When we were asked by Prof. Michael Thompson, editor of the *Phil. Trans. R. Soc. A*, to organize a Theme Issue on the topic of ‘control of chaos’, we were glad to accept the invitation while knowing that the matter is intriguing and fashionable, yet not unequivocally defined in its precise meaning and contents.

Indeed, in the last 15 years, starting with the pioneering paper by Ott, Grebogi and Yorke (1990), the field of control of chaos, and more generally of nonlinear dynamics, has received tremendously increasing attention within a substantially multidisciplinary scientific community. Several books, surveys, journal issues, conference minisymposia, workshops and bibliographies have been devoted to the topic, along with a number of attempts to classify control goals and/or the techniques to attain them.

In this framework, the very basic notion of control of chaos has been given different meanings. In a strict sense, it has been interpreted as the simple removal of unwanted dynamic phenomena to be achieved with some effective control techniques. Such cases are often referred to in the literature as *suppression* of chaos, with the focus being on the effects of control rather than on the underlying skill of the adopted procedure. In contrast, according to a more general and modern viewpoint, it has been considered as the capability to *exploit typical properties of dynamical systems undergoing chaotic behaviours* in order to *control* the system response, where control now means either *suppressing* or *enhancing* chaos, or even *using* it, based on the specific goal of the considered technique, system and area of interest.

Indeed, many applications of chaos control to general mathematical, physical, biological and engineering systems occurring in different technical fields have been pursued, although relatively few studies have dealt with applications in mechanics.

Based on the previous points and interpreting the concept of ‘chaos control’ in the wider sense of exploiting chaotic properties for different possible purposes, with this Theme Issue, we aim at highlighting both the suppression and/or enhancement of complex dynamic phenomena in various systems, by means of different techniques, and the possibility to exploit the richness of dynamical systems theory to the aim of controlling, designing and monitoring engineering systems. All of these are considered, with some major emphasis on mechanics, but with the consciousness of how tight the connections with on-going studies in companion branches of science are.

Accordingly, we have collected (i) a number of review/survey articles highlighting the advancements of chaos control theory and applications in mathematical, physical and engineering sciences, also aimed at establishing a background scenario, and (ii) a number of articles by applied mathematicians, physicists and mechanicians on both methods and their applications to specific technical problems.

Consistent with the main topics appearing in the very title of this Theme Issue, its scientific content is hereinafter presented by organizing the papers into two broad groups.

## 1. Controlling (suppressing or enhancing) chaos

Although a few articles basically address the control of chaos within the classical framework of chaos suppression, most papers are actually based on maturely exploiting typical properties of nonlinear dynamical systems with the aim of *controlling* chaos within the perspective of either *suppressing* or *enhancing* it in the system at hand. This is the reason why they are presented and discussed all together in this section, which includes a group of basically review/survey articles (Fradkov *et al*., Pyragas, Chacon, Boccaletti & Bragard) aimed at summarizing methods/techniques and the relevant applications and a few papers dealing with specific control/anti-control methods or problems (Lenci & Rega, Ditto & Sudeshna Sinha, di Bernardo & Stoten, Subhash Sinha & Dávid, Chen & Shi, Ge *et al*.).

Both the theoretical and practical approaches are pursued, with the complementary aims of providing skilled and rationally consistent methods, and of looking for reliable and effective techniques. Furthermore, specific items, possibly of a technical nature, as well as general questions related to the overall dynamic response are considered. Each technique has its own range of practical interest, and the whole family of methods provides a cornucopia of possibilities suitable to attack many different cases.

*Feedback* (or close-loop) and *non-feedback* (or open-loop) control techniques are addressed. The former basically rely on two major ingredients of a chaotic system, namely (i) its accessibility to many different unstable periodic orbits, any of them possibly corresponding to a desired system's performance according to some criterion, due to system ergodicity and (ii) its sensitivity to small perturbations, due to the system dependence on small changes of its current state. In turn, the latter can exploit different properties of a generic dynamical system, such as the occurrence of global bifurcations with meaning of a prerequisite for chaos or other complex dynamic events, with the aim of influencing the system behaviour by properly modifying, for example, the excitation features.

All of these techniques were originally developed to deal with suppression of chaos. Yet, making an originally non-chaotic dynamical system chaotic, or enhancing the existing chaos of a chaotic system (so-called *anti-control* of chaos) has become a meaningful theoretical and practical goal in the last few years due to the great potential of chaos in many non-traditional applications in electronics, communications, information technology, mechanics, optics, biology and medicine.

Fradkov *et al*. survey the state-of-the-art of control of chaos (in the sense of suppression, but also possibly of enhancement), reporting on both typical (stabilization of an unstable orbit) and more subtle (modifying or delaying a chaotic attractor, changing its location) goals, and stressing the interest towards achieving them by means of small control actions. The first part of the paper addresses major *control methods*, discussing features and some relevant open-problems. (i) Feedforward (or non-feedback) control, which is based on applying a properly chosen periodic signal and is attractive, because it needs no measurement or extra-sensors. (ii) The ‘Ott–Grebogi–Yorke (OGY) method’, which is the most popular feedback control technique, using a discrete system model based on a linearization of Poincaré map for controller design, and applying the control action only at time instants when the motion returns to the neighbourhood of the orbit to be stabilized due to the recurrent property of chaotic dynamics. (iii) The ‘Pyragas’ (or time-delayed feedback) method, which makes use of a control signal obtained from the difference between the current state of the system and the state delayed by one period of an unstable orbit. (iv) Linear, nonlinear and adaptive control approaches, that are grouped into two large classes, namely Lyapunov approaches (speed-gradient, passivity-based methods) and compensation approaches (feedback linearization, geometric methods, etc.), noticing how proper use of modern control theory to handle realistic problems in control of chaos is yet to be undertaken. (v) Oscillation and chaos control in infinite-dimensional (i.e. spatially extended or time-retarded) systems, whose methods are mainly based upon ideas developed for finite-dimensional systems, though specific spatio-temporal control goals also occur. The second part of the paper is devoted to extensively surveying control of chaos problems in *mechanics* and *mechanical engineering*. Reference is made to pendulums, beams and plates, stick-slip friction motion, impacting systems, spacecrafts, vibroformers, microcantilevers, ship oscillations, robot-manipulator arms, earthquake engineering applications, milling, whirling motion under mechanical resonance and mechanical systems with clearance.

In a comprehensive review paper, Pyragas specifically addresses the main features of the time-*delayed feedback* control (DFC), along with some relevant implementations and recent advancements. As already said, DFC makes use of a control signal obtained from the difference between the current state and the state of the system delayed by one period of an unstable orbit, so that the signal vanishes when the target orbit is stabilized. The method does not require exact knowledge of the form of the periodic orbit or the system equations. The system under control automatically settles onto the target periodic motion based on only the knowledge of the relevant period, and the stability of motion is maintained with only tiny perturbations. After listing experimental implementations, applications of the method to theoretical models from different fields and its most important modifications, the author presents the basic theory of DFC, which involves rather intricate nonlinear delay-differential equations and the relevant linear stability analysis. Using the relationship between the Floquet spectra of the system controlled by the proportional and time-delayed feedback techniques allows one to evaluate the optimal parameters of the controller without an explicit integration of time-delay equations. The extended version of DFC can stabilize unstable periodic orbits of higher periods and with larger Floquet exponents. The author also considers the main limitation of the (E)DFC method, namely the inability to stabilize orbits with an odd number of real positive Floquet exponents. This limitation is overcome by artificially enlarging the set of real multipliers greater than unity to an even number, through the introduction of an unstable degree of freedom into a feedback loop.

In the framework of *non-feedback* chaos control, the paper by Chacon summarizes some main theoretical results concerned with the applications of Melnikov's method to control of homoclinic/heteroclinic chaos in low-dimensional, non-autonomous and dissipative oscillators by weak harmonic perturbations. Techniques using either parametric or external chaos-suppressing excitation in some main resonance conditions with the chaos-inducing excitation are addressed in a unified setting, making reference to a generic class of systems satisfying the requirements of Melnikov's method and including constant forces and time-delay terms, as well as to paradigmatic oscillators. The important role played by the initial phase of the chaos-controlling excitation in the suppression scenario is discussed along with the associated range of excitation amplitudes, reporting on the optimal choice of suppressory parameters. Further developments concerned with, for example, the case of incommensurate chaos-suppressing excitations are also addressed. Applications are reported as regards taming chaotic escape from a potential well, spatio-temporal chaos and chaotic solitons in chains of identical chaotic-coupled oscillators and chaotic-charged particles in the field of an electrostatic wave packet. Use of the same techniques for enhancement or maintenance of chaos, and their connections with the results on chaos suppression are also discussed.

Systematic theoretical investigations of *control/anti-control* of the nonlinear dynamics of a system from theoretical and applied mechanics are pursued by Lenci and Rega, who deal with the overturning of a rocking rigid block subjected to a generic periodic excitation at the base. Attention is focused on two meaningful relevant thresholds in the excitation parameters space, namely the heteroclinic bifurcation of the hilltop saddle and the system immediate overturning, which represent lower and upper bounds, respectively, of the region where impending toppling does occur. The control technique, previously investigated in a unified framework of suppression of undesired dynamic/chaotic events in different mechanical systems, is based on properly modifying the shape of the excitation and is herein applied also to the anti-control. For the rigid block, it involves either adding proper superharmonics to a reference harmonic excitation, with the aim of increasing either one of the two thresholds (control), or looking for the worst excitation shape and phase entailing their corresponding decrease, i.e. the enhancement of the undesired dynamical event (anti-control), with the purpose of using it for safe system design against unknown excitations (e.g. earthquake). The two opposite control/anti-control problems are investigated in depth, by determining the optimal excitations allowing for maximum variations of either threshold. The notions of ‘global’ and ‘one-side’ control/anti-control are used, and their different importance is discussed for the various cases. The effects of control (anti-control) of one curve on the uncontrolled (non-anti-controlled) curve are also investigated, both analytically and with detailed numerical charts.

Boccaletti and Bragard discuss some issues related to the control of space–time chaotic states in *spatially extended* systems, where many proposed schemes using space extended perturbations are not straightforwardly implementable for experimental verifications. For such systems, the most important question is to assess whether the perturbation has actually to be extended in space. The authors review some results with reference to the one-dimensional complex Ginzburg–Landau equation, which provides a reduced universal description of weakly nonlinear spatio-temporal dynamics in extended continuous media close to a Hopf bifurcation, and can be considered as a good model equation in many physical situations including laser physics, fluid dynamics, chemical turbulence, bluff body wakes and arrays of Josephson's junctions. With the aim of stabilizing any of the unstable plane wave solutions in a given system regime through the Pyragas control scheme, the authors show how it is unnecessary to apply control to all points of the system, but just to rely on a finite number of local controllers, whose density is not influenced by the system extension. Furthermore, they answer questions about (i) how strong the applied forcing must be in order to drive the system to a regular behaviour; (ii) the required minimal distance between two adjacent controllers; and (iii) the time to wait in order to restore a regular dynamics from a chaotic one. Some open problems are overviewed, e.g. whether a further size increase of the system can compromise the ability of control or whether selecting equally spaced controllers actually represents an optimal choice for achieving stabilization.

Ditto and Sudeshna Sinha discuss an easily implementable control strategy, namely the use of a simple threshold mechanism to limit the dynamic range of a state variable and to control chaotic systems onto stable fixed points and limit cycles of widely varying periodicity. The method only involves monitoring and occasional resetting of a single variable, and leads to clipping desired time sequences and enforcing a periodicity through the thresholding action. The authors' attention is actually devoted towards highlighting flexible applications of the strategy for computer architecture design, as will be discussed in §2.

Di Bernardo and Stoten overview the adaptive *minimal control synthesis* (MCS) algorithm which aims at tracking the evolution of a given reference model and has been successfully used to control chaotic systems, as well as a number of its effective extensions (extended MCS, purely adaptive EMCS) designed to rely on minimal knowledge of plant parameters. These improved control strategies are aimed at effectively exploiting properties such as boundness and ergodicity of chaotic attractors in order (i) to obtain a rigorous proof of global asymptotic stability of the error dynamics under the presence of rapidly varying disturbances; and (ii) to solve global control problems by using local control techniques. After illustrating the methodology on a classical control problem of a chaotic Duffing oscillator to a periodic solution, chaos-based control is used to successfully and robustly swing-up a double pendulum in its unstable equilibrium position (in the absence of any extra torque between the pendulum links) by means of a local adaptive MCS controller complemented with an induced chaotic swing-up transient phase. The ergodicity of the latter is exploited to stabilize the inverted position.

While most work in chaos control literature deals with autonomous systems or the conversion of a non-autonomous system into an autonomous one, Subhash Sinha and Dávid present techniques for local chaos control in nonlinear *time-periodic systems*. Three strategies are addressed and exemplified. Two of them aim at eliminating chaos from a given range of system parameters by driving the system to a desired periodic orbit using either a linear or a nonlinear feedback control technique. In the former, a linear full state feedback controller is designed by symbolic computation of the Floquet transition matrix associated with the linear part of the system. The latter is based on feedback linearization, which has no previous applications to nonlinear periodic systems. A set of coordinate transformations is introduced to convert the original nonlinear control problem to a dynamically equivalent linear problem. The controller can then be designed by the symbolic approach or by other existing linear methods, and transformed back into the original state variable, where it is nonlinear. Advantages of this type of control over linear techniques are that full state feedback is not required, and that it can provide an increased region of stability. The third strategy differs from the previous ones by employing the concept of nonlinear bifurcation control, which is extended to time-periodic systems. The goal is to modify the nonlinear characteristics of a local bifurcation (such as stability, size or rate of growth of a limit cycle) along the route to chaos, such that the onset of chaos is delayed. The method is based on construction of dynamically equivalent time-invariant normal forms, and involves periodic centre manifold reduction, time-dependent normal form theory, and construction of versal deformations. A purely nonlinear feedback controller is designed in the transformed domain for the equivalent autonomous system, and then transformed back to the original coordinates. All the methods can be applied to systems with no restrictions on the size of periodic terms.

Chen and Shi present an introductory mathematical survey of recent works made within the first author's group on the anti-control of chaos in discrete dynamical systems, whose chaotic/bifurcating behaviours have been found useful in a number of real-world applications. In an effort to show that the dynamics generated by anti-control procedures are indeed chaotic in a rigorous mathematical sense, the authors provide some necessary preliminaries on the concept and criteria of discrete chaos in the sense of Devaney and of Li-Yorke (for one-dimensional maps), also reporting on generalizations to *n*-dimensional settings given by Marotto, and to Banach spaces and complete metric spaces made by the authors. Then they describe the state-of-the-art achievement and progress on *chaotification*, by presenting a generic algorithm based on state-feedback control for general finite-dimensional discrete-time dynamical systems (also in Banach spaces), as well as for continuous maps, showing how the controlled system is chaotic according to previous definitions and theorems. In highlighting how chaos anti-control (and control) calls for innovative efforts mostly from scientists and engineers with expertise in both nonlinear dynamics and nonlinear control systems, the authors stress the need to design easily implementable and user-friendly chaotifiers, which must be simpler and less expensive than the system to be controlled.

Along a substantially heuristic and technically oriented research line, Ge *et al*. deal with anti-control of chaos in *electro-mechanics*, presenting an extended computer simulation analysis of the effects of different implementations of a procedure for anti-controlling chaos in a single time-scale brushless DC motor. The procedure consists in adding to one of the system third-order equations an external excitation—which is either constant or periodic with different shapes—aimed at enhancing the occurrence of chaos. Several numerical tools, including phase portraits, bifurcation diagrams and Lyapunov exponents, are used to show that anti-control of chaos can be effectively achieved.

## 2. Using chaos

By interpreting the notion of chaos control/anti-control in the broad sense of exploiting dynamical systems theory with the aim of *controlling, designing and monitoring engineering systems*, this section reports on a number of papers mostly devoted to skilfully using ‘chaotic’ properties for achieving some technical requirement in a given application area. Accordingly, the focus is, on one side, on the most suitable skill embedded in the system dynamics to be profitably exploited, and, on the other side, on the specific technical goal to be accomplished. While pioneering applications of technological use of chaos were mostly concerned with non-mechanical problems (e.g. for synchronization purpose in communications, or for biomedical applications), a number of situations are herein exemplified where complex dynamics is fruitfully used for mechanical purposes.

Flexible and efficient exploitation of the complexity inherent in the chaotic dynamics for technological uses is clearly highlighted by Macau and Grebogi, who deal with the relevancy of the OGY strategy of chaos control to *spacecraft steering*. The authors review how it is possible to have an efficient chaos control and to derive targeting strategies to be applied also to Hamiltonian situations, as those occurring in astrodynamical problems related to spacecraft transfer. Stabilization of chaotic trajectories and chaotic targeting are addressed, the latter being based on intelligently exploiting the exponential sensitivity of chaotic time evolution to perturbations in order to direct trajectories towards a desired state in the shortest possible time. The chaotic targeting method is applied to Hamiltonian systems representing paradigm models for chaos in astrodynamics: the restrict circular three body problem modelling the dynamics of a spacecraft moving in the Earth–Moon system and the Hill problem governing the encounter phenomenon of two light bodies describing circular orbits around a heavy central body. In the former case, the sensitive dependence on initial conditions is exploited for efficient (low-energy) spacecraft transfer by using just small perturbations or very low intense thrust. In the latter case, the unstable periodic orbits embedded in the chaotic invariant saddle related to chaotic scattering are exploited for in-orbit stabilization, namely to keep one satellite in orbit around the other.

Ditto and Sudeshna Sinha report on how thresholding, based on its low complexity and no run-time costs, allows one to exploit the richness of chaos in a direct and efficient way, and is especially useful in situations where one wishes to design controllable components that can switch flexibly between different behaviours. Using varying thresholds on the same chaotic physical element (the basic computing unit) can allow one to emulate all of the different logic gates, which are the necessary and sufficient components of a universal computing machine. These are very desirable features of a *dynamic computer architecture* wherein, unlike the existing paradigms of conventional static architecture that try to achieve flexibility by flexible wiring, the computing elements exhibit inherent flexibility and reconfigurable capability. This paves the way towards a more mature approach for flexibly dynamic computational platforms, based on the principle of large numbers of identical, reconfigurable and reprogrammable units.

Two papers exploit *nonlinear time-series* analysis to investigate *damage problems in structural mechanics*. Yin and Epureanu deal with application of chaos theory to *structural health monitoring*, and use attractor-based features derived from nonlinear time-series analysis for identifying damage. With respect to the currently monitored features based on linear approaches—which often exhibit difficulties in detecting single/multiple damages of small level—approaches exploiting system nonlinearities provide enhanced accuracy and sensitivity of detection. The authors propose a new attractor morphing technique, based on extracting sensitivity vectors determined by analysing short-term variations between two nearby trajectories corresponding to the healthy and damaged systems. The extracted features succeed in indicating the location, level and extent of damage. In addition, proper orthogonal decomposition is used to identify (i) the dominant coherent structures present in the dynamics of high-dimensional systems, which allow a reduction of the number of measurements needed for the relevant analysis and (ii) the number of linearly independent shapes among the extracted features associated with the attractor changes caused by distinct damages, which provides a basis for detecting multiple damages. The viability and performance of the technique is demonstrated for a Duffing oscillator and a thermo-shielding panel forced by unsteady aerodynamic loads.

In turn, Chelidze and Cusumano, motivated by the interest in tracking the real time evolution of damage from its earliest stage, present a new dynamical systems approach to data analysis capable of tracking slowly evolving ‘hidden’ variables responsible for long time-scale non-stationarity in a fast subsystem by using only relevant measurements. Contrary to the attractor-based time-series approaches based on estimating invariant quantities (e.g. Lyapunov exponents or generalized dimensions), which may be adequate for detecting system sudden changes, the method is based on the concept of *phase space warping*. This approach aims at tracking the small distortions occurring in the fast subsystem's vector field as a result of an underlying slowly evolving process, by using the short-time reference model prediction error of a nonlinear reference model. Both the basic theory and the issues associated with its algorithmic implementation are discussed. A vector-tracking version of the procedure, based on smooth orthogonal decomposition analysis, is applied to the study of a nonlinear vibrating beam experiment in which a crack propagates to complete fracture, showing that the damage evolution is governed by a scalar process. The identified damage evolution model is able to provide accurate, real-time estimates of the current damage state and to predict the time of failure well in advance of the actual complete fracture of the beam.

The presentation of the contents of this Theme Issue ends with the modern, nonlinear dynamics based, attack of an old, well-documented, mechanical problem proposed by Moon and Steifel in a paper on coexisting chaotic and periodic dynamics in *clock escapements*. The background authors' position is that proper examination of real machine systems is likely to reveal low levels of self-generated, chaotic-looking, noise that in many cases may have beneficial effects on the dynamic machine performance. They examine the nature of regular and noisy dynamics in a specific class of complex multi-body machines, i.e. the clock escapement, from experimental, historical and analytical points of view. Experiments on two escapement mechanisms from the nineteenth century Reuleaux kinematic collection at Cornell University are used to illustrate chaotic-like noise in clocks, whose vibrations coexist with the periodic dynamics of the balance wheel or pendulum. A mathematical model which exhibits coupling between escapement and supporting structure is used to show how self-generated chaos in clocks can break the dry friction in the gear train and allow for satisfactory system operation, as well as to highlight the occurrence of a strange attractor in the structural vibration of the clock. While referring to the popular image of the clock as of a highly regular machine, the authors report on the blend of mathematical analysis and practical experience historically put into play from several clock experts to address the sources of irregularities in clocks, though, of course in a context of limited scientific consciousness of the observed phenomena. Yet, in the modern perspective, the internal feedback between oscillator and escapement structure, which allows for correct clock operations, looks similar to the anti-control mechanism of chaos models.

To summarize, it is felt that the papers in this Theme Issue do address, in various manners and within different perspectives, several aspects of the matter. Thus, while sharing some common roots, they succeed in covering a broad range of interests and disciplines, well beyond the underlying mechanical framework.

We end by hoping that this Theme Issue, which fruitfully brought together scientists with different backgrounds but common scientific interests, will also be a tool for disseminating interest towards the fascinating matter of control, enhancing and use of chaos in various fields of engineering and science.

Last, but not least, we express our most sincere thanks to all of the authors for accepting to contribute to the issue, as well as to the journal editor for proposing and hosting it.

- © 2006 The Royal Society