controlling low-dimensional chaos: determination and stabilization of unstable periodic orbits by...

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This paper was recommended for publication by editor Roy S. Smith * Correspondence to: R. Der, Institut fu¨r Informatik, Universita¨t Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany. ** Current address: Max-Planck-Institut fu¨r Stro¨mungsforschung, Bunsenstrasse 10, 37073 Go¨ttingen, Germany. Contract grant sponsor: German BMFT Contract grant number: 01 IN 106B/3 CCC 0890 6327/97/060489 11$17.50 Received 17 October 1994 ( 1997 by John Wiley & Sons, Ltd. Revised 1 November 1996 INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 11, 489 499 (1997) CONTROLLING LOW-DIMENSIONAL CHAOS: DETERMINATION AND STABILIZATION OF UNSTABLE PERIODIC ORBITS BY KOHONEN NEURAL NETS MICHAEL FUNKE1,*, MICHAEL HERRMANN2, ** AND RALF DER1 1 Institut fu ( r Informatik, Universita ( t Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany 2 RIKEN, Laboratory for Information Representation, 2-1 Hirosawa, Wako-shi, 351-01 Saitama, Japan SUMMARY A simple but efficient neural-network-based algorithm for non-linear control of chaotic systems is presented. The scheme relies on the method proposed by Ott et al.(Phys. Rev. ¸ett., 64, 1196 (1990)) to stabilize unstable periodic orbits by appropriate small changes in a control parameter. In contrast with this, our approach does not make use of an analytical description of the system evolution. The dynamics is evaluated by a self-organizing Kohonen network with an altered learning rule, which is able to learn the map of the system and to determine the positions of unstable periodic orbits of a given period. At the end of learning, a set of control neurons is generated which target the system along a quasi-optimal path towards the orbit. Besides its intrinsic tolerance against weak noise, the main advantage of the algorithm is its ability to take into account system constraints that occur in practical applications. The mean value of the control parameter and the range of allowed changes can be chosen in advance, and if more than one fixed point exists, the algorithm adapts to the most appropriate one concerning the control effort. ( 1997 by John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process., 11, 489 499 (1997) No. of Figures: 6, No. of Tables: 0, No. of References: 14 Key words: chaos control; unstable periodic orbits; self-organizing maps 1. INTRODUCTION Generally, the control of the chaotic systems serves a twofold purpose: the suppression of the chaotic behaviour and the generation of complex periodic oscillations. Both may rely on the stabilization of unstable periodic orbits (UPOs), which also is believed to be a very special ability for self-regulation of nature. Based on the observation that a chaotic attractor has embedded within it an infinite number of UPOs, Ott et al. have developed in a pioneering work1 (see also

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This paper was recommended for publication by editor Roy S. Smith

*Correspondence to: R. Der, Institut fur Informatik, Universitat Leipzig, Augustusplatz 10/11, D-04109 Leipzig,Germany.**Current address: Max-Planck-Institut fur Stromungsforschung, Bunsenstrasse 10, 37073 Gottingen, Germany.

Contract grant sponsor: German BMFTContract grant number: 01 IN 106B/3

CCC 0890—6327/97/060489—11$17.50 Received 17 October 1994( 1997 by John Wiley & Sons, Ltd. Revised 1 November 1996

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 11, 489—499 (1997)

CONTROLLING LOW-DIMENSIONAL CHAOS:DETERMINATION AND STABILIZATION OF UNSTABLE

PERIODIC ORBITS BY KOHONEN NEURAL NETS

MICHAEL FUNKE1,*, MICHAEL HERRMANN2,** AND RALF DER1

1Institut fu( r Informatik, Universita( t Leipzig, Augustusplatz 10/11, D-04109 Leipzig, Germany2RIKEN, Laboratory for Information Representation, 2-1 Hirosawa, Wako-shi, 351-01 Saitama, Japan

SUMMARY

A simple but efficient neural-network-based algorithm for non-linear control of chaotic systems is presented.The scheme relies on the method proposed by Ott et al. (Phys. Rev. ¸ett., 64, 1196 (1990)) to stabilizeunstable periodic orbits by appropriate small changes in a control parameter. In contrast with this, ourapproach does not make use of an analytical description of the system evolution. The dynamics is evaluatedby a self-organizing Kohonen network with an altered learning rule, which is able to learn the map of thesystem and to determine the positions of unstable periodic orbits of a given period. At the end of learning,a set of control neurons is generated which target the system along a quasi-optimal path towards the orbit.Besides its intrinsic tolerance against weak noise, the main advantage of the algorithm is its ability to takeinto account system constraints that occur in practical applications. The mean value of the controlparameter and the range of allowed changes can be chosen in advance, and if more than one fixed pointexists, the algorithm adapts to the most appropriate one concerning the control effort. ( 1997 by JohnWiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)No. of Figures: 6, No. of Tables: 0, No. of References: 14

Key words: chaos control; unstable periodic orbits; self-organizing maps

1. INTRODUCTION

Generally, the control of the chaotic systems serves a twofold purpose: the suppression of thechaotic behaviour and the generation of complex periodic oscillations. Both may rely on thestabilization of unstable periodic orbits (UPOs), which also is believed to be a very special abilityfor self-regulation of nature. Based on the observation that a chaotic attractor has embeddedwithin it an infinite number of UPOs, Ott et al. have developed in a pioneering work1 (see also

( 1997 by John Wiley & Sons, Ltd.Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

Reference 2) a very general stabilization method by applying small changes to an accessiblecontrol parameter of the system. A different method of influencing a dynamical system exploitsthe history of the system itself. In the case of delayed feedback control3 an additional input isapplied to the system which is proportional to the difference between the present state of thesystem and that at an earlier time. If the proportionality constant is chosen appropriately, thesystem may follow a periodic orbit. This scheme provides a solution only if a UPO of durationequal to the delay time exists. Further, it may happen that such an orbit lies outside the attractor4or that stabilizing parameters produce an ambiguous behaviour. These methods, which representthe present principal approaches to chaos control, are essentially restricted to low-complexityUPOs and low noise which is mainly due to the respective linearity in connecting the controlaction to state variables of the system. Hence a first aim of the present paper is to overcome thisrestriction by a piecewise linear but arbitrarily fine interpolation of the control function.

Whereas the mentioned methods are elegant and conceptually simple, the complexity ofchaotic dynamics has challenged the learning abilities of artificial neural networks which thenhave been successfully applied to chaos control by a number of researchers (see e.g. References5—10). Chaos control using a neural network, which will form the topic of the present paper,allows us to treat cases where the dynamics is not analytically accessible or where it is varying intime. In particular, a non-linear dependence of the control on the state (in the present case anarbitrarily fine piecewise linear dependence) can be constructed in an adaptive manner. Thenon-linear control may increase the range of controllability of the system and thus reduce thetransient chaotic period in the controlled system and allow for an increased stability of thecontrolled behaviour against noise as compared with linear control.

A second field related to chaos control is the targeting problem, i.e. to solve the task of directinga dynamic system from a given state towards another desired state, e.g. a fixed point, under twocontradicting constraints of being as fast as possible while using control actions (parameterchanges) as small as possible. An obvious application is the direction of a satellite towardsa distant planet by a so-called swing-by manoeuvre.

In the case of chaotic systems it means in particular to arrive at the fixed point more quicklythan the uncontrolled system, which—in the case of ergodicity of the chaotic trajectories—will meet every point with probability one after a shorter or longer transient time. The secondaim of our paper is to determine for any given range of the control parameter a good approxima-tion of a theoretically optimal targeting strategy by means of the Kohonen network. Ourapproach relies on the fact that for a given control parameter range a one-shot targeting (i.e. inone step) to the fixed point is only possible from within a fixed interval, which has to bedetermined.

The organization of the paper is as follows. In Section 2 the control problem is specified indetail and a control strategy is described that successively directs the system towards thefixed point. Section 3 describes the self-organized creation of the neural model map and thedetection of the fixed point. The creation of the control neurons is described in Section 4. This isfollowed in Section 5 by an argument concerning the stability against noise. Experimental resultsare given in Section 6 for our chosen example, the logistic map, followed by some concludingremarks in Section 7.

2. THE CONTROL TASK

For a given non-linear system described by a differential equation of the type

x5 (t)"F(x(t), p0) (1)

490 M. FUNKE, M. HERRMANN AND R. DER

( 1997 by John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

where x(t) denotes the time-dependent state of the system, we want to perform the followingtask: stabilize the system on a UPO of a desired period s at a given value of a control parametervector p

0by applying small perturbations (control actions) p(t)"p

0#dp(t) from a previously

fixed interval.More specifically, considering an n-dimensional control parameter p"(p

1, 2, p

n), we assume

here that we have to manage with small control ranges of size *iwith dp

i3[!*

i/2, *

i/2]. This

refers to the problems that in any real system only a finite maximum control action is avail-able—and sometimes a very small one. ’Small’ in our case means that for most states of thesystem the available maximum control action is too small for targeting the system directly intoa desired state, i.e. onto a UPO. We will show how to deal with this restriction.

Let us now consider the discrete map

xn"F(x

n~1, p

0) (2)

which is defined to be equivalent to (1) with xn,x(t

n) at discrete times t

1, t

2, 2. The t

nare

determined by the intersection of the solutions of (1) with a low-dimensional manifold in the statespace, i.e. (2) is the Poincare map of (1).

At the target value p0

of the control parameter a preliminary UPO of (2) with orbit lengths"1, 2, 32 is given by a fixed point of the s—times-iterated map F, i.e. x*s (p0

)"Fs(x*s , p0).

Stabilizing a UPO of period s is equivalent to finding a control function ps , the policy, given by

dp"ps (xn): x*s (p0

)"Fs(xn~s , p0

#ps(xn~s)) (3)

which directs the system from any xn~s in a certain interval around x*s (p0

) in one shot, i.e. oneperiod s, towards x*s (p0

). The known approaches consider a linear Ansatz for the control functiononly. The aim of our approach is to have a neural net discover the non-linear function ps(xn

)required for this purpose.Some remarks are in order.

1. For stabilizing true period—s orbits, our algorithm excludes orbits that are only multiples oflower-order orbits.

2. The above control strategy keeps p constant over a whole orbit. This is done in order tominimize the number of inevitable changes to the control parameters.

3. In turn the imposition of an allowed control range * implies a well-defined interval aroundx*s , only from within which is a one—shot targeting to the fixed point possible. The exactdetermination of this interval (and possibly other intervals apart from x*s for higher-orderorbits) by means of the Kohonen network is an important part of our strategy and isdescribed in the next section.

Henceforth, for simplicity, we take the control parameter p to be a scalar p. The next steps ofour strategy are the following. We discretize the control range * into 2K equidistant intervals oflength *p, i.e. *"2K*p. For each of the 2K#1 values p

0#k*p, k"!K,2, K, of the control

parameter we determine the state x(pk), which is controlled at x*s after one period q. This means in

turn that we know for 2K#1 states x the exact desired control action. Later on for each of the 2Kintervals [x(p

k); x(p

k`1)] a control neuron is created. This performs a linear interpolation of the

x%p correspondence between the exact known boundaries of the interval.We will illustrate (see Figures 1—6) the more general approach by a well-understood one-

dimensional chaotic system, namely the logistic map

xn"k

0xn~1

(1!xn~1

), k03(3·8, 4·0) (4)

CONTROLLING LOW-DIMENSIONAL CHAOS 491

( 1997 by John Wiley & Sons, Ltd.Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

Figure 1. Logistic map. (Top) Map for k0"3·9. The widths of the one-shot intervals result from the allowed control

range *"0·2. The corresponding maps for k.*/

"3·8 and k.!9

"4·0 enclose the area of maps that can be used forcontrol. (Bottom) Control neurons were created at the end of the learning time. Each linear part represents the controlleraction of one neuron depending on the input value x. The projection of its two boundaries to the x-axis gives the domainfor which the neuron is responsible. By using 2K#1"5 one-dimensional Kohonen nets, 2K"4 control neurons covereach one-shot interval. The backward projection of the intervals repeats this covering to other intervals. The horizontalneurons (giving always constant actions) belong to intervals from which a targeting to a desired point (backward

projection of the fixed point) is not possible but a targeting into the succeeding interval is possible

Control is done here by replacing the control parameter p0"k

0for one period—s orbit by

p"k0#n(x

n~s) according to the control function created by the neural net.

3. MODEL OF THE CONTROL-DEPENDENT MAP

The control algorithm to be applied relies essentially on the neural representation of the dynamicsystem, i.e. the chaotic map, by a set of 2K#1 one-dimensional Kohonen nets,11 each of which isresponsible for learning the map Fs(x, p

k) for one of the 2K#1 values of the control parameter p

kfor the desired s. To each neuron r of a one-dimensional net a vector w

r"(w0

r, w1

r)T is assigned

that represents states of the chaotic system at time steps xn

and xn~s , i.e. the states at the

beginning and end of an orbit. The Kohonen map hence approximates the function Fs in (2) by itsvalues in small intervals of the x

n~s domain. The winning neuron r* is determined by its respectivecomponent w1

rbeing closest to x

n~s , whereas the update of the full vectors wris driven by vectors

492 M. FUNKE, M. HERRMANN AND R. DER

( 1997 by John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

Figure 2. Trajectory at time steps 6000#t for control parameter k0"3·9 and variation range *"0·2. The system is

kicked randomly every 40 steps. Stabilization is achieved after about six steps from a random initial value x in the interval[0, 1]. The dots show values of the actions generated by the control neurons. The vanishing control parameter changesshow that the former unstable periodic orbit has been converted into a stable one. The learning time for the set of five

one-dimensional Kohonen nets was 6000 time steps

x"(xn, x

n~s )T according to

w@r"w

r#3 expA!

Dr!r*D22r2 B (x!w

r) (5)

where 3 is the learning rate and r is the width parameter. Both these parameters are annealed tofix the net architecture at the end of learning.

To avoid the system running towards a limit cycle, which may occur for one of the discretecontrol parameter values p

k, the learning procedure of the nets is performed by choosing one of

the pkrandomly, iterating one orbit, feeding the corresponding kth net with initial and last state

and taking this state as initial state for the next orbit with another pk. What we get at the end is

a narrow-spaced stepwise linear representation of the map for fixed values of pk, i.e. Fs,pk

(x). Thefixed points of the map belonging to a period s can be easily determined. Fromall the true period—s fixed points at k"0, i.e. no multiples of lower-order orbits, the fixed pointwith the smallest absolute slope of the map, x*(p

0), possesses the best controllability and is chosen

as the fixed point towards which the system will be stabilized.

4. CREATION OF THE CONTROL NEURONS

Starting from xn~s"x*(p

0), the system will meet the fixed point again only for a chosen value of

p"p0. For all other p

kup to a minimal and maximal k (equal to !K and K for simple maps) the

system has to start from states x@(pk), further and further away from x*(p

0), in order to be directed

exactly to x*(p0) at the end of that step (s periods). The existing minimal and maximal x@(p

k) refer

to the so-called one-shot interval [x@(p.*/

); x@(p.!9

)]. Only from within this interval is a one-shot

CONTROLLING LOW-DIMENSIONAL CHAOS 493

( 1997 by John Wiley & Sons, Ltd.Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

control to the fixed point possible. From the corresponding Kohonen map representation ofFs,pk

(x), states x@(pk) can be easily determined, thus providing us with up to 2K#1 states,

including the fixed point, for which the exact control actions pkare known. Now for any finite

interval [x@(pk); x@(p

k`1)] a control neuron (c0

k, c1

k)T is created, representing a linear interpolation

of the actions pkand p

k`1, so that for any x reached in this interval a control action p is generated

according to

p"c0k#c1

k(x!x@(p

k)),p

k#(p

k`1!p

k)

x!x@(pk)

x@(pk`1

)!x@(pk)

(6)

These up to 2K#1 neurons are the essential ones for controlling the system around the UPOagainst small noise. The growth of neurons from k"0 to k"K or k"!K stops and fewer than2K#1 neurons occur if for some p

ka corresponding x@(p

k) does not exist. The case

x@(pk`1

)"x@(pk) has to be treated separately. This exceptional case means that for an interval of

length zero, i.e. for one point, two different actions pkand p

k`1lead to the same result. Here the

smaller action is always chosen.The next aim is to cover the x

n~s interval step-by-step with additional neurons, having as theonly task the direction of the system in the least possible number of steps into a one-shot intervaland onto the UPO. This is done by back-propagating the one-shot interval in time, looking firstfor all states x

n~s that can reach the one-shot interval by xn. The first candidates are all states for

which also x*(p0)"Fs,p0

(x) holds. The intervals [x@(pk); x@(p

k`1)] around these states, constructed

and covered with control neurons as above, are also true one-shot intervals, although they do notlie around the fixed point. A first group of two-shot intervals can be found adjacent to the lowerand upper boundaries of the one-shot intervals. States x

n~s belonging to these intervals generallycannot be directed into x*(p

0) in one step because of the limited control range. However, by using

control neurons with constant extreme control action output p"p.*/

or p"p.!9

, according tox@(p

.*/), x@(p

.!9) of the adjacent one-shot interval, at least a targeting into a one-shot interval is

possible. Continuing this procedure by successively mapping every m-shot interval back to all(m#1)-shot intervals from which it can be reached, we get a targeting behaviour of the controlneuron that is optimal.

The proof of optimality is very simple. Assume that there is another, faster route to the fixedpoint; then at a certain step this route has to enter at some state x

nthe hierarchy of intervals given

above (at least at the one-shot interval that includes the fixed point). Looking backwards from thism-shot interval, the preceding state x

n~s has to belong to the set of (m#1)-shot intervals accordingto the premise of construction. This contradicts the assumption that the state x

n~s does notbelong to the hierarchy of intervals.

One remark should be made at this point. This argument holds strongly for stabilizing thesystem towards a fixed point at every period s. For period one it is identical with stabilizinga certain ºPO. For higher-order orbits, stabilizing around one of the other fixed points wouldstabilize the same UPO, only shifted in time. Thus for higher-order orbits, instead of directing thesystem from a given state in n steps to the one fixed point chosen first, the fastest way to a certainUPO could be found by mapping backwards the one-shot intervals of all existing fixed points.However, under difficult conditions, e.g. in the presence of noise, the fixed point with the smallestabsolute slope is the easiest one to control. Thus, as mentioned at the end of Section 3, we alwaysdirect the system towards this fixed point, accepting possibly some more steps but gaining a bettercontrollability, i.e. a better stability of the algorithm. It also pays attention to numerical noise, i.e.to the fact that we have to distinguish between a theoretically optimal path and a numericalrealization.

494 M. FUNKE, M. HERRMANN AND R. DER

( 1997 by John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

Figure 3. Covering rate C as a function of both maximal variation in control parameter * and number of control neuronsNc (log—log—log plot). For small * and fixed Nc the covering rate C is related to * by a power law with exponent a"1.The discontinuities occurring for very small * are due to the chosen discretization only. At larger * the power law is no

longer valid; instead, a transition to full covering is observed

The extent to which the targeting problem is solved is expressed by the covering rate,which is defined to be the fraction of the whole state space which is exhausted by the unionof all controllable intervals. For an infinite number of control neurons the covering rate isclearly equal to one. Otherwise it is a function of both the number of control neurons Nc andthe width of the intervals belonging to the present neurons, i.e. maximal control parametervariation *.

Assuming an unlimited number of primary neurons, i.e. a perfect representation of thechaotic map, the covering rate has been determined for the logistic map (see Figure 3). Forsmall * and fixed Nc the covering rate C is related to * linearly, i.e. a power law holds12 withexponent a"1.

Apart from the recovered exponential behaviour for small * or small Nc, it is interesting thatcomplete covering is achieved immediately if * and Nc are beyond certain (interdependent) criticalvalues.

5. STABILITY AGAINST NOISE

The main principle of our targeting strategy is to direct all states of an m-shot interval into statesof a succeeding (m!1)-shot interval.

The control actions at the boundaries of an m-shot interval are always exact. Since it issufficient to hit the corresponding (m!1)-shot interval anywhere, the algorithm is stable againstweak noise.

CONTROLLING LOW-DIMENSIONAL CHAOS 495

( 1997 by John Wiley & Sons, Ltd.Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

Figure 5. Controller with noise. Random noise is added to xn~s before performing one step but after determining

the control action. The mimics a noisy or imperfect controller. The behaviour is more sensitive to perturbation thanin the case of a noisy system, where the controller performed an adequate action for every x

n~s regardless of its origin.Now the route to the fixed point is longer. Near the fixed point an inadequate control action can kick the system out of thevicinity of the fixed point even if the noise amplitude 0·016 is smaller than the width 0·02 of the one-shot interval at the

fixed point

Figure 4. System with noise. Random noise is added to xn

at the end of every step. This mimics a noisy system withnon-deterministic output. Because the targeting procedure is only influenced if the system is directed out of the succeedingm-shot interval, the efficiency of the procedure is only slightly diminished. Near the fixed point the system is nowbeing kicked away randomly but remains in the vicinity of the fixed point. For the considered noise amplitude 0·016 acontrol parameter range *"0·2 is sufficient so that the system is always trapped within the resulting one-shot interval of

width 0·02

496 M. FUNKE, M. HERRMANN AND R. DER

( 1997 by John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

Figure 6. The average number of control actions to guide a state to the target is displayed as a function of the noise level.The initial state was chosen randomly according to a homogeneous distribution in the state space. In the simulation the

noise has been superimposed onto the state after the control action

In the presence of stronger noise the probability Pm~1,m

of hitting any (m!1)-shot intervalfrom an m-shot interval is smaller than one. Further, suppose that a randomly chosen statex arrives after m shots at the fixed point x* in the noise-free case. Under noisy conditions, but withthe control actions chosen as in the noise-free case, the mean value of shots which are needed toreach x* is M(x)"1#+

kkP

k,m. Since the total area covered by the k-shot intervals increases

exponentially with k,12 most of the noisy transitions will be towards k-shot intervals withk'm!1. Hence M(x) will increase with the noise level. However, as long as M(x)!1(m, thenoise will not seriously disturb the targeting process (see Figure 6). Upon further increase in thenoise strength the target is reached more and more by chance.

One of the main advantages of using Kohonen’s algorithm in chaos control is its inherent noisereduction ability, i.e. it is able to approximate the correct chaotic map even from noisy informa-tion on the current state. In Reference 13 it has been proved that the noise reduction in Kohonen’salgorithm is optimal with respect to the transformation on the given state represented by thewinning neuron.

6. RESULTS

6.1. Targeting and one-shot control

The virtues of this approach are demonstrated by the example of the logistic map (4). Thealgorithm has proven to be able to model the map of the chaotic system for an orbit of a desiredperiod in a self-organizing way. The generated control neurons are able to target the system in anoptimized strategy towards the fixed point (see Figure 2) and to perform a non-linear stabiliz-ation at the fixed point with vanishing control actions, i.e. vanishing parameter changes (seeperformed control actions in Figure 2). As can be seen from the k versus x

n~s plot in Figure 1(bottom) the control function learned by the control neurons is a highly non-linear one. In ourexample we limited the number of created control neurons to 100, covering about 30% of theinterval [0, 1], as can be estimated from the intervals in Figure 1 (bottom). By allowing largercontrol ranges, this value increases rapidly for a fixed number of control neurons.

CONTROLLING LOW-DIMENSIONAL CHAOS 497

( 1997 by John Wiley & Sons, Ltd.Int. J. Adapt. Control Signal Process., 11, 489—499 (1997)

6.2. Noise tolerance

The non-linear controller is able to restabilize the system very quickly after it has been kickedout of the vicinity of the fixed point by a single perturbation (see Figure 2). For permanentrandom noise in a controlled system one has to distinguish more precisely between a noisy (orindeterministic) system and a noisy (or imperfect) controller. Both cases have been investigated(see Figures 4 and 5) and it turns out that the system can be stabilized even in the presence ofconsiderable noise amplitudes, which strongly depend on the allowed control range that itselfdetermines the controllable one-shot interval.

The average number of control actions needed to move from a randomly chosen state to thetarget state is considered as a measure of the controller quality. Figure 6 displays the dependenceof this quantity on the noise level. A transition is found from a noise range that is tolerated by thecontroller to a region which more seriously affects the controller quality.

7. CONCLUSIONS

We have found a neural-network-based algorithm to control chaotic systems with possiblyunknown dynamics on an orbit of desired period and for given allowed range of the controlparameter. The algorithm makes use of a self-organizing Kohonen map, modelling the systemdynamics by providing a Ponicare map and a classification of time delay co-ordinates. From thismodel map a set of control neurons is created which control the system in a highly non-linearway. The algorithm shows an intrinsic tolerance against weak noise and a relatively shortlearning time. The explicit extension to systems of higher order is straightforward by usinghigher-dimensional Kohonen nets.

In principle the proposed algorithm should enable one to apply all the control strategiesapplicable for systems with known dynamics to ‘ black-box ’ systems. An interesting problem willbe to investigate more carefully the dependence between the size of the control range * and thetotal size of the m-shot intervals. This is related to the average time to achieve control and to thenumber of controlled trajectories. Here scaling laws are valid, as has been pointed out by Ottet al.1 and discussed in great detail in a recent work by Tel.12 Probably the number of controlneurons that are necessary to achieve control within a given time or to cover a certain part of theinterval [0, 1] will follow similar scaling laws, which would allow the derivation of thoseexponents for an unknown system directly by means of the proposed neural net algorithm.

ACKNOWLEDGEMENT

The reported results are based on work done in the project ‘ LADY ’ sponsored by the GermanBMFT under grant 01 IN 106B/3.

REFERENCES

1. Ott, E., C. Grebogi and J. A. Yorke, Controlling chaos’, Phys. Rev. ¸ett., 64, 1196—1199 (1990).2. Ott, E., Chaos in Dynamical Systems, Cambridge University Press, Cambridge, 1993.3. Pyragas, K., ‘ Continuous control of chaos by self-controlling feedback ’, Phys. ¸ett. A, 170, 421—428 (1992); ‘ Control

of chaos via extended delay feedback ’, in preparation.4. Namaju6 nas, A., K. Pyragas and A. Tamas| evic| ius, ‘ Stabilization of an unstable steady state in a Mackey—Glass

system ’, Phys. ¸ett. A, 204, 255—262 (1995).5. Chen, G. and X. Dong, ‘ Identification and control of chaotic systems: an artificial neural network approach ’, Proc.

IEEE Int. Symp. on Circuits and Systems, Seattle, WA, 1995, IEEE New York, 1995, pp. 1177—1182.

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6. Der, R. and M. Herrmann, ‘ Nonlinear chaos control by neural nets ’, in Marinaro, M. and P. G. Morasso (eds), Proc.Int. Conf. on Artificial Neural Networks (ICANN’94), Sorrento, 1994, Springer, New York, 1995, pp. 1227—1230.

7. Frison, T. W., ‘ Controlling chaos with a neural network ’, Proc. Int. Conf. on Neural Networks, Baltimore, MD, 1992,pp. 75—80.

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