STABILISATION OF UNSTABLE PERIODIC ORBITS FOR CHAOTIC SYSTEMS WITH FRACTAL DIMENSION CLOSE TO AN INTEGER Hugo G. Gonz¶ alez-Hern¶ andez LIDETEA-CGI. Universidad La Salle.Benjamin Franklin 47. M¶exico, D.F. 06140 MEXICO. Fax: (525) 515-7631, Ph: (525) 728-0522. e-mail:
[email protected] ¶ Joaqu¶³n Alvarez CICESE. e-mail:
[email protected]
¶ Jaime Alvarez-Gallegos CINVESTAV-I.P.N.e-mail:
[email protected]
Abstract In this paper we report the use of an extension of the OttGrebogi-Yorke (OGY) approach for controlling chaos, but instead of using a Poincar¶e Map we use the First Return Map (FRM) of the generated °ow. This allows us to deal with systems whose chaotic attractors has a fractal dimension close to an integer value. The method uses only a measurement of one system variable. We take some available parameter as the perturbation parameter, which is changed to force the state trajectory to fall into the stable manifold of the equilibrium point of the FRM. Keywords : Chaos, control, capacity dimension, OGY method.
1
Introduction
Recently, chaos control and anticontrol have been deserved a lot of attention, giving place to di®erent control techniques. The di®erent approaches can be grouped mainly into the following categories [5]: Parametric perturbation methods, entrainment and migration controls, engineering control approaches, and some other approaches like intelligent control. Some parameter perturbation methods derived from the OGY technique [10] have been developed and sometimes successfully applied for the stabilisation of periodic orbits. An important characteristic of this method is the possibility to implement it from a time series, i.e., from a measurement of one of the system variables; therefore, an a priori model is not needed. It is well known that a chaotic attractor has an in¯nite number of unstable periodic orbits embedded in it. The main objective of the parameter perturbation methods, and particularly of the OGY control algorithm, is to stabilise one of such orbits. The application of the OGY method needs a good identi¯cation of the local dynamics around the corresponding saddle ¯xed point in the Poincar¶e section. However, if the contraction rate is much greater than the expansion of the °ow around such
a point, then the classical identi¯cation procedures yield poor models, and a better identi¯cation algorithm must be used, complicating the design. This is normally the case when the chaotic system is of low order, giving attractors with fractal dimension close to an integer. In this paper we propose another procedure that avoids this problem, using the ¯rst return map instead of the Poincar¶e map. The control approach utilised belongs to the parameter perturbation methods; this is applied to the stabilisation of an unstable period-1 orbit obtaining from the First Return Map. The complete procedure is illustrated with the Lorenz system.
2
Chaotic Systems
Consider a system given by x_ = f (x; t; ¯)
(1)
where x 2 Rn is the state, f : Rn ! Rn is a smooth vector ¯eld, and ¯ denotes the system parameters. The solution of (1) is some vector function x = x (t) that describes the trajectories in the state space constructed with its coordinates. Depending on the parameter values the system may display di®erent steady states, ranging from equilibrium points to chaotic attractors. De¯nition 1 (Chaotic Attractor) [11] Consider a C r (r ¸ 1) autonomous vector ¯eld on Rn , de¯ning a system like (1). Denote the °ow generated by (1) as Á (t; x), and assume that ¤ ½ Rn is a compact set, invariant under Á (t; x). Then ¤ is said to be chaotic if 1. Sensitive dependence on initial conditions. There exists " > 0 such that, for any x 2 ¤ and any neighborhood U of x, there exists y 2 U and t > 0 such that jÁ (t; x) ¡ Á (t; y)j > ". 2. Topological transitivity. For any two open sets U , V ½ ¤, there exists t 2 R such that Á (t; U ) \ V 6= ;.
De¯nition 2 (Capacity Dimension) [7] Let A be a bounded subset of Rn . Let N± (A) be the smallest number of sets of maximum diameter ± that cover A. Then the capacity dimension is de¯ned, if it exists, by log N± (A) dimK (A) =lim ¯¯ ¡ 1 ¢¯¯ ±!0 log ±
(2)
Typically, this quantity is not an integer number for a chaotic attractor. When this situation occurs it is said that A has a fractal dimension.
3
Control of Chaotic Systems by Parameter Perturbation Methods
A chaotic attractor can be thought as the closure of an in¯nite number of unstable periodic orbits (UPO) [10]. The OGY method takes this main idea and establishes that a little variation of an accessible system parameter is enough to stabilise the orbit. The system trajectory is forced to approach close to the stable manifold of a ¯xed point in a map representing the system dynamics in a lower dimension (typically its Poincar¶e map). We distinguish two approaches for the implementation of this control method. 1. If the fractal dimension of the attractor is not close to an integer, then it is possible to use the classical approach, described in [2], [3], [4] and [9], among others. This approach consists mainly in reconstructing the attractor [1], [2] and identifying the local dynamics in a neighborhood of an unstable ¯xed point of the Poincar¶e map. 2. The second approach considers systems whose attractors have a fractal dimension close to an integer. In this case, the use of the First Return Map (FRM) yields better results. This FRM is obtained from the local maxima (or minima) of the measured variable, and delays this signal to form a vector of delayed coordinates[6]. The FRM allows us to determine periodic orbits (especially period-1 orbits) and to locate easily the ¯xed point about which we want to control the chaotic system. An advantage of this method is that it is possible to implement it without any prior knowledge about the system equations. This is the feature that has made this method so popular for the experimental control of chaotic systems. The main contribution of this work is to distinguish between both cases and to show an alternative for the case when the fractal dimension of the attractor is close to an integer. According to this scheme it is necessary to identify a periodic unstable orbit embedded in the attractor, characterize locally its dynamics, and determine the
change of the attractor when an external stimulus is applied. We may distinguish three stages for the application of this method. 1. Identi¯cation of the unstable periodic motion by determining the corresponding ¯xed point in the FRM. 2. Characterisation of the local dynamics of the ¯xed point and its displacement due to small external stimulus. To accomplish this, it is needed to ¯nd an accessible parameter of the system, which will play the role of the control action. Although, in some cases depending on the nature of the system,we may introduce a control variable and use it as the accessible control parameter. 3. Design of the control law, i.e., determine the correct change in the available parameter such that the state is forced to fall inside the stable manifold of the periodic orbit. Once it has been determined the unstable ¯xed point we wish to control, the system trajectory is monitored by observing the measured signal coming from the chaotic system. The trajectory eventually approaches a neighborhood of the chosen unstable ¯xed point, and it is in this moment that the control might be activated. Determining the attractor response to external stimulus is a di±cult task. For this, it is necessary to introduce a perturbation into the system, varying the available parameter.
3.1
Embedded unstable periodic motion and ¯xed points
Consider a chaotic system described by equation (1). Let z be a component of the state x = x(t). If the local maxima of z are sampled, this time series can be considered as a new variable »(n). Then, the FRM will be given by »n+1 = P (»n ; ¯)
(3)
with P a nonlinear function. For chaotic attractors with fractal dimension close to an integer value, this map has a ¯xed point and its location depends also on the parameter ¯. We denote this by »F = »F (¯)
(4)
Due to the unstable nature of the periodic orbit embedded in the chaotic attractor, the ¯xed point »F is expected to be a saddle.
3.2
Local dynamics of ¯xed points
The dependence of a ¯xed point with respect to parameter ¯ can be estimated as s=
d »F (¯n+1 ) ¡ »F (¯n ) e »F (¯) j¯=¯n = d¯ ¯n+1 ¡ ¯n
(5)
where ¯n+1 = ¯ + ¢¯, ¢¯ being a small variation in the parameter. Therefore, »F (¯n+1 ) = e »F (¯n ) + (¯n+1 ¡ ¯n ) s
(6)
»n+1 =» e F + M (»n ¡ »F )
(7)
On the other hand, we may represent the map (3) in a neighborhood of the ¯xed point using a square matrix
The matrix M is easily obtained from a standard identi¯cation method. It is characterised by its stable and unstable (¸u ) eigenvalues and the corresponding eigenvectors es and eu , i.e. M eu = ¸u eu ; M es = ¸s es The eigenvectors are normalised such that 1.
(8) eTu eu
=
eTs es
=
Figure 2: Dual base vectors ffu ; fs g. M can be expressed in terms of dual and base vectors, · ¸· T ¸ £ ¤ ¸u 0 fu (12) M = eu es 0 ¸s fsT = ¸u eu fuT + ¸s es fsT
It can be easily found that fu is a left eigenvector of M coresponding to ¸u , i.e. fuT M = ¸u fuT . To achieve the control it is required that the state », in the next iteration, falls into the stable direction. This can be accomplished if the next relation holds, fuT (»n+1 ¡ »F (¯n )) = 0
(13)
This condition means that the vector displacement from the ¯xed point corresponding to a parameter value ¯n to the next iteration of the state » does not have a component a long fuT as can be seen in ¯gure 3.Now, let us consider
Figure 1: Eigenvectors about the ¯xed point We may rewrite equations (8) as £ ¤ £ ¤ Meu M es = ¸u eu ¸s es · £ ¤ £ ¤ ¸u eu es M eu es = 0
0 ¸s
¸
(9)
therefore
M=
3.3
£
eu
es
¤
·
¸u 0
0 ¸s
¸
£
eu
es
¤¡1
(10)
The Control Law
Now consider the dual base vectors ffu ; fs g that satisfy fsT es = fuT eu = 1 and fuT es = fsT eu = 0. These vectors can be observed in ¯gure 2.These dual base vectors are related with the original eigenvectors as: · T ¸ £ ¤¡1 fu = eu es (11) T fs
Figure 3: Graphic representation of an iteration of the map near the unstable ¯xed point. The control is accomplished changing an accessible system parameter such that the projection of xF ¡ xn+1 on fu be zero. equation (7) with »F = »F (¯n+1 ), »n+1 =» e F (¯n+1 ) + M (»n ¡ »F (¯n+1 ))
(14)
Using equation (6), we have »n+1 ¡ »F (¯n ) ¡ (¯n+1 ¡ ¯n ) s » = M (»n ¡ »F (¯n ) ¡ (¯n+1 ¡ ¯n ) s)
(15)
Premultiplying by fuT , fuT (»n+1 ¡ »F (¯n ) ¡ (¯n+1 ¡ ¯n ) s) = e ¸u fuT (»n ¡ »F (¯n ) ¡ (¯n+1 ¡ ¯n ) s)
(16)
Using equation (13) and de¯ning ±¯n = ¯n+1 ¡ ¯n and ±»n = »n ¡ »F (¯n ), that is,
¡±¯n fuT s=¸ e u fuT ±»n ¡ ±¯n ¸u fuT s ±¯n = e
¸u fuT ±»n (¸u ¡ 1) fuT s
(17)
(18)
This is the so-called OGY formula [6].
4
Figure 4: First Return Map for the Lorenz System. ¯ = ¯ ¤ = 2.
Application to the Lorenz System
In 1963, Edward Lorenz [8] reported a deterministic model for a thermal convection process. These equations showed an apparently random behaviour. The report started an explosive growth in scienti¯c e®orts about this kind of complex dynamics. We have chosen this model, despite it is a well-known system, because it has an attractor with a fractal dimension close to an integer, dimK (ALorenz ) = 1:804§0:179. This feature makes the typical OGY method (using attractor reconstruction) di±cult to implement. The so-called Lorenz equations are: x_ = ¾ (y ¡ x) y_ = ½x ¡ y ¡ xz z_ = ¡¯z + xy
(19)
This system is chaotic for the following parameter values ¾ = 10; ½ = 28 and ¯ = 2. We choose ¯ as the parameter to be perturbed and the measured variable is z. The FRM is obtained from the local maxima of the time series. In this case the variable z has been measured from the Lorenz system. We have taken ¯ = ¯¤ = 2 as the nominal parameter. The intersection (»F¤ ; »F¤ ) = (38:2289; 38:2289) is the ¯xed point that we are looking for.If parameter ¯ changes, then the ¯xed point location changes as well. It is required to ¯nd some relationship between ¯ and »F . Using a mean square procedure we found »F = 2:3291¯ + 33:4615 (see Fig. 5). Hence, the s vector used in the OGY formula turns out to be s = (2:3291; 2:3291)T .The last step of this method is to characterise the linearised dynamics about the unstable ¯xed point. Using the measured » (n) and knowing the location of the ¯xed point it is possible to ¯nd the matrix M (Eq. (7)). This matrix M can be obtained using an standard mean squared identi¯cation procedure; then we can
Figure 5: ¯ and »F relationship. easily obtain its eigenvalues ¸u = ¡1:6491; ¸s = 0:2777, T and the left eigenvector fu = (0:8883; 0:5386) . With these values, expression (18) can be calculated. The controlled system was implemented using the Matlab program to show the behaviour of the controlled Lorenz system Figure 6 shows the controlled variable z, the control action is applied after 30 seconds. The parameter ¯ is changed in such a way that ¯min < ¯ < ¯max ; this corresponds to a variation in the ¯xed point location »min < » < »max .The control is enabled when a consecutive pair of maxima » satis¯es this last inequality (see ¯gure 7).Figure 8 shows a projection of the corresponding controlled trajectory in the (x; z)-plane.
5
Conclusions
Parameter perturbation methods are a popular resource for unstable periodic orbits stabilisation due to the pos-
Figure 8: Stabilised orbit of the Lorenz System. Figure 6: Response (z) of the controlled Lorenz System. ¶ ¶ [2] Alvarez-Gallegos, Ja., Alvarez-Gallegos, Jq. & Gonz¶ alez-Hern¶ andez, H. G., \Analysis of the dynamics of an underactuated robot: the forced Pendubot," Proc. of the IEEE 36th International Conference on Decision and Control CDC'97, San Diego, CA. December 10-12, (1997). [3] Auerbach, D.; Grebogi, C.; Ott, E. & Yorke, J. A. \Controlling chaos in higher dimensions", Phys. Rev. Lett. 69, p. 3479 (1992). [4] Barajas, G., M. in Sc. Thesis, (in Spanish) CICESE, M¶exico.(1997) [5] Chen, G. & Dong, X. From Order to Chaos: Methodologies, Perspectives and Applications. World Scienti¯c.(1998). Figure 7: Control ¯. sibility of applying them without any prior knowledge of the equations of the system. This does not mean that it is an easy task, especially when the attractor of the chaotic system to be controlled has a dimension close to an integer, as it is normally the case for low dimension systems. This is due, among other reasons, to the di±cult selection of an appropriate surface of the Poincar¶e map and to the very di®erent expansion and contraction rates of the °ow around a ¯xed point. An alternative procedure is to use the ¯rst return map built with the local maxima of the time series of the measured system variable, as it has been illustrated in this paper.
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