Bifurcation Analysis and Sliding Mode Control of Chaotic Vibrations in an Autonomous System

We study the bifurcations and sliding mode control of chaotic vibrations in an autonomous system. More precisely, a Hopf bifurcation controller is designed so as to control the unstable subcritical Hopf bifurcation to the stable supercritical Hopf bifurcation. Research result shows that the control method can work very well in Hopf bifurcation control. Besides, we controlled the system to any fixed point and any periodic orbit to eliminate the chaotic vibration by means of sliding mode method. And the numerical simulations were presented to confirm the effectiveness of the controller.


Introduction
In 1963, Lorenz discovered chaos in a simple system of three autonomous ordinary differential equations in order to describe the simplified Rayleigh-Benard problem [1].From then on, some other chaotic systems were established, such as Chen system [2], Lü system [3], and Chu system [4].Despite the simplicity of three-dimensional autonomous systems, these systems have a rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations, depending on the parameter values.Moreover, bifurcation analysis and numerical simulation for these systems have been done by many researchers, such as [5][6][7][8][9][10][11][12][13][14].In 2008, a new three-dimensional Lorenz-like chaotic system is reported; nonlinear characteristic and basic dynamic properties of the three-dimensional autonomous system are studied by means of nonlinear dynamics theory, including the stability and the conditions for generating Hopf bifurcation of the equilibria [15].Sotomayor et al. [16] use the projection method described in [17] for the calculation of the first and second Lyapunov coefficients associated with Hopf bifurcations of the Watt governor system, and it was extended to the calculation of the third and fourth Lyapunov coefficients.Dias et al. [18] studied the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters in a Lorenz-like system.Zhang et al. [19] reported the finding of a new simple three-dimensional quadratic chaotic system with three quadratic nonlinearities obtained by adding a cross-product nonlinear term to the second equation of the Rucklidge system.
In recent years, the research of robust control system made great progress not only in theory but also in practical application.As a representative of the nonlinear robust control theory, variable structure control theory has been widely researched throughout the world and also had an increasing number of industrial applications.Wang et al. [20] present two methods to design a single-input/single-output integral variable structure system.Lee et al. [21] present a sliding mode controller with integral compensation for a magnetic suspension balance beam system.The control scheme comprises an integral controller which is designed for achieving zero steady-state error under step disturbances and a sliding mode controller which is designed for enhancing robustness under plant uncertainties.Di-Yi et al. [22] proposed a no-chattering sliding mode control strategy for a class of fractional-order chaotic systems.The designed control scheme guarantees the asymptotical stability of an uncertain fractional-order chaotic system.To ensure the robustness of the system control, Chen et al. [23] stabilized the chaotic orbits to arbitrary chosen fixed points and periodic orbits by means of sliding mode method, and MATLAB simulations were presented to confirm the validity of the controller.(1) In this paper, we investigate Hopf bifurcation of a new chaotic of the form [25]

Linear Analysis of System
where (, , ) ∈ R 3 are the state variables and (, ) ∈ R 2 are real parameters.In this section, we study some of the generalities and linear stability of system (1).In a vectorial notation which will be useful in the calculations, system (1) can be written as   = (, ), where = (, , ) ∈ R 3 and  = (, ) ∈ R 2 .The equilibria of system (1) can be found by solving the following equations simultaneously: From (3), the system (1) has two equilibria  ± = (± √ , ± √ , 0) if  > 0.
Because the system is invariant under the transformation  : (, , ) → (−, −, ), stability of  + and  − can be calculated similarly, so, one only needs to consider the stability of any one of them.The stability of the system at the fixed point  + is analyzed.
then the equilibrium  + is asymptotically stable.
Proof.At the fixed point  + = ( √ , √ , 0), the Jacobian matrix is defined as The characteristic polynomial of the Jacobian matrix of system (1) at  + has the form If  >  0 , according to Routh-Hurwitz criterion, the equilibrium  + is unstable.According to Routh-Hurwitz criterion, the real parts of all the roots  of ( 6) are negative if and only if so the proposition follows.
The equation  =  0 in (4) gives the equation of the Hopf curvein the parameter plane (, ).This equation will be used in Section 4 in the study of Hopf bifurcations which occur at the equilibria  + of system (1).

Outline of the Hopf Bifurcation Methods
This section is a review of the projection method described in [15][16][17][18] for the calculation of the first Lyapunov coefficients and second Lyapunov coefficients associated with Hopf bifurcation, denoted by  1 and  2 , respectively.
Consider the differential equation where Suppose that ( 0 ,  0 ) is an equilibrium point of (8), where the Jacobian matrix A has a pair of purely imaginary eigenvalues  2,3 = ± 0 , ( 0 > 0) and admits no other eigenvalue with zero real part.Let T  be the generalized eigenspace of A corresponding to  2,3 .By this it means the largest subspace invariant by A on which the eigenvalues are  2,3 .
Let ,  ∈ C 3 be vectors such that where A T is the transpose of the matrix A. Any vector  ∈ T  can be represented as  =  +  , where  = ⟨, ⟩ ∈ C. The two-dimensional center manifold associated with the eigenvalues  2,3 = ± 0 can be parameterized by the variables  and  by means of an immersion of the form  = (, ), where  : C 2 → R 3 has a Taylor expansion of the form with ℎ  ∈ C 3 and ℎ  = ℎ  .Substituting this expression into (8) we obtain the following differential equation: where  is given by (9).The complex vectors ℎ  are obtained solving the system of linear equations defined by the coefficients of ( 14), taking into account the coefficients of , so that system (14), on the chart  for a central manifold, writes as follows: with   ∈ C. The first Lyapunov coefficient  1 is defined by where The second Lyapunov coefficient is defined by where  32 = ⟨,  32 ⟩,  32 = 6(ℎ The complex vector ℎ 21 can be found by solving the nonsingular  + 1-dimensional system with the condition ⟨, ℎ 21 ⟩ = 0. Consider the following: A Hopf point ( 0 ,  0 ) of system ( 8) is an equilibrium point where the Jacobian matrix A has a pair of purely imaginary eigenvalues  2,3 = ± 0 , ( 0 > 0), and the other eigenvalue  1 ̸ = 0. From the Center Manifold Theorem, at a Hopf point, a two-dimensional center manifold is well defined; it is invariant under the flow generated by (8) and can be continued with arbitrary high class of differentiability to nearby parameter values.
A Hopf point is called transversal if the parameterdependent complex eigenvalues cross the imaginary axis with nonzero derivative.In a neighborhood of a transversal Hopf point with  1 ̸ = 0 the dynamic behavior of the system (8), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to the following complex normal form: where  ∈ C. ,  and  1 are real functions having derivatives of arbitrary higher order, which are continuations of 0,  0 , and the first Lyapunov coefficient at the Hopf point.When  1 < 0 ( 1 > 0) one family of stable (unstable) periodic orbits can be found on this family of manifolds, shrinking to an equilibrium point at the Hopf point.
A Hopf point of codimension 2 is a Hopf point, where  1 vanishes.It is called transversal if  1 = 0 and  = 0 have transversal intersections, where  = () is the real part of the critical eigenvalues.In a neighborhood of a transversal Hopf point of codimension 2 with  2 ̸ = 0 the dynamic behavior of the system (8), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to   = ( +  0 ) + || 2 +  1 || 4 , where  and  are unfolding parameters.

Hopf Bifurcations at 𝐸 +
In this section we will study the stability of  + under the conditions  =  0 and  > 0.Then, using the notion of the previous section the multilinear symmetric functions corresponding to  can be written as The eigenvalues of A are And from ( 12) one has ) , where where We can get where Now it remains only to verify the transversality condition of the Hopf bifurcation.In order to do so, consider the family of differential equation (1) regarded as dependent on the parameter .The real part , of the pair of complex eigenvalues at the critical parameter  =  0 , verifies Since   ( 0 ) ̸ = 0, the transversality condition at the Hopf point holds.
Using these calculations we prove the next theorem.
Theorem 2. Consider the three-parameter family of differential equations (1).The first Lyapunov coefficient associated with the equilibrium  + is given by where As  > 0 then  1 ( 0 , ) > 0, so the system (1) has a transversal Hopf point at  + when  =  0 and  > 0.More precisely, Hopf point at  + is unstable and for each  >  0 , but close to  0 , there exists an unstable periodic orbit near the asymptotically stable equilibrium point  + .
Next, we will give a numerical example of system (1).Let  = 25; we can calculate the Hopf bifurcation value  0 = −1.The equilibrium is stable when  = −1.4<  0 and unstable when  = −0.8>  0 , as shown in Figures 1 and 3, respectively.From the formulas in previous section, we have   ( 0 ) > 0,  1 > 0. Thus, the periodic solutions bifurcating from the equilibrium point  + are subcritical and unstable (Figure 2).
When the parameter  = 25 is fixed while parameter  is varied in the interval [−1.5, −0.7], some different dynamical behaviors of system (1) are obtained.The bifurcation diagram and the Lyapunov exponent spectrum of system (1) in terms of the parameter  are depicted in Figure 4.With the analysis performed here one can find the Hopf bifurcation at the equilibrium  + does occur when  = −1.

Hopf Bifurcation Control
In this section, we will design control laws such that our feedback system undergoes a supercritical and stable Hopf bifurcation.To accomplish the control of Hopf bifurcation in the system (1), we design the controller which has the following structure: where  is the control gain.In the following, we will study the stability of  + in the controlled system: Then, using the notion of the previous section the multilinear symmetric functions corresponding to  can be written as The first Lyapunov coefficient associated with the equilibrium  + is given by where Obviously, the value of  1 ( 0 , , ) is associated with , .We can adjust the positive and negative of the  1 ( 0 , , ) through the change of the value of .According to the last section, we know that the Hopf bifurcation at the equilibrium point  + is subcritical and unstable.Let  = 25 and obtain  1 ( 0 , , ) = 1.61637931 + 0.01550513.If  < −0.0096, then  1 ( 0 , , ) < 0. That is to say, if we get the value  less than −0.0096, we can control the unstable subcritical Hopf bifurcation to the stable supercritical Hopf bifurcation.Next, we will give a numerical example of system (31).Let  = 25,  = −0.1;we can calculate the first Lyapunov coefficient  1 ( 0 , , ) = −0.1461< 0. Thus, the periodic solutions bifurcating from the equilibrium point  + are supercritical and stable.The time history and phase diagram are shown in Figures 5 and 6, respectively.
Observe that the first Lyapunov coefficient vanishes if and only if  = −1,  = 25,  = −0.0096.In the following theorem we study the sign of the second Lyapunov coefficient where the first coefficient vanishes.

Theorem 3. Consider the differential equations (31). The second Lyapunov coefficient associated with the equilibrium 𝐸 + is given by
As  2 < 0 and the transversality condition at the Hopf point holds then system (31) has a transversal Hopf point of codimension 2 at  + .More specifically, the Hopf point at  + is stable.Re  32 = −0.018869167. (37)

The Global Bifurcation Analysis
For this system, bifurcation can easily be detected by examining graphs of  versus each of the control parameters  and , respectively, if we fix the other one.
When the parameters  = 0.1 are fixed,  varies on the closed interval [0, 6].Figures 7(a)-7(b) show the bifurcation diagrams of the state  and the corresponding Lyapunov exponent spectrum versus increasing , respectively.While  increases the system is undergoing some representative dynamical routes, such as period-doubling bifurcations, chaos, stable fixed points, and stable periodic loops.
When the parameters  = 2 are fixed,  varies on the closed interval [0, 0.6].Figures 8(a)-8(b) show the bifurcation diagrams of the state  and the corresponding Lyapunov exponent spectrum versus increasing , respectively.While  increases the system is undergoing some representative dynamical routes, such as period-doubling bifurcations, chaos, and stable periodic loops.

Sliding Mode Control of Chaotic Vibrations
7.1.The Design of the Controller.We designed a sliding surface with good nature and made the system possess the desired properties when it limitations on the sliding surface.Then to facilitate control, make the system reach the sliding surface and keep sliding.After joining the controller, the system (1) has the following form: where  1 ,  2 , and  3 are control inputs.If we join the reasonable controller, we can control the chaos system to the required range or the fixed point.Define the following matrix: where A is the linear matrix of the system, B is the control matrix, d is the bounded perturbation matrix, and g is the nonlinear matrix of the system.The purpose of control is to let the system state x = [ 1 ,  2 ,  3 ] T to track a time-varying state x  = [ 1 ,  2 ,  3 ] T .So, we can define the tracking error e = x − x  . (40) The error dynamic system can be written as Define the time-varying proportional integral sliding mode surface where K ∈ R 3×3 , det(KB) ̸ = 0.For the convenience of calculation, we get K = diag(1, 1, 1).The additional matrix L ∈ R 3×3 , and A−BL is negative definite matrix.The equation S = Ṡ = 0 must be satisfied under the sliding mode, where In order to satisfy the sliding conditions, the following controller is designed: where sign(S) is symbolic function.
By the same token, we get So the proposition follows.
7.2.The Numerical Simulation.In the case of  1 =  2 =  3 = 0, the time domain charts of the state variables of system (38) are shown in Figure 11. ) .
Select proportional integral sliding mode surface as follows:   the Hopf bifurcation boundaries are analyzed with the change of two parameters.At last, we designed a sliding surface with good nature and made the system possess the desired properties when it limitations on the sliding surface.And we eliminated the chaotic vibration by means of sliding mode method.Compared with other control methods, the  sliding mode method can overcome the uncertainty of the system, has very strong robustness for the interference and unmodeled dynamics, and especially for nonlinear system control has better control effect.And the numerical simulations were presented to confirm the effectiveness of the controller.Apparently there are more interesting problems about this chaotic system in terms of complexity, control, and synchronization, which deserve further investigation.

7 . 3 .
Control to the Fixed Point.We can stabilize the system (38) to any point by this method.In this paper, we select Time domain chart of  after control

Figure 12 :
Figure 12: Time domain charts of state variables after control.
Time domain chart of  3 after control

Figure 13 :
Figure 13: Time domain charts of sliding surfaces after control.

Figure 14 :
Figure 14: Time domain charts of state variables after control.