The Chaos Panaceas

Keynote Address

C. K. Tse

Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hong Kong

Email: encktse@polyu.edu.hk URL: http://chaos.eie.polyu.edu.hk

Abstract — In this talk we review the work in

the field of chaos applications, especially in the

formulation of solutions for practical engineering

problems. Using a few case studies we illustrate the

various common pitfalls in the reporting of chaos

applications in engineering. Our objective is to

promote the prudent use of engineering judgments

and proper exercise of scientific principles when

applying the theory of chaos in analyzing and

solving engineering problems.

Keywords — Chaos, application of chaos, engineer-

ing system, model validity, analysis approach.

1 INTRODUCTION

Use of chaos in solving engineering problems hasbeen one of the most productive areas in scientificand engineering research in recent years [1]–[4]. In-deed, a huge number of research papers with anunprecedented amount of novelty claims have beenproduced in the past two decades. Chaos appli-cations is still one of the trendiest topics of re-search. Proposals, reports and papers addressingthe applications of chaos-based methods in solv-ing practical problems are generally well receivedby funding bodies and journal reviewers. However,in many such proposals, reports and papers, mis-understanding and omission of the real underlyingcauses have led to mistaken analysis and possiblemisuse of chaos. It is therefore of interest to re-view the current state of research in this field andto clarify the various possible misunderstandingsso that chaos-based applications can be developedwith strong engineering judgments and consistencywith scientific principles.

Undeniably, chaos does offer a range of desir-able features for doing certain engineering tasks,but analysis and thorough exposition of the basisof accomplishing certain tasks is still insufficient.The problem often lies in the researchers’ attitudein the way they report their work. Two aspects arenot uncommon among authors in this field. First,annotating the solutions with chaos terminologiesmay divert reviewers’ attention away from compar-ing with conventional methods, hence more readilyjustifying a novelty claim. Second, analysis of theactual cause in a conventional sense may be uninter-

Figure 1: A drunk man was searching under the lamppole for his key which he knew was dropped elsewhere.When asked why, he replied, “because it’s bright here.”

esting or could honestly make the solutions soundtoo trivial.

Another very fruitful area of research related tochaos applications is the discovery of new (unre-ported) exotic phenomena from engineering sys-tems. This is a very worthwhile research area asbetter understanding of the system behavior canlead to higher reliability and improved functional-ity of the system under study. However, the lackof engineering judgment may drastically reduce thecredibility of the reported phenomena. For in-stance, if the models are invalid for those condi-tions under which exotic behaviors are found fromsimulations or numerical studies, the entire bodyof results can be meaningless and the real systemsactually never behave in the ways predicted by sim-ulations or numerical studies.

Also commonly pursued in this field is the re-examination of old solutions from a chaos theoryviewpoint. Controlling chaos is among the mostwidely studied areas. Turning some theoreticalwork in the chaos control literature into engineer-ing use seems to create a never-ending publicationcycles. This is undoubtedly an area of research ofmuch worthiness because conventional viewpointsmay fall short of novel insights into some complexproblems. However, again, the field still suffers

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from inadequate engineering judgment. Some au-thors put their blind faith on oversimplified or in-valid models, while some actually end up with thesame old solutions that have been used for yearsin practice, of course with the old solutions hiddenunder new chaos terminologies.

Notwithstanding the many working blunders, thefield of chaos applications is still a very worthwhilearea of research as many potential applications areyet to be discovered and investigated. In the pro-cess of investigating new applications, however, ahealthy amount of engineering judgment is needed,especially in terms of identifying the underlyingmechanisms that lead to the apparent successfulapplications. With the theoretical foundation ofchaos being a mature scientific discipline, it makesno sense that engineers are only using it as a gim-mick in getting their projects funded or papers pub-lished.

In this talk, we will review the problems men-tioned above, using a few case studies to illustratethe various common pitfalls in the reporting ofchaos applications. Our objective is to promote theprudent use of engineering judgments and properexercise of scientific principles when applying thetheory of chaos in analyzing and solving engineer-ing problems

2 PITFALLS AMID APPARENT BEAU-

TIES

Chaos provides a lot of free features for many prac-tical applications. Researchers were able to ex-ploit those features and pioneered the early de-velopments, but some soon realized the pitfallsthat could be hard or practically impossible to getaround. However, being ill informed, many newresearchers continued to drill in the wrong direc-tion. Researchers in the field still prefer to presentthemselves very positively, as many reports tend toemphasize the potential advantages rather than tohighlight the difficult problems that are virtuallyunsolvable. The following case studies may serveto illustrate a typical course of development in thisfield.

Case Study 1:

The recent research efforts in applying chaos intelecommunications have clearly exposed an inter-esting course of development. By virtue of itsextremely high sensitivity to parameter changes,chaos provides an attractive solution to secure cod-ing and communications. However, the same fea-ture creates a technical difficulty in coherently de-coding and demodulation under realistic channel

conditions, namely the impossible task of synchro-nizing chaos in real world environment. We shouldnote that digital communication systems are some-times required to perform satisfactorily under noisycondition typically down to a signal-to-noise ratioof 0 dB. Obviously, there exists no method to syn-chronize chaos when the noise level is as large asthe signals. Neither can any noise-cleaning algo-rithm separate the useful chaotic signal from thenoisy sample under such a condition. The prob-lem should be clearly known to engineers [5]. De-spite the efforts of some (perhaps unpopular) re-searchers who pointed out this virtually unsolvableproblem and instead promoted noncoherent chaos-based methods, the application of coherent meth-ods in telecommunication based on chaos synchro-nization is still being pursued!

Case Study 2:

Deterministic chaos has been used in cryptography,again, taking advantage of the inherent parametricsensitivity. However, the determinism may itselfbe the biggest pitfall that allows attack algorithmsto be derived for each specific encryption scheme.Typically, a deterministic (possibly chaotic) func-tion is designed to encrypt a message using a cer-tain system parameter as the key. The decodingalgorithm “reverses” the process in some way andrecovers the message. In practice, both encrypt-ing and decrypting algorithms are open to the pub-lic. The method is secure provided that an in-truder can recover messages only through an ex-haustive search of the key, as is true for existingcryptographic methods. Thus, for digital imple-mentations, the number of bits used to represent akey is the ultimate security. So far, although therehave been no proofs of the general attackability ofchaos-based cryptographic methods, deterministicschemes have kept being broken by specifically de-signed attack algorithms (not through exhaustivekey search). The average time delay between ascheme proposal and its reported attack is about11 months. It is counter-intuitive that research ac-tivities are directed more to deriving new chaos-based schemes rather than to a rigorous study ofthe possible general attackability of deterministiccryptographic methods.

3 RANDOMNESS VERSUS CHAOS

Chaos has been compared with randomness: beatit, publish it! Very often, the use of chaos hasbeen unscientifically understood or explained. Es-sentially, the important question of “what makes itwork” was not clearly addressed.

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We should point out that “randomness” is a sta-tistical concept, but its actual generation must relyon a specific form of determinism. In this sense,comparison between chaos and randomness is noth-ing but a comparison of different statistical prop-erties of two different deterministic processes (if acomparison has to be made at all). Further, itmakes no sense to compare chaos with randomnessbecause there are many kinds of chaos. If a partic-ular chaotic function (based on Lorenz’s or Chua’s)is used in an application, such comparisons can atbest be between the particular chaotic function andthe random function used.

Nonetheless, chaos does offer the necessary fea-tures for doing a range of useful tasks, but it is notchaos in general that makes it work! Obviously,certain features of the specific chaotic function usedhave made the crucial contribution. The problem,however, lies in the researchers attitude in the waythey report their work. Typical paper titles arelike “Solving ... by chaos”, giving an impressionthat chaos worked exclusively! This is somewhatmisleading. If the reason for it to work is clearlyidentified, chaos could be irrelevant.

Case Study 3:

In EMI reduction, chaos is used to randomize theswitching frequency so as to avoid high fixed spikesin the frequency spectrum. The same trick was usedmany years ago in RF engineering, though the wordchaos was not mentioned. As argued above, com-paring chaotization and randomization of switchingfrequency is meaningless. (Recently, more sophisti-cated and focused research has been performed infinding the relationship between the resulting fre-quency spectrum and the form of chaotic functionsused for randomizing the switching frequency [6].)

Case Study 4:

In fluid mixing, chaos has been shown to leadto better mixing with reduced power consumption[7, 8]. The real problem could be the distribu-tion of the frequency components of the fluid ve-locity and the frequency of its change of sign thatare necessary to guarantee good mixing. Here, re-searchers have to be careful when comparing “achaos method” with a regular mixing, since thespecific chaotic driving signal used can give spe-cific conclusion which is not necessarily true in gen-eral. Claiming that chaos uses less power has tobe carefully reviewed since for the same rms powerthe question really is how the power is effectivelydistributed to facilitate mixing. Intuitively chaoticdriving signals that have strong periodic and/or DC

components would not give very good results. So,the real question is how to get the appropriate mix-ture of high frequency and low frequency signals tostir a particular kind of fluid. Chaos may not be thekey point here. Anyway, as chaos contains a widespectral mix, it would be a good starting point toinvestigate [8].

Case Study 5:

Chaos was thought to help eliminate striation in flu-orescent lamps effectively. Specifically it has beenfound that a good low frequency mix of the spec-trum of the driving signal is crucial. If the par-ticular driving signal used contains sufficient low-frequency mix, striation in fluorescent lamps canbe eliminated. Chaos is basically irrelevant. Someexperiments have been reported [9].

4 MODELS FOR THE SAKE OF CON-

VENIENCE

In some engineering systems, exotic behaviors arefound theoretically and by simulations. Many au-thors managed to package them with enough chaosand bifurcation terminologies and surely get thempublished. The trouble is that these behaviorscould turn out to be only observable from the math-ematical models used to find them. In the param-eter ranges under study, however, the real systemsare not consistent with the mathematical modelsand hence do not exhibit such behaviors.

The use of invalid models for controlling engi-neering systems is common, especially in the con-trol literature. Control researchers first got themodels from engineers, assumed that the modelswere good, and derived control strategies based onthe models. However, very often, when real systemsexhibit chaotic behavior, the underlying mechanismcould be more complex than it would have been ob-served from the mathematical models. Typically,amplifier saturations leading to some forms of bor-der collision can create totally different kinds ofchaotic behavior. In some cases, the predicted bi-furcations simply do not show up in the real sys-tems. The following case studies illustrate somecommon inconsistencies between models and obser-vations.

Case Study 6:

Some simple voltage feedback dc-dc converters wereshown analytically and by simulation to exhibitperiod-doubling as a fast-scale bifurcation phe-nomenon [10]. But such a phenomenon never showsup in practice! The problem is with the assumptionof the use of proportional feedback in the analysis

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and simulation. In practice, feedback compensationnetwork is also designed based on some form of lowpass characteristic. This low pass characteristic de-stroys any high frequency signal that goes aroundthe loop. Period doubling at the switching periodis essentially a high frequency phenomenon whichrelies on signal feedback around the loop at a fre-quency as high as the switching frequency. Clearly,no period doubling would occur in practice becauseof the low pass characteristic of the feedback loop.

5 REPACKAGING WITH CHAOS

Some theories have been beautifully written forcontrol of chaos. The novelty of their use in real sys-tems is sometimes questionable, however, despitethe proliferating rate of publication in claiming thatchaos has been controlled in specific engineeringsystems. In fact, many engineering systems, whenunstable, are operating chaotically. The fact thatthey are controlled to operate in a regular regime ina normal industrial setting clearly shows that theexisting control schemes are controlling chaos quitewell. In many cases, the existing methods turn outto be most elegant. On the other hand, lack of en-gineering judgment and blind application of chaoscontrol methods may end up with absurd proposals.

Case Study 7:

The method of parametric resonance has beenshown in the mathematics and physics communi-ties to be an effective chaos control method. Basi-cally, application of the method involves injectinga small sinusoidal oscillation to a critical param-eter which causes the system to stabilize. Switch-ing dc-dc converters under a common current-modecontrol scheme are known to exhibit subharmonicsand chaos for duty cycles which are greater than0.5. Study has shown that applying the methodof parametric resonance to the reference inductorcurrent can stabilize the system. However, closerexamination reveals that the method of parametricresonance is actually equivalent to a ramp compen-sation which has been proposed some 30 years inthe power electronics community. Essentially, themethod of parametric resonance is simply a varyingramp compensation and can be very sensitive if thephase angle of the applied sine signal is not adjustedappropriately. On the other hand, the traditionalmethod of ramp compensation is a lot more robust.

Case Study 8:

It is not uncommon to see absurd chaos controlsolutions being derived for specific engineering sys-tems. When dc-dc converters are openloop, i.e.,

operating at a fixed duty cycle, they are alwaysperiod-1 stable. However, openloop systems arenever used in practice because of their inferiorityof transient response and line regulation. Someseemingly sophisticated mathematical derivation ofthe control equation, blending the system equationsand some chaos control methods, ends up with ascheme whereby the duty cycle is given by an ex-pression which is essentially constant. The claim ofstability is obviously valid because it is just equiv-alent to an openloop system; however, the methodis totally unacceptable by engineers.

6 CONCLUSION

In this talk we point out several problems in the formu-lation of solutions for practical engineering problems.Using a few case studies we illustrate the various com-mon pitfalls in the reporting of chaos applications inengineering. Researchers tended to “sell” chaos as a“panacea”, but often failed to look into the actual wayin which the particular chaos works in the particularsituation. We conclude this talk with the cartoon ofFig. 1: A drunk man was searching under the lamp polefor his key chain which he knew was dropped elsewhere.When asked why, he replied, “because it’s bright here.”

References

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