Research Article | Open Access
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.
In 1963, Lorenz discovered chaos in a simple system of three autonomous ordinary differential equations in order to describe the simplified Rayleigh-Benard problem . From then on, some other chaotic systems were established, such as Chen system , Lü system , and Chu system . 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 [5–14]. 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 . Sotomayor et al.  use the projection method described in  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.  studied the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters in a Lorenz-like system. Zhang et al.  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.  present two methods to design a single-input/single-output integral variable structure system. Lee et al.  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.  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.  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. Chen et al.  studied the nonlinear dynamics behavior of hydroturbine governing system. In order to eliminate the chaotic vibration, the author used sliding mode method and MATLAB simulations and controlled the system to any fixed point and any periodic orbit. The results show that using sliding mode method can make the system track target orbit strictly and smoothly with short transition time. In this paper, we mainly consider a new chaotic system proposed by Gao . Gao constructed a three-dimensional continuous autonomous chaotic system, which is different from classic Lorenz system, Chen system, and Lü system. Some basic dynamical properties of the new system are analyzed by means of theoretical analysis and nonlinear techniques, such as phase diagram, time response diagram, Lyapunov exponent spectrum, and bifurcation diagram. However, the relationship between the Hopf bifurcation and the system parameters has not been clarified yet. And there is not any analysis of chaos control.
The main purpose of this paper is to investigate the bifurcations and sliding mode control of chaotic vibrations in an autonomous system (1). The rest of this paper is organized as follows. In Section 2, we present the linear analysis of equilibrium of system (1). A brief review of the method used to study codimensions one and two Hopf bifurcations is presented in Section 3. The direction of Hopf bifurcation and the stability of bifurcating periodic solutions are analyzed in detail in Section 4. In Section 5, we designed a controller such that the equilibrium undergoes a controllable Hopf bifurcation. In Section 6, nonlinear dynamic properties of this chaotic system are studied by means of bifurcation diagrams. We controlled the system to any fixed point and any periodic orbit to eliminate the chaotic vibration by means of sliding mode method in Section 7. Section 8 concludes the paper.
2. Linear Analysis of System (1)
In this paper, we investigate Hopf bifurcation of a new chaotic of the form  where are the state variables and 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 and .
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.
Proposition 1. If , the equilibrium is always unstable. If and then the equilibrium is asymptotically stable.
Proof. At the fixed point , the Jacobian matrix is defined as The characteristic polynomial of the Jacobian matrix of system (1) at has the form If , 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.
3. Outline of the Hopf Bifurcation Methods
This section is a review of the projection method described in [15–18] for the calculation of the first Lyapunov coefficients and second Lyapunov coefficients associated with Hopf bifurcation, denoted by and , respectively.
Consider the differential equation where , are, respectively, vectors representing phase variables and control parameters. Assume that is of class in . Suppose that (8) has an equilibrium point at and, denoting the variable also by , write as where , and, for ,
Suppose that is an equilibrium point of (8), where the Jacobian matrix has a pair of purely imaginary eigenvalues , and admits no other eigenvalue with zero real part. Let be the generalized eigenspace of corresponding to . By this it means the largest subspace invariant by on which the eigenvalues are .
Let be vectors such that where is the transpose of the matrix . Any vector can be represented as , where . The two-dimensional center manifold associated with the eigenvalues can be parameterized by the variables and by means of an immersion of the form , where has a Taylor expansion of the form with and . Substituting this expression into (8) we obtain the following differential equation: where is given by (9). The complex vectors 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 .
The first Lyapunov coefficient is defined by where , , , , and is the unit 3 × 3 matrix.
The second Lyapunov coefficient is defined by where , .
The complex vector can be found by solving the nonsingular -dimensional system with the condition . Consider the following:
A Hopf point of system (8) is an equilibrium point where the Jacobian matrix has a pair of purely imaginary eigenvalues , , and the other eigenvalue . 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 parameter-dependent complex eigenvalues cross the imaginary axis with nonzero derivative. In a neighborhood of a transversal Hopf point with 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 . , and are real functions having derivatives of arbitrary higher order, which are continuations of 0, , and the first Lyapunov coefficient at the Hopf point. When 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 vanishes. It is called transversal if and have transversal intersections, where is the real part of the critical eigenvalues. In a neighborhood of a transversal Hopf point of codimension 2 with the dynamic behavior of the system (8), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to , where and are unfolding parameters.
4. Hopf Bifurcations at
In this section we will study the stability of under the conditions and . Then, using the notion of the previous section the multilinear symmetric functions corresponding to can be written as
The eigenvalues of 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 , verifies Since , 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 then , so the system (1) has a transversal Hopf point at when and . More precisely, Hopf point at is unstable and for each , but close to , there exists an unstable periodic orbit near the asymptotically stable equilibrium point .
Next, we will give a numerical example of system (1). Let ; we can calculate the Hopf bifurcation value . The equilibrium is stable when and unstable when , as shown in Figures 1 and 3, respectively. From the formulas in previous section, we have , . Thus, the periodic solutions bifurcating from the equilibrium point are subcritical and unstable (Figure 2).
When the parameter is fixed while parameter is varied in the interval , 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 .
5. 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
By direct calculation, we can get The first Lyapunov coefficient associated with the equilibrium is given by where
Obviously, the value of is associated with , . We can adjust the positive and negative of the 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 and obtain . If , then . 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 , ; we can calculate the first Lyapunov coefficient . 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 , , . 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 is given by As 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.
Proof. Following the notation introduced in Section 3 and by direct calculations one has
6. 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 are fixed, varies on the closed interval . 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.
As increases, system (1) is undergoing the flowing dynamical routes:(1)if , , , , the system is stable;(2)if , , , , the system is chaotic. But there are several periodic windows in the chaotic band;(3)if , , , , there is a reverse period-doubling bifurcation route with a flip bifurcation;(4)if , , , , the system is chaotic. But there are several periodic windows in the chaotic band;(5)if , , , , there is a very long reverse period-doubling bifurcation window.
When the parameters are fixed, varies on the closed interval . 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.
As increases, system (1) is undergoing the flowing dynamical routes:(1)if , , , , the system is chaotic;(2)if , , , , there is a reverse period-doubling bifurcation route with a flip bifurcation;(3)if , , , , the system is chaotic. But there are several periodic windows in the chaotic band;(4)if , , , , there is a very long reverse period-doubling bifurcation window.
Let , , ; we use MATLAB to draw the stability region on the parameter plane , as shown in Figure 9. In the figure, the symbol , represents , . If , and , then the equilibrium is asymptotically stable. If , and , then system (1) has a transversal Hopf point at . And in regions (I) and (IV): , , , in regions (II) and (III): , , , in region (V): , , , and in region (VI): , , . All of the points are stable in regions (II) and (III) and in the other regions are unstable (Figure 10).
7. 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 , , and 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 is the linear matrix of the system, is the control matrix, is the bounded perturbation matrix, and is the nonlinear matrix of the system. The purpose of control is to let the system state to track a time-varying state . So, we can define the tracking error
The error dynamic system can be written as
Define the time-varying proportional integral sliding mode surface where , . For the convenience of calculation, we get . The additional matrix , and is negative definite matrix. The equation must be satisfied under the sliding mode, where
In order to satisfy the sliding conditions, the following controller is designed: where is symbolic function.
Proposition 4. The controller (44) can make the system (38) reach the sliding mode in a limited time if the constant satisfied the inequality , where , are arbitrary small positive numbers. The state variables and the selected reference state are identical.
7.2. The Numerical Simulation
(a) Time domain chart of before control
(b) Time domain chart of before control
(c) Time domain chart of before control
In order to control the system (38) to the target state, we select the eigenvalues of which are . The pole-placement method is adopted to get the following matrix:
Select proportional integral sliding mode surface as follows:
Set the initial value and the reference state . The control signal is as follows:
7.3. Control to the Fixed Point
We can stabilize the system (38) to any point by this method. In this paper, we select the fixed point , reference state , small parameter , and the initial value of the sliding mode surface . The controller is activated at , the time domain charts of state variables and sliding surfaces as shown in Figures 12 and 13, respectively.
(a) Time domain chart of after control
(b) Time domain chart of after control
(c) Time domain chart of after control
(a) Time domain chart of after control
(b) Time domain chart of after control
(c) Time domain chart of after control
Figures 12 and 13 show that after joining the controller, the system (38) tracks reference state ultimately, and the sliding mode surface becomes 0. It is proved that the system (38) reached the sliding mode. The output curve of the controller will swing between upper and lower bounds in a certain base value; this suggests that the system error has reached zero. The reason of the fluctuations in the output of the controller is only that the small changes of have caused the jump of the .
7.4. Control to the Periodic Orbit
We can stabilize the system (38) to the periodic orbit. In this paper, we select the reference state . The controller is activated at , the time domain charts of state variables as shown in Figure 14.
(a) Time domain chart of after control
(b) Time domain chart of after control
(c) Time domain chart of after control
Obviously, the system (38) tracks reference state to the periodic orbit ultimately.
8. Concluding Remarks
In this paper, a new autonomous system is studied in detail. Dynamical behaviors of the new system are analyzed, both theoretically and numerically, including some basic dynamical properties. By choosing an appropriate bifurcation parameter, we prove that Hopf bifurcation occurs when the bifurcation parameter passes through the critical value. The direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are analyzed in detail. Besides, we designed a Hopf bifurcation controller so as to control the unstable subcritical Hopf bifurcation to the stable supercritical Hopf bifurcation. The stable region and 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.
Conflict of Interests
The authors have declared that no conflict of interests exists.
The authors express their gratitude to the referee for valuable comments on the first version of the paper. The authors also gratefully acknowledge support from the National Natural Science Foundation (no. 11161027 and no. 61364001), Key Project of Chinese Ministry of Education (no. 212180), the Fundamental Research Funds for the Universities of Gansu Province (no. 620023), and the Natural Science Foundation of Gansu Province (no. 1010RJZA067), Government of China.
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