Abstract
We choose the delay as a variable parameter and investigate the Lorentz-like system with delayed feedback by using Hopf bifurcation theory and functional differential equations. The local stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. After that the direction of Hopf bifurcation and stability of periodic solutions bifurcating from equilibrium is determined by using the normal form theory and center manifold theorem. In the end, some numerical simulations are employed to validate the theoretical analysis. The results show that the purpose of controlling chaos can be achieved by adjusting appropriate feedback effect strength and delay parameters. The applied delay feedback control method in this paper is general and can be applied to other nonlinear chaotic systems.
1. Introduction
Since the discovery of Lorenz chaotic attractor in 1963, chaos has been studied and developed by many scholars, for instance, Rössler system [1], Chua circuit [2], and Chen system [3] et al. However, due to the extremely sensitive characteristics of chaos to the system environment, it is sometimes necessary to suppress or stimulate chaos phenomena in real applications. Therefore, chaos control has attracted more and more attention, and the applications of dynamical systems and chaos involve mathematical biology, financial systems, chemistry, electronic circuits, secure communications, cryptography, and neuroscience research [4–9].
The main goal of chaos control is to eliminate chaotic behavior and stabilize the chaotic system. Especially, when it is useful, we want to generate chaos intentionally. So far, many advanced theories and methodologies have been developed in order to be better in controlling chaos. The existing control method can be classified, mainly, into two categories. The first one, the OGY method [10], which has completely changed the chaos research topic, is based on the invariant manifold structure of unstable orbits. The second one is DFC method, proposed by Pyragas [11, 12], using time-delayed controlling forces. It provides an alternative effective method for feedback control of chaos. In sharp contrast to the formal one, the second is a simple and convenient method of controlling chaos in continuous dynamical system. In order to make further study of the control of chaos via time-delayed feedback, in this paper, we aim to investigate the dynamical behaviors of Lorenz-like system with time delayed.
A new butterfly-shaped Lorenz-like system, which was proposed by Liu [13], is described by the following simple three-dimensional autonomous system with three quadratic terms: where are state variables and are real valued parameters. When , system (1) admits a chaotic attractor (see Figure 1). The conditions of Hopf bifurcation occurring and the stability and analysis of the equilibrium points have been studied in detail in [14].
The study of [15] shows that the chaotic behavior can be stabilized on various periodic orbits by use of Pyragas time-delayed feedback control. The results of the existence of Hopf bifurcation and effectiveness of delayed feedback have been given [16–23]. Following the idea of Pyragas [12], we add a time-delayed force to the second equation of Lorenz-like system (1), and we have where Regarding the time delay as the bifurcation parameter, when passes through some certain critical values, the equilibrium will lose its stability and Hopf bifurcation will take place. By tuning the feedback effect strength , we can implement the control of chaos phenomena of system.
The remainder of this paper is organized as follows. In Section 2, the local stability and the existence of Hopf bifurcation of the control system (2) are determined. In Section 3, some explicit formulas determining the direction and stability of periodic solutions bifurcating from Hopf bifurcations points are demonstrated by applying the normal form theory and the center manifold theorem [24, 25]. In Section 4, The numerical simulations are carried to demonstrate our theorem analysis. Concluding comments are given in Section 5.
2. Bifurcation Analysis of the Chaotic System
In this section, we will investigate the effect of delay on the dynamic behavior of system (2). More specifically, we study the local stability and the existential conditions of Hopf bifurcation. Obviously, when system (2) becomes the Lorenz-like system (1). System (1) is linearized at the equilibrium to obtain the Jacobian matrix as follows: The associated characteristic equation of (3) is given by When , (4) has three roots Obviously, if , the equilibrium is stable. Therefore, when , system (2) undergoes a pitchfork bifurcation. Else, the equilibrium is unstable.
The Lorenz-like system (2) is symmetric about the -axis, so and have the same stability. It is sufficient to analyze the stability of . By linear transformationsystem (2) will be The Jacobian matrix of (7) at the any point is written as The characteristic equation of (7) is given by Thus, we will study the distribution of the roots of the third degree exponential polynomial equation where
First of all, we will introduce the results of [26].
Lemma 1. Consider the exponential polynomialwhere and are constants. As vary, the sum of the order of the zeros of on the open right half plane can change only if a zero appears on or crosses the imaginary axis.
Obviously, if is a root of (10) if and only if satisfies separating the real and imaginary parts, we have which is equivalent to Let and denote , , and , and then (15) becomes From (16), we have and since , we conclude that if , then (16) has at least one positive root.
From (17) we have
Clearly, if , then the function is monotone increasing in . Thus, when and , (17) has no positive real roots. On the other hand, when and , thenhas two real roots We introduce the following results which were proved by [27].
Lemma 2. For the polynomial (16), we have the following results:
(1) If and , then (16) does not have positive real root.
(2) If and , then (16) has positive real roots if and only if and
(3) If , then (16) has at least one positive root.
Suppose that (16) has positive roots. Without loss of generality, we assume that it has three positive roots, which are denoted , and Then (15) has three positive roots Now from (14), we have Then we can write where and . Then is a pair of purely imaginary roots of (10) when DefineNote that when , (10) becomes which is equivalent to Therefore, applying Lemmas 1 and 2 to (10), we obtain the following lemma.
Lemma 3. For (10) we have
(1) If and , then all roots with positive real parts of (10) have the same sum to those of (24) for all ;
(2) If , then all roots with positive real parts of (10) have the same sum to those of (24) for
Now let us consider the behavior of the roots of (10) near To do this, we assume that is a solution of (10) satisfying Then we have the following transversality condition.
Lemma 4. If , then , , and have the same sign.
Proof. Substituting into (10) and taking the derivative with respect to , we obtain From (14), we obtain where , and since , we can conclude that and have the same sign. This completes the proof.
Theorem 5. Suppose and Routh-Hurwitz criterion for (10) hold and system (2) undergoes Hopf bifurcation at the equilibria when ( and , ). In addition, if , then there exists such that , and the equilibria of system (2) are asymptotically stable for , and they are unstable for ∈, . Furthermore, system (2) undergoes Hopf bifurcation at the equilibria when ( and , ).
3. Direction and Stability of Hopf Bifurcation Period Solution
In previous section, we obtained the conditions when the Hopf bifurcation occurs. In this section, we shall study the direction and the stability of the bifurcations with normal form theory and central manifold theorem. During this section we always assume that system (2) undergoes Hopf bifurcation at the equilibrium for .
Let , , , , and , dropping the bars for simplifications of notations. The nonlinear system (2) can be transformed into an functional differential equation (FDE) in as where and are given, respectively, byand By the Riesz representation theorem, there exists a function of bounded variation for , such that If we choose where is a Dirac delta function, for , define and For convenience, we can write system (29) into an operator equation where
For , define and a bilinear inner product where Obviously and are adjoint operators. By the discussion in Section 2, we know that are eigenvalues of Thus they are also eigenvalues of . We need to calculate the eigenvectors of and corresponding to and , respectively.
Let be the eigenvector of A(0), i.e., and then we have Then it is easy to obtain Similarly, we suppose that is the eigenvector of corresponding to From the definition of , we have Where is a constant such that =1, by (38) we get Therefore, we can choose as In the following, we apply the idea in Hassrd and Kazarinoff and Wan [24] to compute the coordinates describing the center manifold at . Let be the solution of (29) when
Define On the center manifold , we get where and are local coordinates for center manifold in the direction of and . It is clear that is real if is real. Again, we are only considering real solutions. Therefore, to get the solution of of (29), since , We have Let , and then We write this equation as where Since and we have In view of (49), we get Comparing the coefficients with (49), we get where Since and in are unknown, we still need to compute them. By (36) and (44), we drive Let We can rewrite where From (56) and (59) and the definition of , expanding the above series and comparing the coefficients, we getAnd from (56), we know that, for , Comparing the coefficients with (59), we obtain By (60), (62), and the definition of , it follows that Substituting into the last equation, we can obtain the solution of it, which reads where is a constant vector.
Similarly, in view of (60), (63), and the definition of , we get where is a constant vector.
Next, we shall seek appropriate , in (65) (67), respectively, by the definition of and (60), we haveand where By (56), we get and For is the eigenvalues of and is the corresponding eigenvector, we obtain By substituting (65) and (68) into (70), we obtain which leads to It follows thatwhere Similarly, by substituting (67) and (71) into (69), we have and, hence, where
Consequently, we can determine and by (65) and (67). Furthermore, using them we can compute Therefore, all are determined by the parameters and delay in (29). After that, we can easily compute the following values: Thus, we have the main results of this section.
Theorem 6. For (80), we have the following:
(1) The sign of determines the direction of the Hopf bifurcation; if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating period solutions exist for ;
(2) The sign of determines the stability of the bifurcating period solutions; if , then the bifurcation period solutions are orbitally stable (unstable);
(3) The sign of determines the period of the bifurcating periodic solutions; if , then the period increases (decreases).
4. Application to Control of Chaos
In this section, we perform some numerical simulations to verify the results of theoretical analysis in the previous section. Now we consider the following system: which has three equilibrium points , , and .
For the equilibrium point , the characteristic equation of system (81) will be It is clear that (82) will be have a positive root and two negative roots. By Lemma 2 and for all it is clear that (82) has at least one root with positive real part for which means that is always unstable. Furthermore, we conclude that the bifurcating periodic solution is also unstable in the phase space even though they are stable in the center manifold.
For the equilibria and , the characteristic equation of system (81) will be
Clearly, when (83) has a negative root and a pair of complex roots with positive real parts. Following Section 2, we can obtain , , and , for all When , Thus we know that (82) has two roots with positive real parts for all , and the equilibrium points of system (81) is unstable. Therefore, for the purpose of control of chaos, we only consider In particular, we take , that is, In this case, we can compute
In addition, notice that From the arithmetic in Section 3, when , we can compute Thus the equilibrium point is asymptotically stable as is illustrated in Figure 2. When passes through the critical values , loses its stability and a Hopf bifurcation occurs. Since and , the Hopf bifurcation is supercritical and the direction of the Hopf bifurcation is and these bifurcating period solutions from are stable; see Figures 3 and 5. On the other hand, the numerical simulations show that the bifurcating periodic solutions disappear when the delay and chaos occurs again; see Figures 4 and 6.
5. Conclusions
In this paper, we aim to investigate delay feedback control of a new butterfly-shaped Lorenz-like system. Taking the time delay as a bifurcation parameter, we explore the effect of time delay on its dynamics. The local stability and the existence of Hopf bifurcation of the chaotic system is investigated by analyzing the characteristic equations. After that, we also investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, which are determined by applying the normal form theory and the center manifold theorem. Our theoretical results and numerical simulations show that when passes through some certain critical values, the stability of the system will change from unstable to stable, or the stable periodic orbit will be branched out from the equilibrium, which will lead to the chaos phenomenon in the system disappear, and achieve the purpose of controlling chaos.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 61761002) and the Graduate Innovation Project of North Minzu University (no. YCX18091) and the First-Class Disciplines Foundation of Ningxia (Grant no. NXYLXK2017B09).