Hopf bifurcation analysis of the generalized Lorenz system with time delayed feedback control
Abstract
In this work we propose a feedback approach to regulate the chaotic behavior of the whole family of the generalized Lorenz system, by designing a nonlinear delayed feedback control. We first study the effect of the delay on the dynamics of the system and we investigate the existence of Hopf bifurcations. Then, by using the center manifold reduction technique and the normal form theory, we derive the explicit formulas for the direction, stability and period of the periodic solutions bifurcating from the steady state at certain critical values of the delay.
1 Introduction
Because the fact that several dynamical systems exhibit a chaotic behavior, there has been much interest in the study of chaos. In recent years, the trend of analysing the chaos moved to the new phase consisting of its control and utilization: this means on one hand to design suitable controls to eliminate the chaos, and on the other hand to generate it intentionally.
Our goal in this work is to carry out a rigorous mathematical analysis of dynamic behavior of the whole family of the generalized Lorenz system in its chaotic regime by using time delayed feedback controlling forces in the same spirit of [10]. The controller is a nonlinear function of the state variables of the system, therefore the results obtained in this paper can be considered in some way an improvement of the results of [8], where the authors study the generalized Lorenz system with a linear version of the control proposed here. The global dynamics of the system depends on the parameter , which characterizes the particular system of the whole family. In particular, we obtain the Lorenz system for and the Chen system for . Indeed, the systems belonging to the family of the generalized Lorenz system have a similar mathematical structure but they are not topologically equivalent.
In the first part of the paper we introduce the controlled system and we consider the effects of the delay on the steady states also investigating the occurrence of stability switches. Then we show the existence of Hopf bifurcations and so we estabilish that bifurcating periodic solutions exist for each value of the parameter . In the last section of the paper we use the same technique pointed out in [3], based on the center manifold reduction and the normal form theory, in order to determine the direction, stability and period of these periodic solutions which bifurcate from the steady state. This strategy permits to derive the explicit formulas for the properties of the Hopf bifurcation. Moreover, we give numerical simulations of the controlled system, which indicate that when the delay passes through certain critical values, the chaotic behavior is converted to stable periodic orbit for thw whole family of systems.
2 Nonlinear delayed feedback control for the generalized Lorenz system
The generalized Lorenz system is described by the following system of ordinary differential equations for the state variables x ,y, z:
(1) 
where , and it has been introduced for the first time in [9]. By varying the parameter , we obtain different but structurally similar systems: precisely, they have the same kind of equilibria stability, but they are not topologically equivalent. In particular, the system reduces to the Lorenz system for , to the Lu system for and to the Chen system for , in their chaotic regime. The system 1 has the following three equilibrium points:
;
and they are all unstable for all . In the chaotic regime, the system exhibits an irregular dynamics which makes its evolution unpredictable. For this reason, our aim is to design a suitable control which regulates the system behaviour to any given point of the form =, that is the form of the two nontrivial fixed points of the uncontrolled system 1. Namely, by designing the control
,
where
, , , ,
the above system is transformed into the closedloop one:
(2) 
and it can be proved that and both converge to whereas converges to .
Unfortunately, the above proposed control does not take into account that the feedback physically enters into the system at a later time, thus in order to avoid this drawback, we consider the delayed feedback controller
where is the time lag. Since the delay may be destabilizing, our aim is to investigate the stability of the resulting delayed system:
(3) 
The characteristic equation associated with the linearization of system 3 around the point is:
(4) 
where and .
Without any delay , Eq. 4 becomes
(5) 
and, noticing that
and by the RouthHurwitz criterion, Eq. 5 has three roots with negative real part for all , as aspected since the control stabilizes the system [7]. Thus we have the following result.
Theorem 2.1
The equilibrium point of system 1 is globally asymptotically stable when , for all .
Proof. The equilibrium is locally asymptotically stable when because all the roots of the characteristic equation have negative real part. To prove the global stability, we consider a Lyapunov functional given by:
where and , and this functional is positive definite and . In the same way, it’s easy to show that is negative definite . By the Lyapunov stability theorem, we can conclude that the equilibrium point is globally asymptotically stable.
Time delays are known to cause destabilization of equilibria and produce oscillations through Hopf bifurcations. Moreover, it has been observed that further increase in the delay may result in restabilization. This phenomenon is called stability switch: by increasing the delay , it can occur that zeroes of the characteristic equation 4 cross the imaginary axis and the system may change from stable to unstable or vice versa. In order to discuss the existence of such phenomena in system 3, we look at the characteristic equation 4 as a function of the delay and examine the location of roots and the direction of motion as they cross the imaginary axis [1].
It’s crucial to determine the condition to obtain destabilization, that is the critical value at which there is the existence of purely imaginary characteristic values. Indeed, if the roots of 4 are in the lefthalf plane , the equilibrium is asymptotically stable for all . Otherwise, there could be values of for which a pair of complex conjugate roots of 4 crosses the imaginary axis, and if the cross is from left to right, the equilibrium is destabilized, instead if the cross is from right to left an unstable equilibrium is stabilized when increases.
Following the ideas of [1, 2] and making use of Theorem 1 [1], we assume that , , is a root of 4 for some positive and we define the auxiliary function
(6) 
where and have been already defined in 4 and .
It is clear that the existence of purely imaginary eigenvalues for system 3 is equivalent to the existence of positive roots of . If has a positive simple root , then there exists a pair of of purely imaginary eigenvalues with and, for this , we find a sequence of of delays for which stability switches can occur (at least a finite number); furthermore, there exists a positive such that the system is unstable for all .
Note that
and
(7) 

If , then and is monotonically increasing. Therefore, when and , has no positive roots and all the characteristic roots will remain to the left of the imaginary axis for all .

If , since , there is at least one positive root of and the characteristic roots can cross the imaginary axis.
According to Theorem 1 [1], stability switches are possible for each positive root of 6 and the cross is from left to right if , and from right to left is .
The characteristic quasipolynomial 4 for has the form
(8) 
where:
Let , , be a positive root of , and . Then satisfies 8, that is equivalent to the following system:
(9) 
By setting
(10) 
where
after simplification the above system implies that
(11) 
Thus, for each positive root , it yields the following sequence of delays for which there are pure imaginary roots of 4:
(12) 
We numerically find that for all there exist three real roots of 6, of which only two are positive, , and crossing is from left to right with increasing occurs whenever assumes a value corresponding to , as , and so crossing from right to left occurs for values of corresponding to . Moreover, since the zero solution is stable for , then . We observe that
therefore there can be only a finite number of stability switches, if they occur. In our case, we see that for all the smallest value is the critical value at which stability switch occurs from stable to unstable, so that the stability is lost at and for the solution remains unstable.
Remark. By solving system 9 numerically, we find for each value of the critical delay , in particular for , for and for . Actually, as we can see in Fig. 1, is a decreasing function of on the interval .
Therefore, if we consider the trascendental equation 4 as a complex variable mapping problem from the plane to the plane [4]
(13) 
then the critical value is such that has solutions with for . Hence, is the bifurcation value for which . Without any delay, we already showed that all the solutions of have . This implies that, if one considers the contour in the plane consisting of the imaginary axis and of a semicircle of infinite radius, its image under the map given by 13 does not enclose the origin of the plane (see Figs.24(a)). Now, when , if we consider the mapping , as soon as the trasformed curve passes through the origin in the plane this gives the critical value (see Figs.24(b)). For , the mapping is shown in Figs. 24(c) and the origin in the plane is crossed.
Numerical simulations of the ODEs system show that for the solution, after some transient oscillations, stabilizes to the equilibrium position (see Figs.57 (a)).
However, with an increasing delay, solutions exhibit an oscillatory behavior for all , as shown in Figs.57 (b),(c), and this dynamic suggests that the system exhibits Hopf bifurcation, even if there is a qualitative difference due to the fact that the threshold value of delay changes.
We obtain the following result.
Theorem 2.2

If and , then the equilibrium point remains asymptotically stable for all .

If either

, or

, and ,
then there exist and as defined above such that the equilibrium point is asymptotically stable for . Furthermore, if , then the system 3 undergoes a Hopf bifurcation at the equilibrium when .

Proof. It remains to show the transversality condition for the Hopf bifurcation holds at .
We first set
So, differentiating Eq.4 with respective to , we obtain
(14) 
Using Eq.8, we obtain
(15) 
Therefore
(16) 
If , the transversality condition holds and a Hopf bifurcation occurs at .
Remark. If , then . If , then characteristic equation has roots with positive real parts for and close to , but this contradicts the fact that is asimptotically stable for as in Theorem 2.2. Thus, , then .
The expression of as a function of is quite cumbersome, but numerical simulations show that for all (see Fig. 8(a)), and in the same way we show that (see Fig.8(b)). Moreover, it’s easy to prove that and , thus the hypothesis ii is always verified and the system 3 undergoes a Hopf bifurcation at the equilibrium when .
Transversality condition defines the direction of motion of as varies. It is also the necessary condition for the existence of periodic orbits: by varying , the critical characteristic roots cross the imaginary axis with nonzero velocity.
In this section, we have identified the conditions under which delay can destabilize the equilibrium and leads to Hopf bifurcation. We have shown that, when , periodic solutions exist. Now we have to investigate the direction, stability and period of these periodic solutions bifurcating from the steady state.
3 Direction and stability of the Hopf bifurcation
The aim of this section is to derive the explicit formulas determining in particular the direction, stability and period of the periodic solutions bifurcating from the steady state at the critical value , using the normal form and the center manifold theory pointed out in [3] and [6].
Throughout this section, we assume that the system 3 undergoes Hopf bifurcation at the steady state for and that are the corresponding purely imaginary roots of the characteristic equation at the steady state .
We first traslate the equilibrium to the origin through the change of variables , , , and we drop the bars for semplification of notation, thus system 3 is transformed into
(17) 
Let , and we use as the bifurcation parameter with the Hopf bifurcation value. We scale the time t in system 17 and set
We define an operator as
(18) 
and
(19) 
for . So, we can rewrite the system 17 as an FDE in as
(20) 
where . Now, by the Riesz representation theorem, there exists a function of bounded variation for , such that the operator can be represented in an integral form as follows
(21) 
and we can choose
(22) 
where is the Dirac delta function.
The next step is to define operators A and R so that system 20 can be written as an abstract ODE in the Banach space . So, we define for
and
Then system 20 is equivalent to
(23) 
where for .
For , we can define the operator
and the bilinear inner product
(24) 
where . Then and are the adjoint operators with respect to the above bilinear form. By the discussion of the previous section, we know that are eigenvalues of , thus they are also eigenvalues of . We now need to compute the eigenvectors of and corresponding to and , respectively.
Suppose that , is the eigenvector of A(0) corresponding to . Then, . It follows from the definition of and from that
(25) 
We can easily obtain
(26) 
Similarly, let , , is the eigenvector of corresponding to and we find:
(27) 
We can easily obtain
(28) 
The ’orthonormality’ condition , helps us determining the value of . By the definition 24 we have
(29) 
Furthermore, we also have that .
The next step is to compute the coordinates to describe the center manifold at . Let be a solution of Eq.20 when . We define
(30) 
On the center manifold we have , where
(31) 
and and are the coordinates for the center manifold in the direction of and . Since is real if is real, we consider only real solutions. For , since , we have
(32) 
with
(33) 
Noticing , we have
and similarly we have for .
Thus, from Eq.33 we get