A Technique for Higher Dimensions

In this section we outline an efficient method of targetting that can be applied to systems of higher dimension [9]. This method has been successfully applied to the double-rotor map, which is four-dimensional, and the two-dimensional Henon map [9] . The numerical method employed attempts to direct iterates on a chaotic attractor to a predetermined target point on the attractor. This is done by finding a sequence of small perturbations to the available system parameter(s) in order to direct the trajectory of a given initial condition to the stable manifold of a point on the trajectory leading to the target. The number of such iterations may be reduced by constructing a heirarchy of paths leading to the target. The method may, in principle, be applied to chaotic attractors with any number of positive Lyapunov exponents.We outline the control procedure below:

We assume, as in the cases for lower-dimensional systems, that we are given a point Z t on the attractor which is to be targetted. We assume further that Z t is not a periodic point and that it has two-dimensional stable and unstable manifolds associated with it.

Let Zo, Z 1 , .. Zt_ 1 be the t preimages of Z t . This set of preimages, together with the Lyapunov basis associated with each, is called the path to the target, Z t . If, contrary to the initial assumptions, Z t is indeed periodic, then a non-periodic point sufficiently close to the actual target, must be selected.

An arbitrary set of initial conditions is chosen within the basin of attraction and the map is iterated.

We suppose that one of the iterates, Yo, falls within a suitably small neighbourhood of the point Z o in (6.5)

(6.6)

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the path to Z t . If Yo falls close to one of the points Zi (0 < j < t — 2), then it is relabelled Zo and t —4 t — j. Without such control, the points Y1, Y2, , rapidly diverge from the trajectory Z 1 , Z2>

starting at Zo.

The core idea behind this targetting algorithm is to apply two successive perturbations in order to direct Yo to a new point Y2, lying on the stable manifold, E2^, associated with Z2. The stable manifold

is associated with Z2 for the unperturbed map at the nominal system parameter value p = P. Two values, po and pi, close to p, are desired such that Y2 = f(f(YO)P0),P1) E E. Whenever this procedure succeeds, the trajectory starting at Y2 approaches the one leading to the target. The approach of the trajectory starting from Y2, to the one leading to the target, may not necessarily be uniform. On the average, however, it should approach the target trajectory at a rate proportional to exp(L3) • t , where L3 is the largest negative Lyapunov exponent and t is the number of iterations, counted from Y2.

The difficulty in finding the intersection point -2.2 is that the stable manifold E2 associated with Z2 is approximate to a plane only in a small neighbourhood about Z2. Although the distance

Zo — You may be small, 11Z2 — ..11.211 generally is not. That is, the intersection of the plane through Y2, spanned by the gradient vectors go and g i , with the stable manifold .E.1 may be relatively far from Z2. The stable manifold EZ may be approximated away from Z 2 , by considering the inverse images of the stable manifold of an arbitrary point further down the path. Let this point be Z t with the associated stable manifold E. Under the inverse map, f -1 , Ef is an expanding set. Consider a point Z near to Zt on the tangent plane to Ef . Although Z does not lie exactly on ET, the inverse images, f -1 (Z), f —2 (Z), ..., approach the corresponding sets Ef_ 1 , Ef_ 2 , ..., because, under the inverse map, errors are suppressed along the directions spanned by the Lyapunov vectors associated with the positive Lyapunov exponents. Lyapunov basis vectors so and s 1i associated with the negative Lyapunov exponents at Zt, are introduced. It follows that so and s i span the tangent plane to .Ef at Z i . If a0 and a1 are small numbers, then Z = Zt + uoso+ 0- 1s1 should be close to Si, and its inverse images should approach the corresponding stable sets quickly. For instance, f — (z -2)(Z) should lie very close to the stable set E2 associated with Z2 even though the distance Ilf — (i-2)(Z) — Z 2 II may not be small. The basic control

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step then, is to calculate values cro and 0 1 together with parameter values po and p i such that

f—(i-2)(ze + aoso + aisi) = f i—(i-2)-1 (f(Yo,P0),P1)• (6.7) This equation can be solved numerically. For the double-rotor map [39], co and o - i were found to be of the order 10 -6 even when IN' 2 — Z2Il — 1. Typically, the values of o-0 and a1 depend on the negative Lyapunov exponents associated with the points on the path. More negative exponents lead to smaller values for go and a1 . Although the value of i is arbitrary, it should be noted that inverse images of points on El further down the path, typically yield a point whose appropriate inverse image is closer to the stable set .q The numerical solution of (6.7) is rendered more ill-conditioned in this way. If we consider the inverse images on

Et ,

Df-1 (Zi )Df -1 (Zi_1)Df -1 (Zi_2)...Df -1 (Z3).

If we consider Si-2 instead, then it becomes necessary to evaluate

2) Df —i.( zi

The matrix products become more singular as more terms are added and there is thus a trade-off between numerical precision and approximation errors arising from the system dynamics. Because of small errors in the initial approximation of Si together with numerical rounding errors, the described control must be reapplied from time to time in order to keep the new trajectory close to the path leading to the target.

It is not always possible to solve equation (6.7). Sometimes the chosen numerical method diverges because good starting values of the parameters cannot be obtained by linearizing (6.7). In such cases, it is not possible to start the control process at Yo in order to bring the trajectory close to the target.

If the procedure fails then the process must be repeated only after the trajectory again approaches a neighbourhood of Z 1 .

The procedure described above works well for a variety of maps, but is has the disadvantage that the map must be iterated a large number of times before reaching a neighbourhood of one of the points in the path leading to the target. A long path increases the likelihood that a given iterate lies near a point on it, but then many control steps are needed to reach the target. It has been shown, for the double-rotor

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map [39], that some refinements to the basic control procedure can be implemented to steer a typical iterate to a target point in as few iterations as possible. Such adjusments are however, often made with specific reference to a particular map or set of experimental data.