- Email: [email protected]
Steering dynamical trajectories to target a desired state
Chaos.
Sohrons
& Fractals Vol. 4. No. IO. pp. 1931-1942. 1994 Copyright 0 1994 Elsener Science Ltd Printed in Great Britain. All rights reserved 0960Hl779/94$7.00 + [email protected]
0960-0779(94)EOOOS-A
Steering Dynamical Trajectories
to Target a Desired State
DAOLIN XU and STEVEN R. BISHOP Centre
for Nonlinear
Dynamics,
Civil Engineering Building, University London WCIE 6BT, UK
College
London,
Gower
Street,
(Received 20 January 1994)
Abstract-This desired state time H&on stable fixed attractor into
paper describes a method that can direct a chaotic dynamical system to achieve a by adjusting parameters within an optimal control process. The well-known discretesystem is chosen to illustrate the method which can steer a chaotic trajectory towards a point, force a stable fixed point into a chaotic attractor, or switch from one chaotic another chaotic attractor.
1.
INTRODUCTION
Many nonlinear systems have been shown to exhibit chaotic dynamics as well as periodic oscillations [l]. In practice, though, chaos is more usually viewed as unwanted behaviour with the desired state being a stable fixed point or periodic solution. Thus, one task is either to prevent chaotic motions or adapt the system to enable it to be controlled into a desired state. Parameter perturbation control [2], developed by Ott, Grebogi and Yorke (OGY) in Maryland University, has been applied with this control objective in mind. The method utilizes the observation that a chaotic attractor has embedded within it an infinite number of unstable periodic orbits. The authors brilliantly make use of the ergodic nature of chaotic dynamics. This enables the defined neighbourhood at an unstable periodic orbit (or fixed point) embedded in the chaotic attractor to capture the randomly wandering chaotic trajectory. Small, carefully chosen, temporal perturbations to an accessible parameter of the system are carried out so that a stabilizing local feedback control law can play the control role once the chaotic trajectory falls in the pre-designated neighbourhood. Consequently, a chaotic motion can be controlled, leading to a stable periodic motion. Since this early work there have been several clever advancements on the OGY method including a modification proposed by Dressler and Nitsche using time-delay coordinates, leading to better performance [3]. Additionally there have been parameter perturbation methods to a higher-dimensional dynamical system [4, 51. The OGY method has drawn growing attention. and has been employed in various scientific disciplines including fluid mechanics, chemical, electric diode and laser systems [6-lo] (see also related references cited therein). An alternative method [ll] proposed by Chen and Dong adopts conventional feedback control strategies to control a chaotic system towards an unstable equilibrium state that is embedded in a chaotic attractor. By using state-variable feedback, the desired unstable periodic orbit can evolve into a stable periodic orbit so that the chaotic system can be stabilized on this periodic orbit. Due to the local linearization, these methods have some limitations. Firstly, neither methods can direct a chaotic trajectory towards a fixed point which is not embedded within a chaotic attractor. This problem has been partly solved by Schwartz and Triandaf [12, 131. Secondly, the methods cannot 1931
1932
DAOLIN Xu and S. R. BISHOP
control the chaotic system before the chaotic trajectory lies in the small zone around the chosen unstable fixed point, which means that it might take a long time to achieve control. This limitation has been overcome to a great extent by Shinbrot et al. [14-161. In this paper, we present a different parameter-perturbation control method for directing a chaotic trajectory onto an unstable periodic orbit embedded within a chaotic attractor, and stabilizing this unstable periodic orbit. The control process is based on the strategy that the revision of accessible parameters of the system is determined by minimizing the distance function between the current controlled state and the objective state. Furthermore, we will study a class of nonlinear systems which possesses the property that periodic orbits exist in a continuous subspace corresponding to an adjustable parameter space. In this subspace, by setting up sub-targets, we can achieve a distant target such that the system can be directed to any one of the solutions of this system in the subspace. In the following description of the method, for reasons of simplicity, we shall only consider discrete-time systems. For continuous-time systems, the method can be implemented in a similar manner in the dynamics on a Poincare section. 2. THE CONTROL METHOD A dynamical discrete time system can be described in the form Z(i + 1) = F(Z(i),
u)
(1)
where Z E 9.” is an n-dimensional state vector, u E 3”’ is an m-dimensional externally controllable parameter vector, and F is a sufficiently smooth function. We define t,, is the initial time and for t 2 to the system is controlled. The whole control procedure is completed within N time units. A time unit corresponds to one mapping time. Let integer k represent the mapping time of the system under control. Then Z(k) is the kth mapping state of the controlled system and u(k) is the value of controlled parameters in the system in the kth map. The initial system state is Z(t,) = Z(0) and initial parameter vector is u(0). Our purpose is to convert the initial state Z(0) into the target state Z* E Ph.”by means of varying the parameters u step by step from u(0) to u(N). In each control stage, the adjustment of parameters is SU~+~= u(k + 1) - u(k)
(2)
1%+1l ZZc
(3)
and is restricted within the range where E is the maximal allowed adjustment of parameters. As u(k) is altered to u( k + l), the (k + 1)th map of the system (1) under control is described by Z(k
+ 1) = F(Z(k),
Equation (4) can be linearized at the point approximate linear map is given by Z(k
+ 1) = Z(k
u(k + 1)). Z(k)
+ 1) + G,(u(k
where 2 (k + 1) = F( Z (k), u(k)) is the uncontrolled Gk = +‘(Z(k)>
(4)
and the parameters + 1) - u(k))
u(k).
The (5)
map and
u(k))1
is an 12x m matrix evaluated at position Z = Z(k) and parameters (1) is from the class of nonlinear system with linear controllable
u = u(k). If the system parameters u, such as
1933
Steering dynamical trajectories
H&on map, then equation (5) is an exact expression of the equation (4). When the parameters are modified by SUk+l, the trajectory is steered from the point Z(k + 1) to the point Z(k + 1). The goal here is to find a series of typical controllable parameter values u(l), u(2), u(3), * * *, u(N) that force the trajectory from Z(O), Z(l), Z(2), . . ., Z(N) and finally to the objective Z*. If the system is controllable, the controlled trajectory should asymptotically approach the target 2 *. It is necessary to establish a measure that can evaluate if the controlled trajectory is approaching the target state Z*. Accordingly, we define the distance measure J/( = l\Z(k) - z*))* (6) where (1011is the norm of the vector Z(k) - Z *. The optimal principle can be applied to determine the best option for u(k) that results in the minimum distance min Jk. The first optimal control stage is carried out by setting J1 = (z(l)
+ G,(u(l)
- u(0)) - Z*)T(Z(l)
The minimum cost is satisfied by aJ,/&(l) control stage is determined by u(1) = -(Go’GJ’G;@(l)
+ (G,,(u(l)
- u(0)) - Z*).
= 0, and the parameter - Z*) + u(0)
(7)
vector u in the first
if IG$G~[ f 0.
(8)
Investigating the second partial of J1, we see that a*J1
= G;Go > 0.
(9)
&l(l)2 Because the second partials is positive, the feedback absolute or global minimum of J1. The adjustment of the parameters u(2) is u(2) = -(G;G,)-‘G;@(2)
- Z*) + u(1)
and in a similar way, the kth control parameters
(IO)
are given by
u(k) = -(Gk’-lGk-,)-‘G~-l(~(k) The adjustment of parameters
of control defined in (8) yields the
- Z*) + u(k - 1).
(11)
is
C& = -(Gk’-lGk-l)-‘G~-l(~(k)
- Z*).
(12)
If the dimension of a controllable parameter vector is the same as that of a system (Gk_l is an n x n matrix), then equation (12) can be rewritten as Buk = -G,!,@(k) Obviously, &Q is a time-varying feedback Z(N) -+ Z* and SuN + 0, as N + 00.
- Z*). and
if the
(13) system
is controllable,
then
3. SUB-TARGETS
It should be noted that the restriction (3) is not involved in the optimal control process. If the condition (3) is taken into consideration within the control routine (12), then the following constraint results j(G;Gk)-‘G;@(k
+ 1) - Z*)I < E.
(14)
Equation (14) implies that there is a parameter-controllable region around the target point in phase space. If Z(k + 1) is within the region, fluctuations of parameters during the
1934
DA~LIN XU and S. R. BISi*OP
control are smaller than the allowed value in (3). The method works when Z(k + 1) is outside this region, but large changes in parameters may occur. The magnitude of change in parameters depends on Gk and the distance between z(k + 1) and Z*. If it is wished that parameters are to be smoothly adjusted, then it is necessary that the target point Z* should be near point z(k + 1). Note here the choice of an initial control point should be carefully determined. The point should be one at which the next mapping point remains close by. In this way the controlling parameters vary gently. For a class of nonlinear systems which possesses the property that periodic orbits exist in a continuous subspace corresponding to an adjustable parameter space, when Z* is far from the initial point Z(O), we can set up a sequence of sub-targets ZT, ZT, . . ., Zz which are scattered along a path that links the initial start point Z(0) with the final target Z”, = Z*. By using the parameter control (12), we first steer Z(0) to ZT and then control trajectory moving along the designated path and eventually reaching the objective ZT. From the kth sub-target to the (k + 1)th sub-target, the adjustment of parameters is mainly determined by the distance between two sub-targets Zz and Zz+l. Therefore, sub-targets should be set up sufficiently near to each other such that the parameter limiting condition (14) can be satisfied. For a system with linear parameters, any point can be controlled onto a sub-target Zx within two mappings by employing the strategy of equation (12). Let the start point be Z(0) and parameters be u(O), the target point be Zx with parameters U; which are unknown. In the first mapping, the adjustment of parameters is hu, = -(G,;G,,)-‘G:(F(Z(O), Substitute
6u, into (5), if IGiG Z(1)
= F(Z(O),
f 0, the controlled
u(O)) -
u(O))
-
trajectory
G,,(G,TG,,)-‘G,:(F(Z(O)),
Z,*).
(15)
is u(O)) -
Z:)
= Z;.
(16)
Parameters are revised from u(0) to u(1) and trajectory at Z(0) is steered towards Z$, but the following mappings are unstable or even unpredictable if the parameters are fixed at the value of u(0) + 6~~ (i.e. we switch off the control). In the second controlled mapping, the parameter adjustment is buz = -(GfG,))‘G;(F(Z;, If (G:G,I Z(2)
# 0, the controlled = F(Z;,
This last equation
mapping
u(0) + bu,) (18) implies
-
u(0) + SU,) -
Z;).
(17)
is
G,(G:G,)-‘G:(F(Z;,
u(O) + iiu,)
-
Z;)
= Z:.
(18)
that:
(i) The sub-target Zt is a fixed point and parameter u = [4(O) + hu, + Ouz is the parameter ut relevant to the fixed point Z 2. This indicates that the target point should be a fixed point. A sub-target path is set on a sequence of fixed points in phase space. (ii) Any point, except points that lead to lGrGl = 0, can be brought onto a fixed point within two mappings. This suggests that control process in linear parameter systems is stable. The control speed depends on the number of sub-targets. (iii) Equation (12) can automatically find the value of parameters of a fixed point (desired state). This means that we can also control a system when we do not know the parameters corresponding to the target state. Meanwhile equation (12) shows, in each control stage, that the adjustment of parameters follows a certain rule. In fact, simply and directly switching the parameters onto the parameter value of the target state (if we know the value) does not guarantee that the trajectory will land onto the target state.
Steering dynamical trajectories
1935
For systems with nonlinear parameters, the stability of the control method becomes more complex, but if perturbations of the parameters are sufficiently small in the sense that approximation (5) of system (1) can be satisfied, the control may still be stable. Speed of control is normally slower than that for linear parameter systems. However, there is a limitation that the parameter control method cannot be applied when jGr Gl = 0, termed a singularity, on which the adjustment of the parameters becomes infinite. The singularity may appear in the form of a point, a line or a region. For singular points, sub-targets can be applied to establish a path far from singular points. This scenario is schematically illustrated in Fig. 1. The ‘+’ sign denotes the singular point and the ‘0’ signs the sub-target points. The sub-target path, shown in the Fig. 1, can be used to avoid singular points and direct a trajectory towards the final target without losing control. One exciting application is that a periodic orbit can be created by a suitable choice of sub-target path. In Fig. 1, the sub-target path is looped over itself and the final target Z*, is superimposed on ZE_/r. Thus a loop is formed from ZTPp to Z*, (in Fig. 1, /3 = 6). If the trajectory is controlled continuously then it will go around the sub-target loop generating a periodic orbit. For a system with linear parameters, this orbit will be 2/3-periodic.
4. NUMERICAL
EXAMPLE
To illustrate the method we choose the well-known H&on system. H&on map is a nonlinear system with linear parameters defined by X(k
+ 1) = 1 - ax(k)2
Y(k
+ 1) = bX(k)
where (X, Y) E CR x % and (a, 6) are real controllable map are determined by x
=
b -
1 + j/[(b - 1)2 + 4a]
The two-dimensional
+ Y(k) (19) parameters.
3
The fixed points
Y = bX.
of the
(20)
By continuously varying parameters a and b in equation (20), there exists a continuous subspace of fixed points in the phase space. This means that sub-targets can be set up anywhere in the space. The fixed points may be stable or unstable depending upon the
+ /’ Singular point
% Start
point
Fig. 1,
DAOLIN Xv and S. R. BISHOP
1936
eigenvalues of the characteristic
equation -2aX -A b
1
-A
=O
(21)
+ b].
(22)
where il is the eigenvalue given by &
= -aX
f 2/[(aX)’
The fixed point is stable only if (23) According to equations (20), (22) and (23), stable fixed points exist provided that a<
$1 - b)’
b < 1.
and
(24)
In the H&on system
q-f2
;I.
(25)
Thus there is a singular line along X = 0. We next investigate system. 4.1.
the control of the H&on
Directing a chaotic trajectory towards a stable fixed point
For a = 1.4, b = 0.3, the Henon map undergoes chaotic motion as shown in Fig 2. According to equations (20), (22), (23) and (24) the location of a stable fixed point in the phase space is known and our aim here is to steer the chaotic trajectory onto a stable fixed point. We choose the stable fixed point located at (0.9274, 0.1855). To avoid a sub-target path crossing the singular line (see the broken line in Fig. 2), we pick the initial control point whose value of X(0) is greater than zero. As shown in Fig. 2, any of the points marked ‘0’) can be selected as the initial control point since these points also possess the property that the next mapping is close by. Here the 26th mapping the point (0.6099, 0.0926) is picked up as the initial control point. Figure 3 shows the chaotic attractor at parameter values of a = 1.4 and b = 0.3 and the line between the initial point (marked ‘0’ and the target point (marked ‘*‘) indicates the sub-target path. During the whole control process, the initial point is directed to the sub-target path and the controlled trajectory moves along the path gradually approaching
40 Mapping times
Fig. 2
60
80
Steering dynamical trajectories 0.4 *
1937
-. *,.y.__
0.3 . 0.2 r J
0.1.
P .$ 5 3 rz
O-0.1 -0.2
-0.41 -1.5
_-1.0
-0.5
0
0.5
State variable
X
1.0
i 1.5
Fig. 3.
the target. Figure 4 illustrates the evolution of the trajectory from a chaotic state to a stable equilibrium state. After the 26th mapping, the chaotic trajectory is controlled and oscillates around the sub-target path. When the controlled trajectory is brought to the position of the stable fixed point, the parameters are synchronously revised towards the value of parameters with respect to the fixed point. Since the stable fixed point is stable, the trajectory remains on the equilibrium position even if the control is switched off. Figure 5 demonstrates the variation of parameters throughout the control process. The final revised parameters in Fig. 5 are automatically detected to be the value of a* = 0.3 and b* = 0.2; the value at the stable fixed point (0.6099, 0.0926). It should be noted that Fig. 5 only indicates one of way of directing a chaotic orbit to a periodic orbit. The choice of initial start points and sub-target path greatly affects the parameter adjustment. 4.2.
Forcing a stable fixed point towards chaos
If we now assume that the H&on system is operating in an equilibrium state, then we may select this point as an initial point and convert the system into a chaotic trajectory. However, we can not directly control a trajectory to be a chaotic one since the target point must be a fixed point. Thus, we must first locate a fixed point that can be easy to lead to chaos. As we know, there exist such unstable fixed points which are embedded within a chaotic attractor and have the same parameter value as the chaotic attractor. When these unstable fixed points are disturbed, they will exponentially diverge away from their initial 1.5 Chaotic
-1.5 ’ 0
10
trajectory
20
30
40
Mappingtimes Fig. 4.
50
60
I 70
1938
DAOLIN Xu and S. R. BISHOP
0-
25
30
35
40
45
Mapping
50
55
60
times
Fig. 5.
position and enter the chaotic attractor. Obviously, these fixed points can be viewed as windows from which the system can be sent into chaotic state. Thus, we select one of the unstable fixed points embedded within the chaotic attractor as the target point. After the target point is achieved, it is easy to ‘kick’ the unstable fixed point into chaotic attractor by a small perturbation. As seen in the Fig. 6, the start point chosen is at the position (0.9274, 0.1855) with parameters a = 0.3, b = 0.2. The target point selected is (0.6314, 0.1894) an unstable fixed point embedded in the chaotic attractor. The line between the start point and the target point is the sub-target path. Figure 7 shows the process of the transition from the equilibrium state to the chaotic state. The controlled path is a zigzag-like route. When the controlled trajectory is locked onto the target point, then a small disturbance is employed The history of adjustment of the and the system is kicked into the chaotic motion. parameters is shown in the Fig. 8. The initial parameter values are a = 0.3, b = 0.2, and the final parameters are searched at values of a* = 1.4 and b* = 0.3.
4.3.
Switching
between
IWO chaotic attractors
In the H&on system, we can choose two chaotic attractors and switch from one to another. We denote the A chaotic attractor (shown in Fig. 9) to be with respect to the parameters a = 1.4 and b = 0.3, while the other, with parameters a* = 1.07 and b* = 0.3, is called the B chaotic attractor (see Fig. 10). We select a start point in the A chaotic
._.
0.2 0.1
t
_.-.-
-1.0
0
4.5
State variable
Fig. 6
0.5 X
1.0
I
1.5
Steering
dynamical
1939
trajectories
chaotic 40
60
100
80
trajectory I 120
140
55
60
Mapping times Fig. 7.
1.4 1
1.2
I
0.4 1
,
’
-b
_,’ 0.2
y
25
30
35
40
50
45
Mapping times Fig. 8.
0.4 -
0.3 0.2 h P
..:-.
-.-
.___ ..
. .--. --.
:--_
I
:. .-. _
0.1
” . . .. . ‘. ‘%,_ *. *.
2 ? S S v)
OI -0.1 -0.2
:
:.
.! *. I :
I
-0.3
I
-0.4
1 -1.5
_ ._..-1.0
-0.5
0
r
..-
0.5
1.0
1 1. 5
State variable X
Fig. 9.
attractor and the target point is set as an unstable fixed point embedded in the B chaotic attractor. When the controlled trajectory from the A chaotic attractor lands on the target, we use the perturbation technique to enable the target to develop into the B chaotic attractor. The transition is shown in Fig. 11. Before the control is activated, the system wanders
1940
DAOLINXu and S. R. 0.4
BISHOP
_
I
0.31
‘\_,
0.2 * 0 .B 2B ;
1 t
0.1 I 0
9
1
I
-0.1 m
-0.2t -0.3’ -1.0
)
/’
: I
-0.5
0
0.5
State variables
1.0
1.5
X
Fig. 10.
40 60 Mapping times Fig. 11.
attractor. At the 44th mapping, the point at the position of (0.6219, attractor is picked up as the start point. The controlled trajectory is forced to move along the zigzag-like path between the start point (S.P.) and the target point (T.P.), and finally hits at the target point (0.6915, 0.2075) which is embedded within the B chaotic attractor. Then, in a similar way as before, the system can be pushed into the B chaos from this window (i.e. the unstable fixed point). Figure 12 shows the modification of parameters for the control process. within
the A chaotic
0.1281) in the A chaotic
5. CONCLUSIONS
In this paper, we develop a parameter-control method for a discrete-time chaotic system. In a numerical example the special case of two accessible linear parameters in the H&on system is considered. A sequence of simulation results of the control procedure is shown to illustrate the transition between chaotic states and stable equilibrium states. The investigation clearly indicates that the unstable fixed points that are embedded within chaotic attractors can become windows from which a system can be kicked into chaos. The numerical experiments presented in the paper illustrate only some of the possible applications of the method. The method can automatically detect the parameters of a fixed or periodic state in controllable phase-space. Additionally the method can create new
Steering
i P
dynamical
trajectories
1941
’ ---______ -
1.2 t I
--____
--___
pamm~ter a 1.0 /
b
P-eter 48
46
Mapping
50
-~
times
Fig. 12
multi-periodic orbits. For nonlinear systems with linear parameters, the control process is stable. To steer one point to another, the control process is completed within only two mappings. However, there are some limitations of the method. Namely, some points cannot be controlled because of inherent singularities. Another is that the initial control point should be a point at which the next mapping point remains close, such that the parameter-limiting condition is satisfied. These two limitations will both restrict the option for initial control points. Nevertheless, there are still many points, as shown, that can be conveniently selected as an initial control point. The method is simple and efficient in controlling chaotic dynamical systems such as the Henon system. Further work will continue to examine the behaviour when the parameters of systems are nonlinear. Acknowledgements-The author D. L. Xu is supported by Sit-to-British Friendship Scholarship Scheme and would like to note his appreciation to the British Council. S. R. Bishop is supported by the Science and Engineering Research Council of the UK. In addition, the authors are grateful to the reviewers for their comments.
REFERENCES 1. .I. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. Springer, Berlin (1983). 2. E. Ott, C. Grebogi and J. A. Yorke, Controlling chaos, Phys. Rev. Lett. 64, 1196-1199 (1990). 3. G. Nitsche and U. Dressler, Controlling chaotic dynamical systems using time delay coordinates, Physica D58, 153-164 (1992). 4. F. J. Romeiras, C. Grebogi, E. Ott and W. P. Dayawansa, Controlling chaotic dynamical systems, Physica DSS, 165-192 (1992). 5. D. Auerbach, C. Grebogi, E. Ott and J. A. Yorke. Controlling chaos in high-dimensional systems, Phys. Rev. Lett. 69, 3479-3482 (1992). 6. W. L. Ditto, S. N. Rauseo and M. L. Spano, Experimental control of chaos, Phys. Rev. Left. 65, 3211-3214 (1990). 7. E. Ott, C. Grebogi and J. A. Yorke, in Chaos/XAOC Soviet-American Perspectives on Nonlinear Science, edited by D. K. Campbell, pp. 153-172. American Institute of Physics, New York 1990. 8. Y. C. Lai, M. 2. Ding and C. Grebogi, Controlling Hamiltonian chaos, Phys. Rev. E47, 86-92 (1993). 9. Y. C. Lai and C. Grebogi, Synchronization of chaotic trajectories using control, Phys. Rev. E47, 2357-2360 (1993). 10. Y. C. Lai, T. Tel and C. Grebogi, Stabilizing chaotic-scattering trajectories using control, Phys. Rev. E48, 709-717 (1993). 11. G. R. Chen and X. N. Dong, On feedback control of chaotic nonlinear dynamic systems, Int. J. Bifurcation & Chaos, 2, 407-411 (1992). 12. I. B. Schwartz and I. Triandaf, Tracking unstable orbits in experiments, Phys. Rev. A46, 7439-7444 (1992). 13. 2. Gills, C. Iwata, R. Roy, I. B. Schwartz and I. Triandaf, Extending the stability regime of a multimode laser system, Phys. Rev. Lett. 69, 3169-3172 (1992).
1942
DAOLIN Xu and S. R. BISHOP
14. T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, Using chaos to direct trajectories to targets, Phys. Rev. Left. 65, 3215-3218 (1990). 15. T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, Using chaos to target stationary states of flows, Phys. Left. A169, 349-354 (1992). 16. E. J. Kostelich, C. Grebogi, E. Ott and J. A. Yorke, Higher-dimensional targeting, Phys. Rev. E47, 305-310 (1993).
Sohrons
& Fractals Vol. 4. No. IO. pp. 1931-1942. 1994 Copyright 0 1994 Elsener Science Ltd Printed in Great Britain. All rights reserved 0960Hl779/94$7.00 + [email protected]
0960-0779(94)EOOOS-A
Steering Dynamical Trajectories
to Target a Desired State
DAOLIN XU and STEVEN R. BISHOP Centre
for Nonlinear
Dynamics,
Civil Engineering Building, University London WCIE 6BT, UK
College
London,
Gower
Street,
(Received 20 January 1994)
Abstract-This desired state time H&on stable fixed attractor into
paper describes a method that can direct a chaotic dynamical system to achieve a by adjusting parameters within an optimal control process. The well-known discretesystem is chosen to illustrate the method which can steer a chaotic trajectory towards a point, force a stable fixed point into a chaotic attractor, or switch from one chaotic another chaotic attractor.
1.
INTRODUCTION
Many nonlinear systems have been shown to exhibit chaotic dynamics as well as periodic oscillations [l]. In practice, though, chaos is more usually viewed as unwanted behaviour with the desired state being a stable fixed point or periodic solution. Thus, one task is either to prevent chaotic motions or adapt the system to enable it to be controlled into a desired state. Parameter perturbation control [2], developed by Ott, Grebogi and Yorke (OGY) in Maryland University, has been applied with this control objective in mind. The method utilizes the observation that a chaotic attractor has embedded within it an infinite number of unstable periodic orbits. The authors brilliantly make use of the ergodic nature of chaotic dynamics. This enables the defined neighbourhood at an unstable periodic orbit (or fixed point) embedded in the chaotic attractor to capture the randomly wandering chaotic trajectory. Small, carefully chosen, temporal perturbations to an accessible parameter of the system are carried out so that a stabilizing local feedback control law can play the control role once the chaotic trajectory falls in the pre-designated neighbourhood. Consequently, a chaotic motion can be controlled, leading to a stable periodic motion. Since this early work there have been several clever advancements on the OGY method including a modification proposed by Dressler and Nitsche using time-delay coordinates, leading to better performance [3]. Additionally there have been parameter perturbation methods to a higher-dimensional dynamical system [4, 51. The OGY method has drawn growing attention. and has been employed in various scientific disciplines including fluid mechanics, chemical, electric diode and laser systems [6-lo] (see also related references cited therein). An alternative method [ll] proposed by Chen and Dong adopts conventional feedback control strategies to control a chaotic system towards an unstable equilibrium state that is embedded in a chaotic attractor. By using state-variable feedback, the desired unstable periodic orbit can evolve into a stable periodic orbit so that the chaotic system can be stabilized on this periodic orbit. Due to the local linearization, these methods have some limitations. Firstly, neither methods can direct a chaotic trajectory towards a fixed point which is not embedded within a chaotic attractor. This problem has been partly solved by Schwartz and Triandaf [12, 131. Secondly, the methods cannot 1931
1932
DAOLIN Xu and S. R. BISHOP
control the chaotic system before the chaotic trajectory lies in the small zone around the chosen unstable fixed point, which means that it might take a long time to achieve control. This limitation has been overcome to a great extent by Shinbrot et al. [14-161. In this paper, we present a different parameter-perturbation control method for directing a chaotic trajectory onto an unstable periodic orbit embedded within a chaotic attractor, and stabilizing this unstable periodic orbit. The control process is based on the strategy that the revision of accessible parameters of the system is determined by minimizing the distance function between the current controlled state and the objective state. Furthermore, we will study a class of nonlinear systems which possesses the property that periodic orbits exist in a continuous subspace corresponding to an adjustable parameter space. In this subspace, by setting up sub-targets, we can achieve a distant target such that the system can be directed to any one of the solutions of this system in the subspace. In the following description of the method, for reasons of simplicity, we shall only consider discrete-time systems. For continuous-time systems, the method can be implemented in a similar manner in the dynamics on a Poincare section. 2. THE CONTROL METHOD A dynamical discrete time system can be described in the form Z(i + 1) = F(Z(i),
u)
(1)
where Z E 9.” is an n-dimensional state vector, u E 3”’ is an m-dimensional externally controllable parameter vector, and F is a sufficiently smooth function. We define t,, is the initial time and for t 2 to the system is controlled. The whole control procedure is completed within N time units. A time unit corresponds to one mapping time. Let integer k represent the mapping time of the system under control. Then Z(k) is the kth mapping state of the controlled system and u(k) is the value of controlled parameters in the system in the kth map. The initial system state is Z(t,) = Z(0) and initial parameter vector is u(0). Our purpose is to convert the initial state Z(0) into the target state Z* E Ph.”by means of varying the parameters u step by step from u(0) to u(N). In each control stage, the adjustment of parameters is SU~+~= u(k + 1) - u(k)
(2)
1%+1l ZZc
(3)
and is restricted within the range where E is the maximal allowed adjustment of parameters. As u(k) is altered to u( k + l), the (k + 1)th map of the system (1) under control is described by Z(k
+ 1) = F(Z(k),
Equation (4) can be linearized at the point approximate linear map is given by Z(k
+ 1) = Z(k
u(k + 1)). Z(k)
+ 1) + G,(u(k
where 2 (k + 1) = F( Z (k), u(k)) is the uncontrolled Gk = +‘(Z(k)>
(4)
and the parameters + 1) - u(k))
u(k).
The (5)
map and
u(k))1
is an 12x m matrix evaluated at position Z = Z(k) and parameters (1) is from the class of nonlinear system with linear controllable
u = u(k). If the system parameters u, such as
1933
Steering dynamical trajectories
H&on map, then equation (5) is an exact expression of the equation (4). When the parameters are modified by SUk+l, the trajectory is steered from the point Z(k + 1) to the point Z(k + 1). The goal here is to find a series of typical controllable parameter values u(l), u(2), u(3), * * *, u(N) that force the trajectory from Z(O), Z(l), Z(2), . . ., Z(N) and finally to the objective Z*. If the system is controllable, the controlled trajectory should asymptotically approach the target 2 *. It is necessary to establish a measure that can evaluate if the controlled trajectory is approaching the target state Z*. Accordingly, we define the distance measure J/( = l\Z(k) - z*))* (6) where (1011is the norm of the vector Z(k) - Z *. The optimal principle can be applied to determine the best option for u(k) that results in the minimum distance min Jk. The first optimal control stage is carried out by setting J1 = (z(l)
+ G,(u(l)
- u(0)) - Z*)T(Z(l)
The minimum cost is satisfied by aJ,/&(l) control stage is determined by u(1) = -(Go’GJ’G;@(l)
+ (G,,(u(l)
- u(0)) - Z*).
= 0, and the parameter - Z*) + u(0)
(7)
vector u in the first
if IG$G~[ f 0.
(8)
Investigating the second partial of J1, we see that a*J1
= G;Go > 0.
(9)
&l(l)2 Because the second partials is positive, the feedback absolute or global minimum of J1. The adjustment of the parameters u(2) is u(2) = -(G;G,)-‘G;@(2)
- Z*) + u(1)
and in a similar way, the kth control parameters
(IO)
are given by
u(k) = -(Gk’-lGk-,)-‘G~-l(~(k) The adjustment of parameters
of control defined in (8) yields the
- Z*) + u(k - 1).
(11)
is
C& = -(Gk’-lGk-l)-‘G~-l(~(k)
- Z*).
(12)
If the dimension of a controllable parameter vector is the same as that of a system (Gk_l is an n x n matrix), then equation (12) can be rewritten as Buk = -G,!,@(k) Obviously, &Q is a time-varying feedback Z(N) -+ Z* and SuN + 0, as N + 00.
- Z*). and
if the
(13) system
is controllable,
then
3. SUB-TARGETS
It should be noted that the restriction (3) is not involved in the optimal control process. If the condition (3) is taken into consideration within the control routine (12), then the following constraint results j(G;Gk)-‘G;@(k
+ 1) - Z*)I < E.
(14)
Equation (14) implies that there is a parameter-controllable region around the target point in phase space. If Z(k + 1) is within the region, fluctuations of parameters during the
1934
DA~LIN XU and S. R. BISi*OP
control are smaller than the allowed value in (3). The method works when Z(k + 1) is outside this region, but large changes in parameters may occur. The magnitude of change in parameters depends on Gk and the distance between z(k + 1) and Z*. If it is wished that parameters are to be smoothly adjusted, then it is necessary that the target point Z* should be near point z(k + 1). Note here the choice of an initial control point should be carefully determined. The point should be one at which the next mapping point remains close by. In this way the controlling parameters vary gently. For a class of nonlinear systems which possesses the property that periodic orbits exist in a continuous subspace corresponding to an adjustable parameter space, when Z* is far from the initial point Z(O), we can set up a sequence of sub-targets ZT, ZT, . . ., Zz which are scattered along a path that links the initial start point Z(0) with the final target Z”, = Z*. By using the parameter control (12), we first steer Z(0) to ZT and then control trajectory moving along the designated path and eventually reaching the objective ZT. From the kth sub-target to the (k + 1)th sub-target, the adjustment of parameters is mainly determined by the distance between two sub-targets Zz and Zz+l. Therefore, sub-targets should be set up sufficiently near to each other such that the parameter limiting condition (14) can be satisfied. For a system with linear parameters, any point can be controlled onto a sub-target Zx within two mappings by employing the strategy of equation (12). Let the start point be Z(0) and parameters be u(O), the target point be Zx with parameters U; which are unknown. In the first mapping, the adjustment of parameters is hu, = -(G,;G,,)-‘G:(F(Z(O), Substitute
6u, into (5), if IGiG Z(1)
= F(Z(O),
f 0, the controlled
u(O)) -
u(O))
-
trajectory
G,,(G,TG,,)-‘G,:(F(Z(O)),
Z,*).
(15)
is u(O)) -
Z:)
= Z;.
(16)
Parameters are revised from u(0) to u(1) and trajectory at Z(0) is steered towards Z$, but the following mappings are unstable or even unpredictable if the parameters are fixed at the value of u(0) + 6~~ (i.e. we switch off the control). In the second controlled mapping, the parameter adjustment is buz = -(GfG,))‘G;(F(Z;, If (G:G,I Z(2)
# 0, the controlled = F(Z;,
This last equation
mapping
u(0) + bu,) (18) implies
-
u(0) + SU,) -
Z;).
(17)
is
G,(G:G,)-‘G:(F(Z;,
u(O) + iiu,)
-
Z;)
= Z:.
(18)
that:
(i) The sub-target Zt is a fixed point and parameter u = [4(O) + hu, + Ouz is the parameter ut relevant to the fixed point Z 2. This indicates that the target point should be a fixed point. A sub-target path is set on a sequence of fixed points in phase space. (ii) Any point, except points that lead to lGrGl = 0, can be brought onto a fixed point within two mappings. This suggests that control process in linear parameter systems is stable. The control speed depends on the number of sub-targets. (iii) Equation (12) can automatically find the value of parameters of a fixed point (desired state). This means that we can also control a system when we do not know the parameters corresponding to the target state. Meanwhile equation (12) shows, in each control stage, that the adjustment of parameters follows a certain rule. In fact, simply and directly switching the parameters onto the parameter value of the target state (if we know the value) does not guarantee that the trajectory will land onto the target state.
Steering dynamical trajectories
1935
For systems with nonlinear parameters, the stability of the control method becomes more complex, but if perturbations of the parameters are sufficiently small in the sense that approximation (5) of system (1) can be satisfied, the control may still be stable. Speed of control is normally slower than that for linear parameter systems. However, there is a limitation that the parameter control method cannot be applied when jGr Gl = 0, termed a singularity, on which the adjustment of the parameters becomes infinite. The singularity may appear in the form of a point, a line or a region. For singular points, sub-targets can be applied to establish a path far from singular points. This scenario is schematically illustrated in Fig. 1. The ‘+’ sign denotes the singular point and the ‘0’ signs the sub-target points. The sub-target path, shown in the Fig. 1, can be used to avoid singular points and direct a trajectory towards the final target without losing control. One exciting application is that a periodic orbit can be created by a suitable choice of sub-target path. In Fig. 1, the sub-target path is looped over itself and the final target Z*, is superimposed on ZE_/r. Thus a loop is formed from ZTPp to Z*, (in Fig. 1, /3 = 6). If the trajectory is controlled continuously then it will go around the sub-target loop generating a periodic orbit. For a system with linear parameters, this orbit will be 2/3-periodic.
4. NUMERICAL
EXAMPLE
To illustrate the method we choose the well-known H&on system. H&on map is a nonlinear system with linear parameters defined by X(k
+ 1) = 1 - ax(k)2
Y(k
+ 1) = bX(k)
where (X, Y) E CR x % and (a, 6) are real controllable map are determined by x
=
b -
1 + j/[(b - 1)2 + 4a]
The two-dimensional
+ Y(k) (19) parameters.
3
The fixed points
Y = bX.
of the
(20)
By continuously varying parameters a and b in equation (20), there exists a continuous subspace of fixed points in the phase space. This means that sub-targets can be set up anywhere in the space. The fixed points may be stable or unstable depending upon the
+ /’ Singular point
% Start
point
Fig. 1,
DAOLIN Xv and S. R. BISHOP
1936
eigenvalues of the characteristic
equation -2aX -A b
1
-A
=O
(21)
+ b].
(22)
where il is the eigenvalue given by &
= -aX
f 2/[(aX)’
The fixed point is stable only if (23) According to equations (20), (22) and (23), stable fixed points exist provided that a<
$1 - b)’
b < 1.
and
(24)
In the H&on system
q-f2
;I.
(25)
Thus there is a singular line along X = 0. We next investigate system. 4.1.
the control of the H&on
Directing a chaotic trajectory towards a stable fixed point
For a = 1.4, b = 0.3, the Henon map undergoes chaotic motion as shown in Fig 2. According to equations (20), (22), (23) and (24) the location of a stable fixed point in the phase space is known and our aim here is to steer the chaotic trajectory onto a stable fixed point. We choose the stable fixed point located at (0.9274, 0.1855). To avoid a sub-target path crossing the singular line (see the broken line in Fig. 2), we pick the initial control point whose value of X(0) is greater than zero. As shown in Fig. 2, any of the points marked ‘0’) can be selected as the initial control point since these points also possess the property that the next mapping is close by. Here the 26th mapping the point (0.6099, 0.0926) is picked up as the initial control point. Figure 3 shows the chaotic attractor at parameter values of a = 1.4 and b = 0.3 and the line between the initial point (marked ‘0’ and the target point (marked ‘*‘) indicates the sub-target path. During the whole control process, the initial point is directed to the sub-target path and the controlled trajectory moves along the path gradually approaching
40 Mapping times
Fig. 2
60
80
Steering dynamical trajectories 0.4 *
1937
-. *,.y.__
0.3 . 0.2 r J
0.1.
P .$ 5 3 rz
O-0.1 -0.2
-0.41 -1.5
_-1.0
-0.5
0
0.5
State variable
X
1.0
i 1.5
Fig. 3.
the target. Figure 4 illustrates the evolution of the trajectory from a chaotic state to a stable equilibrium state. After the 26th mapping, the chaotic trajectory is controlled and oscillates around the sub-target path. When the controlled trajectory is brought to the position of the stable fixed point, the parameters are synchronously revised towards the value of parameters with respect to the fixed point. Since the stable fixed point is stable, the trajectory remains on the equilibrium position even if the control is switched off. Figure 5 demonstrates the variation of parameters throughout the control process. The final revised parameters in Fig. 5 are automatically detected to be the value of a* = 0.3 and b* = 0.2; the value at the stable fixed point (0.6099, 0.0926). It should be noted that Fig. 5 only indicates one of way of directing a chaotic orbit to a periodic orbit. The choice of initial start points and sub-target path greatly affects the parameter adjustment. 4.2.
Forcing a stable fixed point towards chaos
If we now assume that the H&on system is operating in an equilibrium state, then we may select this point as an initial point and convert the system into a chaotic trajectory. However, we can not directly control a trajectory to be a chaotic one since the target point must be a fixed point. Thus, we must first locate a fixed point that can be easy to lead to chaos. As we know, there exist such unstable fixed points which are embedded within a chaotic attractor and have the same parameter value as the chaotic attractor. When these unstable fixed points are disturbed, they will exponentially diverge away from their initial 1.5 Chaotic
-1.5 ’ 0
10
trajectory
20
30
40
Mappingtimes Fig. 4.
50
60
I 70
1938
DAOLIN Xu and S. R. BISHOP
0-
25
30
35
40
45
Mapping
50
55
60
times
Fig. 5.
position and enter the chaotic attractor. Obviously, these fixed points can be viewed as windows from which the system can be sent into chaotic state. Thus, we select one of the unstable fixed points embedded within the chaotic attractor as the target point. After the target point is achieved, it is easy to ‘kick’ the unstable fixed point into chaotic attractor by a small perturbation. As seen in the Fig. 6, the start point chosen is at the position (0.9274, 0.1855) with parameters a = 0.3, b = 0.2. The target point selected is (0.6314, 0.1894) an unstable fixed point embedded in the chaotic attractor. The line between the start point and the target point is the sub-target path. Figure 7 shows the process of the transition from the equilibrium state to the chaotic state. The controlled path is a zigzag-like route. When the controlled trajectory is locked onto the target point, then a small disturbance is employed The history of adjustment of the and the system is kicked into the chaotic motion. parameters is shown in the Fig. 8. The initial parameter values are a = 0.3, b = 0.2, and the final parameters are searched at values of a* = 1.4 and b* = 0.3.
4.3.
Switching
between
IWO chaotic attractors
In the H&on system, we can choose two chaotic attractors and switch from one to another. We denote the A chaotic attractor (shown in Fig. 9) to be with respect to the parameters a = 1.4 and b = 0.3, while the other, with parameters a* = 1.07 and b* = 0.3, is called the B chaotic attractor (see Fig. 10). We select a start point in the A chaotic
._.
0.2 0.1
t
_.-.-
-1.0
0
4.5
State variable
Fig. 6
0.5 X
1.0
I
1.5
Steering
dynamical
1939
trajectories
chaotic 40
60
100
80
trajectory I 120
140
55
60
Mapping times Fig. 7.
1.4 1
1.2
I
0.4 1
,
’
-b
_,’ 0.2
y
25
30
35
40
50
45
Mapping times Fig. 8.
0.4 -
0.3 0.2 h P
..:-.
-.-
.___ ..
. .--. --.
:--_
I
:. .-. _
0.1
” . . .. . ‘. ‘%,_ *. *.
2 ? S S v)
OI -0.1 -0.2
:
:.
.! *. I :
I
-0.3
I
-0.4
1 -1.5
_ ._..-1.0
-0.5
0
r
..-
0.5
1.0
1 1. 5
State variable X
Fig. 9.
attractor and the target point is set as an unstable fixed point embedded in the B chaotic attractor. When the controlled trajectory from the A chaotic attractor lands on the target, we use the perturbation technique to enable the target to develop into the B chaotic attractor. The transition is shown in Fig. 11. Before the control is activated, the system wanders
1940
DAOLINXu and S. R. 0.4
BISHOP
_
I
0.31
‘\_,
0.2 * 0 .B 2B ;
1 t
0.1 I 0
9
1
I
-0.1 m
-0.2t -0.3’ -1.0
)
/’
: I
-0.5
0
0.5
State variables
1.0
1.5
X
Fig. 10.
40 60 Mapping times Fig. 11.
attractor. At the 44th mapping, the point at the position of (0.6219, attractor is picked up as the start point. The controlled trajectory is forced to move along the zigzag-like path between the start point (S.P.) and the target point (T.P.), and finally hits at the target point (0.6915, 0.2075) which is embedded within the B chaotic attractor. Then, in a similar way as before, the system can be pushed into the B chaos from this window (i.e. the unstable fixed point). Figure 12 shows the modification of parameters for the control process. within
the A chaotic
0.1281) in the A chaotic
5. CONCLUSIONS
In this paper, we develop a parameter-control method for a discrete-time chaotic system. In a numerical example the special case of two accessible linear parameters in the H&on system is considered. A sequence of simulation results of the control procedure is shown to illustrate the transition between chaotic states and stable equilibrium states. The investigation clearly indicates that the unstable fixed points that are embedded within chaotic attractors can become windows from which a system can be kicked into chaos. The numerical experiments presented in the paper illustrate only some of the possible applications of the method. The method can automatically detect the parameters of a fixed or periodic state in controllable phase-space. Additionally the method can create new
Steering
i P
dynamical
trajectories
1941
’ ---______ -
1.2 t I
--____
--___
pamm~ter a 1.0 /
b
P-eter 48
46
Mapping
50
-~
times
Fig. 12
multi-periodic orbits. For nonlinear systems with linear parameters, the control process is stable. To steer one point to another, the control process is completed within only two mappings. However, there are some limitations of the method. Namely, some points cannot be controlled because of inherent singularities. Another is that the initial control point should be a point at which the next mapping point remains close, such that the parameter-limiting condition is satisfied. These two limitations will both restrict the option for initial control points. Nevertheless, there are still many points, as shown, that can be conveniently selected as an initial control point. The method is simple and efficient in controlling chaotic dynamical systems such as the Henon system. Further work will continue to examine the behaviour when the parameters of systems are nonlinear. Acknowledgements-The author D. L. Xu is supported by Sit-to-British Friendship Scholarship Scheme and would like to note his appreciation to the British Council. S. R. Bishop is supported by the Science and Engineering Research Council of the UK. In addition, the authors are grateful to the reviewers for their comments.
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DAOLIN Xu and S. R. BISHOP
14. T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, Using chaos to direct trajectories to targets, Phys. Rev. Left. 65, 3215-3218 (1990). 15. T. Shinbrot, C. Grebogi, E. Ott and J. A. Yorke, Using chaos to target stationary states of flows, Phys. Left. A169, 349-354 (1992). 16. E. J. Kostelich, C. Grebogi, E. Ott and J. A. Yorke, Higher-dimensional targeting, Phys. Rev. E47, 305-310 (1993).