λ of the matrix A to have negative real part is that the expressions 2a d with 0 ω > .To find the value of ω , it is observed that the coefficients of ( ) 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 P 1 and P 2 .Using the central manifold theory and the same steps that in the proof of Theorem 2, below is shown that the periodic orbit which arises in P 1 is unstable.Journal of Applied Mathematics and Physics When solving the system ( ) 0 A iI q ω − = , with A the Jacobian matrix of the system (12) evaluated in P 1 , the complex eigenvector ( ) ( ) ( ) ( ) . When solving the system point P 1 is moved to the origin via the transformation x y z = , resulting the system (12) in ( ) where A is given by ( 14), the non-linear part ( ) ( ) , , 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: is founded using the formula (10), below the elements of this formula are described.
ω − and her inverse are: ( )  Hopf bifurcation.By Lemma 1, the conclusion is the same to the equilibrium P 2 .

Conclusions
Via the Theorem 2, the existence of the supercritical and non-degenerate Hopf Bifurcation to the system (3) in the equilibrium points P 1 and P 2 was proved.
It was proved that at perturbing the system (3) in the system (12), the α * x , the Hopf Bifurcation Control consists in to determine a control law a way that the equilibrium point ( ) ,α * x of (1) moves to an equilibrium point ( ) 0 0

Figure 1 b
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 0 bd < : There is only one equilibrium point ( ) 0 0, 0, 0 P = .• Case 0 bd = : If 0 b = , all point of form ( ) 0, 0, z is an equilibrium and for 0 d = the equilibrium is 0 P .• Case 0 bd > : There are three equilibrium points
and the part 2 of the Theorem 1 guaranties that the equilibrium points P 1 and P 2 are unstable, as is illustrated in the Figure2(b).

Corollary 1
The control law in feedback of states bifurcation of the system (3) from supercritical to subcritical.Example 2 In the Figure3, a numerical simulation is presented to illustrate the Theorem 4 results.Taking the parameter values and the Theorem 4 ensures the presence of an unstable periodic orbit around each equilibrium P 1 and P 2 .On the graph, the orbit with initial condition ( ) of each equilibrium point, like a Lorenz attractor.

Lemma 2
Polynomial with real coefficients ( ) couple of conjugate imaginary pure roots, if and only if, 3 1 2 l l l = with 2 0 l > .
it follows that the transversality value is positive.And by the Hopf Theorem, in the equilibrium point P 1 of the system (12) arises a periodic orbit with period

When 9 k
= , the expression(20) takes the form, Due to the first Lyapunov coefficient is positive, the periodic orbit is unstable.Hence, in the equilibrium P 1 of the system(12) arises sub-critical non-degenerated P. E. Calderon-Saavedra et al.DOI: 10.4236/jamp.2018.681461718 Journal of Applied Mathematics and Physics P 1 of the perturbed system is stable.