DOUBLE HOPF BIFURCATION AND CHAOS IN LIU SYSTEM WITH DELAYED FEEDBACK

In this paper, we consider the stability of equilibria, Hopf and double Hopf bifurcation in Liu system with delay feedback. Firstly, we identify the critical values for stability switches and Hopf bifurcation using the method of bifurcation analysis. When we choose appropriate feedback strength and delay, two symmetrical nontrivial equilibria of Liu system can be controlled to be stable at the same time, and the stable bifurcating periodic solutions occur in the neighborhood of the two equilibria at the same time. Secondly, by applying the normal form method and center manifold theory, the normal form near the double Hopf bifurcation, as well as classifications of local dynamics are analyzed. Furthermore, we give the bifurcation diagram to illustrate numerically that a family of stable periodic solutions bifurcated from Hopf bifurcation occur in a large region of delay and the Liu system with delay can appear the phenomenon of “chaos switchover”.

where a, b, c, h and k are parameters. They discussed some basic dynamical properties of Liu system and numerically studied the continuous spectrum and chaotic behaviors of this new butterfly attractor. From then on, a lot of researches have been done on the Liu system. For example, Matouk [20] used linear feedback control technique to stabilize and synchronize the chaotic Liu system. Zhu and Chen [34] proposed three feedback control strategies of Liu system to its unstable equilibria and compared the effectiveness of these strategies. Xu et al. [32] analyzed the stability of impulsive control Liu system and gave some sufficient conditions which guaranteed the global asymptotical stability for the controlled system.
Delayed feedback control (DFC) method has been receiving considerable attention recently since it was proposed by Pyragas [21,22]. It provides an alternative effective method for feedback control of chaos. The basic idea of DFC is to realize a continuous control for a dynamical system by applying a feedback signal which is proportional to the difference between the dynamical variable X(t) and its delayed value. In other words, we use a perturbation of the form F (t) = K[X(t) − X(t − τ )], where K is feedback strength and τ is time delay. DFC method has been successfully applied to many practical chaotic systems [12][13][14][15][16][17].
In this paper, we consider Liu system with delay as follows using the DFC method: (1.2) There exist Hopf bifurcations and double Hopf bifurcations in system (1.2). Using the normal form theory and center manifold theorem, we obtain the stability of bifurcating periodic solutions and the direction of Hopf bifurcation, and derive the normal forms of double Hopf bifurcation and their unfolding with perturbation parameters. There are the same characteristic equation of linearized system for two nontrivial symmetrical equilibria, therefore, for fixed feedback strength, we obtain the region of delay in which the two unstable equilibria can be controlled to be stable at the same time, and we also obtain the critical values of delay near which stable bifurcation periodic solutions occur at the two equilibria at the same time. Furthermore, we analyze the dynamical behaviors in Liu system (1.2) near the double Hopf bifurcation point. We show the coexistence of a pair of stable periodic solutions, a pair of unstable periodic solutions, a pair of stable quasiperiodic solutions, or a pair of unstable quasi-periodic solutions.
Moreover, our concern is that if the family of stable bifurcating periodic solutions bifurcated from Hopf bifurcation occur in a large region of delay and the controlled system can appear chaos again. By numerical simulation, we show the specific regions in which a family of stable bifurcating periodic solutions occur, and the phenomenon of "chaos switchover" existed in Liu system (1.2).
The rest of the paper is organized as follows. In section 2, we consider the stability of equilibria and bifurcating periodic solutions, and the direction of Hopf bifurcation in Liu system (1.2). In section 3, we derive the normal forms of double Hopf bifurcation and their unfolding with perturbation parameters near the double Hopf bifurcation point.

Stability of equilibria
In this section, system (1.2) is considered. First of all, we determine the equilibria of this system. System (1.2) has three equilibria: We obtain the characteristic equation of linearized system for E 1 as follows: When τ = 0, (2.1) becomes When a, b, c > 0, λ = 0 is not the root of (2.1), and (2.2) have two roots with negative real parts and one root with positive real part, and the equilibrium E 1 is unstable for all τ ≥ 0. We obtain the same characteristic equation of linearized system for E 2 and E 3 as follows: where a 2 = c + a + K, a 1 = ac + cK, a 0 = 2abc. When τ = 0, (2.3) becomes all the roots of equation (2.4) have negative real parts, and the equilibria E 2 and E 3 are stable. Let λ = iω (ω > 0) be a root of (2.3). Applying the analysis results of [4], we give the following assumptions: (H1) a + c ≤ 0 or a 2 c + ac 2 − 2abc ≤ 0; abc = 0; a + c − 2b = 0.

Stability and direction of Hopf bifurcation
In this section, according to the theories of [11,10], we obtain the direction of Hopf bifurcation and the stability of bifurcating periodic solutions on the center manifold. Without loss of generality, we denote the critical value τ = τ * , at which system (1.2) undergoes a Hopf bifurcation at equilibrium (x * , y * , z * ).
Theorem 2.2. If (H2) holds, then system (1.2) undergoes Hopf bifurcations at the equilibria E 2 and E 3 at the same time when τ = τ 1 , the bifurcating periodic solutions are forward (backward), and they are unstable (stable) on the center manifold.
2 , the bifurcating periodic solutions are backward (forward), and they are unstable (stable) on the center manifold.
For the typical parameter set: a = 10, b = 40, c = 2.5, h = 4, k = 1. When K = 15, we can work out Re(C 1 (0)) < 0 at τ = τ 2 ) the bifurcating periodic solutions are backward (forward), and they are stable. Therefore, when we choose appropriate regions of delay, a pair of asymptotically stable equilibria or a pair of stable periodic solutions are coexisted in system (1.2). We give the numerical simulation when τ = 0.38 < τ When τ = 0.38, a periodic solution with small amplitude is bifurcated from E 2 and E 3 respectively. The amplitudes of periodic solutions increase with the delay far away from the critical point τ (0) 1 in the region which the equilibria E 2 and E 3 are unstable. A pair of stable periodic solutions with large amplitudes are coexisted when τ = 0.32 (see Figure 5), and the pair of periodic solutions is reversed phase. We concern the large region of delay in which the family of stable periodic solutions occur. Therefore, we give the bifurcation diagram for equilibrium E 2 when τ ∈ [0, 0.55] and K = 15 (see Figure 6). It is plotted by Matlab software, which depicts the maximum value of periodic solutions in z-axis following the change of time delay. From this figure, we have that stable bifurcating periodic solutions occur at E 2 when τ ∈ (0.22, 0.3882) and τ ∈ (0.4664, 0.47), and the equilibrium E 2 is stable when τ ∈ (0.3882, 0.4664). These numerical results are in accordance with the above theoretical analysis given by Theorem 2.3 and 2.4, and there exists the phenomenon of "chaos switchover" in system (1.2) when τ ∈ [0, 0.55]. Because there are the same characteristic equation of linearized system for E 2 and E 3 , equilibrium E 3 has the same behaviors as E 2 .

Double Hopf bifurcation
From the bifurcation diagram for the feedback strength K and the time delay τ (see Figure 2), we know that there are some points at which two Hopf bifurcation critical curves intersect. For example, (K c , τ c ) = (41.7843, 0.46337) (see point R in Figure  2). From the discussion given in Section 2, when K = K c , τ = τ c , the characteristic equation (2.3) has two pairs of pure imaginary eigenvalues Λ = {±iω 1 , ±iω 2 }, and all the other eigenvalues have negative real parts. We denote that system (1.2) undergoes a double Hopf bifurcation at equilibrium (x * , y * , z * ) when K = K c , τ = τ c . There is only a implicit expression of frequencies h(ω 2 ) = 0 for the Liu system with parameters. Aiming to a group of determinate parameters, we can solve the values of frequencies, here, we only consider the normal form of non-resonant case.
In this section, in order to determine the bifurcation direction and the stability of bifurcating periodic solutions near the double Hopf bifurcation critical point (K c , τ c ), we have to compute the normal forms on the center manifold. The method we use is based on the center manifold reduction and normal form theory due to Faria and Magalhaes [5,6](see Appendix).
In polar coordinates η 1 = r 1 e iθ1 , η 2 = r 2 e iθ2 , the amplitude and phase equations on the center manifold can be derived from (A.8) as where P ij are given by (A.8); where K ε , τ ε are perturbation parameters and d j i (i = 1, 2; j = 1, 2) are given by (A.6).
For (3.1), let ξ j = |Re(P jj )|r j , (j = 1, 2), we have the following planar system: We note that M 0 = (0, 0) is always an equilibrium of (3.2). The two semi-trivial equilibria given by perturbation parameters are which bifurcate from the origin at the bifurcation lines respectively. There may also exist a nontrivial equilibrium For this expression to be valid, we need to assure that ϑδ − 1 = 0. The nontrivial equilibrium M 3 collides with a semi-trivial one and from the positive quadrant on the bifurcation curves )τ ε > 0, the fixed point M 3 is sink, else M 3 is source. Therefore, we need consider the following bifurcation curve.
According to the conclusion of [9], we give the bifurcation diagram (see Figure  7).
Since there exists no unstable manifold containing equilibrium, according to the center manifold theory, (3.1) on the center manifold determine the asymptotic behavior of solutions of the full equations (1.2). Therefore, if (3.2) has one or two asymptotically stable (unstable) semi-trivial equilibria M 1 and M 2 , then (1.2) has one or two asymptotically stable (unstable) periodic solutions in the neighborhood of (x * , y * , z * ). If (3.2) has an asymptotically stable (unstable) equilibrium M 3 , then (1.2) has an asymptotically stable (unstable) quasi-periodic solution in the neighborhood of (x * , y * , z * ). So, we shall call the periodic solution the source (respectively, saddle, sink) periodic solution of (1.2) when the semi-trivial equilibrium of (3.2) is a source (respectively, saddle, sink), and call the quasi-periodic solution the source (respectively, saddle, sink) quasi-periodic solution of (1.2) when the nontrivial equilibrium of (3.2) is a source (respectively, saddle, sink). For the original system (1.2) in the neighborhood of (x * , y * , z * ), the above bifurcation criteria divide the parameters plane (K ε , τ ε ) into seven regions (see Figure  7). In region D 1 , there is only one trivial equilibrium which is saddle; when the parameters vary across the line L 1 from region D 1 to D 2 , the trivial equilibrium becomes a sink point, and an unstable periodic solution O 1 (saddle) appears by Hopf bifurcation from the trivial solution; with the variation of the parameters from region D 2 to D 3 , another sink periodic solution O 2 appears by Hopf bifurcation from the trivial solution; in region D 4 , a sink quasi-periodic solution appears Figure 7. The bifurcation diagram with parameters K ε and τ ε near (K c , τ c ).
by Neimark-Sacker bifurcation from the periodic solution O 2 , and O 2 becomes a saddle from a sink; when the parameters vary across line T 3 from region D 4 to D 5 , the sink quasi-periodic become a source one; in region D 6 ; quasi-periodic solution collides with the periodic solution O 1 and then disappears, and O 1 becomes a source solution; when the parameters vary across line L 1 from region D 6 to D 7 , the periodic solution O 1 collides with the trivial solution and then disappears, and the trivial solution become a source from a saddle; when the parameters vary across line L 2 from region D 7 to D 1 , the saddle periodic solution O 2 collides with the trivial solution and then disappears, and the trivial solution become a saddle from a source.
Because system (1.2) has the same characteristic equation of linearized system for equilibria E 2 and E 3 , there are the same dynamical behaviors in the neighborhood of E 2 and E 3 . Therefore, for system (1.2), there are the coexistence of a pair of unstable periodic solutions in region D 2 and D 7 , the coexistence of a pair of stable periodic solutions and a pair of unstable periodic solutions in region D 3 , the coexistence of two pairs of unstable periodic solutions in region D 6 , the coexistence of a pair of unstable periodic solutions and a pair of stable quasi-periodic solutions in region D 4 , and the coexistence of a pair of unstable periodic solutions and a pair of unstable quasi-periodic solutions in region D 5 .
We give the numerical simulation for E 2 and E 3 when τ ε = −0.0001, K ε = −0.9, τ ε = −0.00337, K ε = 0 and τ ε = −0.001, K ε = 0.52 respectively. The two parameters belong to regions D 2 , D 3 and D 4 respectively (see Figures 8-10), and there exists a pair of stable fixed points, a pair of stable periodic solutions and a pair of stable quasi-periodic solutions respectively. Figure 11 is the bifurcation diagram for equilibrium E 2 when τ ∈ [0, 0.48] and K = K c . From this figure, we know that a family of stable bifurcating periodic solutions occur when τ ∈ (0.1, 0.42), and the inverse period doubling bifurcation   approaches to chaos is found in (1.2). When τ approaches to τ c = 0.46337, the stable Hopf bifurcating periodic solutions disappear, and chaos occurs again with the increase of delay. There does not exist the region in which the equilibrium E 2 is stable, which is different from Figure 6 and is in accordance with Figure 2. Similarly, equilibrium E 3 has the same dynamical behaviors as equilibrium E 2 .

Conclusion
In this paper, we discuss the stability of equilibria and bifurcating periodic solutions, the direction of Hopf bifurcation and dynamical behaviors near the double Hopf bifurcation critical point in Liu system with delay feedback.
We propose a realizable technique to control chaotic Liu system (1.1) to be stable using delayed feedback control method. Namely, if delay τ and feedback strength K satisfy the conditions of Theorems 2.3 and 2.4, then the system (1.1) can be controlled in new states, which have a pair of stable equilibria or a pair of stable bifurcating periodic solutions, and the chaos disappears. Furthermore, we derive the normal forms of double Hopf bifurcation and their unfolding with perturbation parameters, and then give the bifurcation diagram for the perturbation parameters and analyze the dynamical behaviors near the double Hopf bifurcation point. Therefore, with the change of delay and feedback strength, we can obtain the coexistence of a pair of stable periodic solutions, a pair of unstable periodic solutions, two pairs of unstable periodic solutions, a pair of stable quasi-periodic solutions, or a pair of unstable quasi-periodic solutions.
We give the bifurcation diagrams (Figures 6 and 11) to illustrate numerically that a family of stable bifurcating periodic solutions occur in a large region of delay  Figure 11. Bifurcation diagram for equilibrium E 2 when τ ∈ [0, 0.48] and K = K c . and the Liu system can appear the phenomenon of "chaos switchover". It is a helpful trying for studying the complex phenomena of delay differential equation.
Since the time delay is easy to be controlled and realized in real applications, our paper proves once again that DFC method is an effective method to control or generate chaos.
Then we can derive the normal form up to the third ordeṙ and f 1 3 (η, 0, 0) is the function giving the cubic terms in (η, ε, v t ) for ε = 0, v t = 0 defined by the first equation of (A.5), therefore, the normal on the manifold form arising from (A.5) becomes the following form where d j i (i, j = 1, 2) are given by (A.6),