Control of Chaos

Duffing oscillator with delayed feedback is widely used in engineering. Chaos in such system plays an important role in the dynamic response of the system, which may lead to the collapse of the system. Therefore, it is necessary and significant to study the chaotic dynamical behaviors of such systems. Chaotic dynamics of the Duffing oscillator subjected to periodic external and nonlinear parameter excitations with delayed feedback are investigated both analytically and numerically in this paper. With the Melnikov method, the critical value of chaos arising from heteroclinic intersection is derived analytically. The feature of the critical curves separating chaotic and nonchaotic regions on the excitation frequency and the time delay is investigated analytically in detail. Under the corresponding system parameters, the monotonicity of the critical value to the excitation frequency, displacement time delay, and velocity time delay is obtained rigorously. The chaos threshold obtained by the analytical method is verified by numerical simulations.

We report a new type of chaos synchronization:inverse anticipating synchronization, where a time delay chaotic system x can drive another system y in such a way that the driven system anticipates the driver by synchronizing with its inverse future state:y(t) = -x(t + τ ),τ > 0. We extend the concept of inverse anticipating chaos synchronization to cascaded systems. We propose means for the experimental observation of inverse anticipating chaos synchronization in external cavity lasers.

The control of transport due to electrostatic turbulence is investigated using test-particle simulations. We show that a barrier for the transport, that is, a region where transport is reduced, can be generated through the randomization of phases of the turbulent field. This corresponds to the annihilation of coherent structures which are present at all scales, without actually suppressing turbulence. When the barrier is active, a flux of particles towards the center of the simulation box is present inside the region where the barrier is located.

We predict theoretically that it is possible to stabilize the steady state in multimode, intracavity doubled, diode pumped Nd:YAG ͑neodymium-doped yttrium aluminum garnet͒ lasers using two output signals, namely, the sum intensities of the infrared laser modes polarized in two different orthogonal directions (X and Y ) and one feedback input parameter, the pump rate. The stabilization is possible for arbitrarily large numbers of modes polarized in the X and Y direction. Different strategies of stabilization based on proportional feedback, derivative control, and their combination are discussed. The analytical and numerical results of the linear control theory are illustrated with numerical simulations of the underlying nonlinear differential equations. We show that one can maintain the stable steady state of the laser output for an arbitrarily large pump rate by taking advantage of a tracking procedure.

The widespread phenomena of multistability is a problem involving rich dynamics to be explored. In this paper, we study the multistability of a generalized nonlinear forcing oscillator excited by f (x)cosωt. We take Doubochinski's Pendulum as an example. The so-called "amplitude quantization", i.e., the multiple discrete periodical solutions, is identified as self-adaptive subharmonic resonance in response to nonlinear feeding. The subharmonic resonance frequency is found related to the symmetry of the driving force: odd subharmonic resonance occurs under even symmetric driving force and vice versa. We solve the multiple periodical solutions and investigate the transition and competition between these multi-stable modes via frequency response curves and Poincare maps. We find irreversible transition between the multistable modes and propose a multistability control strategy.

Time reversal of quantum dynamics can be achieved by a global change of the Hamiltonian sign (a hasty Loschmidt daemon), as in the Loschmidt Echo experiments in NMR, or by a local but persistent procedure (a stubborn daemon) as in the Time Reversal Mirror (TRM) used in ultrasound acoustics. While the first is limited by chaos and disorder, the last procedure seems to benefit from it. As a first step to quantify such stability we develop a procedure, the Perfect Inverse Filter (PIF), that accounts for memory effects, and we apply it to a system of coupled oscillators. In order to ensure a many-body dynamics numerically intrinsically reversible, we develop an algorithm, the pair partitioning, based on the Trotter strategy used for quantum dynamics. We analyze situations where the PIF gives substantial improvements over the TRM.

This is a review of global sensitivity indices that were introduced in I.M. Sobol ' (1990). These indices allow to analyze numerically the structure of a nonlinear function defined analytically or by a "black box". As an example the Brownian bridge is considered and an example of the application of global sensitivity indices in finance is presented.

We analyze nonlocally coupled networks of identical chaotic oscillators with either time-discrete or time-continuous dynamics (Henon map, Lozi map, Lorenz system). We hypothesize that chimera states, in which spatial domains of coherent (synchronous) and incoherent (desynchronized) dynamics coexist, can be obtained only in networks of nonhyperbolic chaotic systems and cannot be found in networks of hyperbolic systems. This hypothesis is supported by analytical results and numerical simulations for hyperbolic and nonhyperbolic cases. 1 These systems are reputed hyperbolic because of non-existence of stable regimes and its bifurcations.

This paper is meant to serve as a tutorial describing the link between symbolic dynamics as a description of a chaotic attractor, and how to use control of chaos to manipulate the corresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenz attractor, in the form of the discrete successive maxima map of the z-variable time-series, as our main example. For the ¯rst time, here, we use this oscillator as a chaotic signal carrier. We review the many previously developed issues necessary to create a working control of symbol dynamics system. These include a brief review of the theory of symbol dynamics, and how they arise from the °ow of a di®erential equation. We also discuss the role of the (symbol dynamics) generating partition, the di±culty of ¯nding such partitions, which is an open problem for most dynamical systems, and a newly developed algorithm to ¯nd the generating partition which relies just on knowing a large set of periodic orbits. We also discuss the importance of using a generating partition in terms of considering the possibility of using some other arbitrary partition, with discussion of consequences both generally to characterizing the system, and also speci¯cally to communicating on chaotic signal carriers. Also, of practical importance, we review the necessary feedback-control issues to force the °ow of a chaotic di®erential equation to carry a desired message.

Targeting control of chaos is concerned with taking advantage of sen- sitive dependence to initial conditions to coax a dynamical system to following a desirable trajectory. In other words, it is taking advantage of the butterfly effect so that the rich spectrum of possible trajectories embedded within a chaotic attractor can be selected with extremely small energy input. We review historical and pop- ular approaches which fall under this general area in an attempt to reveal these techniques in a useful manner for applied scientists.

We use control of chaos to encode information into the oscillations of the Belousov-Zhabotinsky reaction. An arbitrary binary message is encoded by forcing the chaotic oscillations to follow a specified trajectory. The information manipulating control requires only small perturbations to vary the binary message. In this paper we extend our recent theoretical work [Bollt and Dolnik, Phys. Rev. E 64, 1196 (1990)] by introducing a new and simplified encoding technique which can be utilized in the presence of experimental noise. We numerically and theoretically study several practical aspects of controlling symbol dynamics including: modeling noisy time-series, learning underlying symbol dynamics, and evaluation of derivatives for control by observing system responses to an intelligent and deliberate sequence of input parameter variations. All of the modeling techniques incorporated here are ultimately designed to learn and control symbol dynamics of experimental data known only as an o...

This paper is meant to serve as a tutorial describing the link between symbolic dynamics as a description of a chaotic attractor, and how to use control of chaos to manipulate the corresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenz attractor, in the form of the discrete successive maxima map of the z-variable time-series, as our main example. For the first time, here, we use this oscillator as a chaotic signal carrier. We review the many previously developed issues necessary to create a working control of symbol dynamics system. These include a brief review of the theory of symbol dynamics, and how they arise from the flow of a differential equation. We also discuss the role of the (symbol dynamics) generating partition, the difficulty of finding such partitions, which is an open problem for most dynamical systems, and a newly developed algorithm to find the generating partition which relies just on knowing a large set of periodic orbits. We ...

This research deals with the synthesis of control law by means of Analytic Programming (AP) for the Hénon Map, which is discrete chaotic system. The tool for symbolic regression (AP) is used for the purpose of stabilization of stable state and higher periodic orbits, which represent oscillations between several values of chaotic system. For experimentation, Self-Organizing Migrating Algorithm (SOMA) with AP and Differential Evolution (DE) as the second algorithm for meta-evolution were used.

This keynote speech discusses mutual intersection of evolutionary techniques and deterministic chaos. Three possible application of evolutinary computation on deterministic chaos are discussed here. The first one discuss chaos synthesis by means of evolutinary computation and the second one chaos control of simple chaotic system. The last, third, part discus use of evolutionary computation on spatiotemporal chaos control.

This contribution deals with optimization of the control of chaos by means of evolutionary algorithms. The main aim of this work is to show that evolutionary algorithms are capable of optimization of chaos control and to show several methods of constructing the complex targeting cost function leading to satisfactory results. As a model of deterministic chaotic system the two dimensional Henon map was used. The optimizations were realized in several ways, each one for another cost function or another desired periodic orbit. The evolutionary algorithm Self-Organizing Migrating Algorithm (SOMA) was used in four versions. For each version, simulations were repeated several times to show and check robustness of used method and cost function. At the end of this work the results of optimized chaos control for each designed targeting cost function are compared.

An inherent part of evolutionary algorithms, that are based on Darwin theory of evolution and Mendel theory of genetic heritage, are random processes. In this participation, we discuss whether are random processes really needed in evolutionary algorithms. We use n periodic deterministic processes instead of random number generators and compare performance of evolutionary algorithms powered by those processes and by pseudo-random number generators. Deterministic processes used in this participation are based on deterministic chaos and are used to generate periodical series with different length. Results presented here are numerical demonstration rather than mathematical proofs. We propose that a certain class of deterministic processes can be used instead of random number generators without lowering of evolutionary algorithms performance.

The chaotic behavior of a nonlinear damped three-well F 6 -Van der Pol oscillator under external and parametric excitations is studied in detail by means of Melnikov's analysis. The predictions are tested against numerical simulations based on the basin of attraction of initial conditions. It is found that the threshold for the onset of chaos in the system increases as the amplitude of the parametric excitation decreases and increases as the amplitude of the external forcing increases. Additionally, the onset of chaos is studied through analysis of the Poincareḿ ap, a construction of the bifurcation diagrams and a computation of the maximal Lyapunov exponent.

We study in this paper the dynamics and chaos control of a nonlinear electromagnetic seismometer system consisting of an extended Duffing electrical oscillator magnetically coupled with a natural Duffing mechanical oscillator. The singular perturbation method is used to find slow solutions. Some bifurcation structures and the variation of the corresponding Lyapunov exponent are obtained. Transitions from a regular behavior to chaotic orbits are seen to occur for large amplitudes of the external excitation. We also examine the application of a simple adaptive damping feedback controller to eliminate the chaotic behavior in a controlled extended Duffing system. The main idea is to regulate the chaotic motion of the electromagnetic seismometer system around less complex attractors, such as equilibrium points and periodic orbits. The effectiveness and efficiency of the proposed feedback control strategy is illustrated by means of numerical simulations.

The present work aims to employ evolutionary algorithms (EAs) to optimize an industrial chemical process. A unique combination of the simplified fundamental theory and direct hands-on computer simulation is used to present the modeling of a dynamic chemical engineering process in a highly understandable way. The main aim is to use them for analysis of dynamical system behaviour, especially of a given chemical reactor. A non-linear mathematical model is required to describe the dynamic behaviour of batch process; this justifies the use of evolutionary method of the EAs to deal with this process. Two algorithms -differential evolution and self-organizing migrating algorithm are used in this investigation. Differential Evolution is an evolutionary optimization technique which is exceptionally simple, significantly faster & robust at numerical optimization and is more likely to find a function's true global optimum. SOMA is also robust algorithm in sense of global extreme searching. In this way, in order to optimize the process, the EAs code is coupled with the rigorous model of the reactor. Both algorithms (SOMA, DE) have been applied 100 times in order to find the optimum of process parameters and the reactor geometry. The results show that the EAs are used successfully in the process optimization.

We present an experimental and theoretical study of the chaotic ionization of quasi-onedimensional potassium Rydberg wavepackets via a classical phase-space turnstile mechanism. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study explicitly illuminates their relevance to atomic ionization. We create time-dependent Rydberg wavepackets, subject them to alternating applied electric-field pulses, and measure the electron survival probability. Ionization depends not only on the initial electron energy, but also on the classical phase-space position of the electron with respect to the turnstile -that part of the electron packet inside the turnstile ionizes after the applied ionization sequence, while that part outside the turnstile does not. The survival data thus encode information on the geometry and location of the turnstile, and are in good agreement with theoretical predictions.

We investigate complex spatio-temporal dynamics in the framework of a generic model describing nonlinear semiconductor or electrochemical systems with S-shaped current-voltage characteristics. Global time-delayed feedback control is applied to stabilize unstable spatio-temporal patterns which are created in a subcritical Hopf bifurcation. This demonstrates the invalidity of the alleged odd-number limitation of time-delayed feedback control in systems with spatial degrees of freedom.

The research on synchronization and anti-synchronization of fractional order chaotic oscillators have innumerable applications in various aspects of life. So, this research paper examines synchronization and anti-synchronization of fractional order chaotic memristive oscillators evolving from different initial conditions using active control techniques. Using the chaotic fractional order memristive oscillators as a paradigm, stable synchronization control variables are obtained via the active control technique. The obtained synchronization control variables enable the drive and response state variables to achieve identical dynamics despite starting from different initial conditions. The stability of the synchronization control variables are validated through numerical simulations. The result obtained has application in secure information transmission and separation of particles.

Chaos synchronization using a continuous chaos control method was studied in two identical chaotic laser systems consisting of semiconductor lasers and optical feedback from an external mirror. Numerical calculations for rate equations indicate that the stability of chaos synchronization depends significantly on the external mirror position. We performed a linear stability analysis for the rate equations. Our results show that the stability of the synchronization is much influenced by the mode interaction between the relaxation oscillation frequency of the semiconductor laser and the external cavity frequency. Due to this interaction, an intensive mode competition between the two frequencies destroys the synchronization, but stable synchronization can be achieved when the mode competition is very weak.

For low-dimensional chaotic attractors there is usually a single number of unstable dimensions for all of its periodic orbits and we can say such attractors exhibit "mono-chaos". In high-dimensional chaotic attractors, trajectories are prone to travel through quite different regions of phase space, some far more unstable than others. This heterogeneity makes predictability even more difficult than in low-dimensional homogeneous chaotic attractors. A chaotic attractor is "multi-chaotic" if every point of the attractor is arbitrarily close to periodic points with different numbers of unstable dimensions. We believe that most physical systems possessing a high-dimensional attractor are of this type. We make three conjectures about multi-chaos which we explore using three twodimensional paradigmatic examples of multi-chaotic attractors. They can be thought of as smallscale examples that give insight for real high-dimensional phenomena. We find a single route from mono-chaos to multi-chaos if an attractor changes continuously as a parameter is varied. This multichaos bifurcation (MCB) is a periodic orbit bifurcation; one branch of periodic orbits is created with a number of unstable dimensions that is different from the mono-chaos.

This paper presents several phase synchronization phenomena on multi-state oscillators coupled by inductors as a ring. Each oscillator circuit can individually behave some periodic oscillations (limit cycles) in the same parameters asynchronously. In this study, a coupled oscillator circuit which is composed by some oscillators is proposed and classification of phase synchronization modes is investigated. In numerical simulation, many types of phase synchronization modes are confirmed in the proposed systems.

In this study, we investigate behaviors of two identical coupled Chua's circuits whose parameters periodically varying in associated with each of the period of internal state values. For the periodically varying of the parameters, odd-order cycles do not exist in this system. We observed coexistence of lots of attractors which have different orders of cycles and different synchronization states.

In this study, we propose two chaotic oscillators coupled via switch. The chaotic oscillators are auto gain controlled oscillators containing time delay. The oscillators have feedback systems which control the gain. The coupling switch is placed between the feedback systems which control both of the oscillators. The coupling switch connects alternately with one subcircuit and the other with a fixed time interval. We carry out computer calculations and circuit experiments for two coupled auto gain controlled oscillators containing time delay and investigate synchronization phenomena. In the computer calculations, we observe coexistence phenomena of in-phase synchronization and opposite-phase synchronization. In the circuit experiments, we observe coexistence phenomena of in-phase quasi-synchronization and oppositephase quasi-synchronization. Finally, to investigate the difference between the numerical results and the experimental results, we investigate effects of parameter mismatch and noise.

In our previous study, we observed the interesting phenomenon. It is that the synchronous rate of the whole system increases in spite of increasing parameter mismatches in the system. In this study, in order to verifier the this phenomenon, an asymmetrical coupled system is proposed and investigated. The system is realized by connecting chaotic subcircuits and van der Pol oscillators. In the case of five subcircuits, we confirmed synchronous phenomena in computer calculations. Additionally, It was confirmed that synchronous rates of chaotic subcircuits are increased by increasing a parameter mismatch rate of van der Pol oscillators. We consider that this result is corresponding to results of previous study.

In this study, two Chua's circuits with lossless transmission lines placed in parallel are investigated. The effect of the crosstalk is modeled by the connections via coupling capacitors or mutual inductors. By computer simulations, we confirm that two Chua's circuits are synchronized in in-pahse or anti-phase by the crosstalk effect of the transmission lines.

An oscillator including a chaotic system is an important device and one of essential component in the natural world to solve a mechanism of a nonlinear dynamics of network. In this study, a simple chaotic circuit with three states of both chaotic oscillation and limit cycles which is named multi-state chaotic circuit is proposed. Synchronization phenomena and complex behavior on a network system of the multi-state chaotic circuits coupled by inductors are investigated. Several interesting chaotic phenomena of spatio-temporal behavior have been observed in the coupled network system.

A large number of coupled oscillators are useful as models for a wide variety of systems in natural fields. In this study, wave propagation generated in a ladder of cross-coupled chaotic circuits are investigated. For the case of 20 circuits, interesting wave propagation phenomena of phase states are found. Computer simulations show that this coupled system produces several wave propagation.

This paper presents several phase synchronization phenomena on multi-state oscillators coupled by inductors as a ring. Each oscillator circuit can individually behave some periodic oscillations (limit cycles) in the same parameters asynchronously. In this study, a coupled oscillator circuit which is composed by some oscillators is proposed and classification of phase synchronization modes is investigated. In numerical simulation, many types of phase synchronization modes are confirmed in the proposed systems.

In this study, phase synchronization observed in a ring of chaotic circuits coupled by resistors is analyzed. For the case that an odd number (2N +1) of the circuits are coupled, a kind of frustration is caused and (2N +1)-phase synchronization of chaos is stably generated.

In our previous study, we observed the interesting phenomenon. It is that the synchronous rate of the whole system increases in spite of increasing parameter mismatches in the system. In this study, in order to verifier the this phenomenon, an asymmetrical coupled system is proposed and investigated. The system is realized by connecting chaotic subcircuits and van der Pol oscillators. In the case of five subcircuits, we confirmed synchronous phenomena in computer calculations. Additionally, It was confirmed that synchronous rates of chaotic subcircuits are increased by increasing a parameter mismatch rate of van der Pol oscillators. We consider that this result is corresponding to results of previous study.

In this study, we investigate synchronization of parametrically excited van der Pol oscillators. By carrying out computer calculations for two or three subcircuits case, we confirm that various kinds of synchronization phenomena of chaos are observed. In the case of two subcircuits, we confirm that synchronization phenomena are related to phase difference of the functions corresponding to the parametric excitation. Then the two subcircuits are synchronized at the opposite-phase. In the case of three subcircuits, we confirm self-switching phenomenon of synchronization states when there is not phase difference of the functions corresponding to the parametric excitation. On the other hand, when there is phase difference, two of the three subcircuits are synchronized at the opposite-phase.

Ventricular tachycardia and fibrillation are potentially lethal cardiac arrhythmias generated by high frequency, irregular spatio-temporal electrical activity. Re-entrant propagation has been demonstrated as a mechanism generating these arrhythmias in computational and in vitro animal models of these arrhythmias. Re-entry can be idealised in homogenous isotropic virtual cardiac tissues as spiral and scroll wave solutions of reaction-diffusion equations. A spiral wave in a bounded medium can be terminated if its core reaches a boundary. Ventricular tachyarrhythmias in patients are sometimes observed to spontaneously self-terminate. One possible mechanism for self-termination of a spiral wave is meander of its core to an inexcitable boundary. We have previously proposed the hypothesis that the spatial extent of meander of a re-entrant wave in the heart can be directly related to its probability of self-termination, and so inversely related to its lethality. Meander in two-dimensional ...

The dynamics of a resistively coupled nonidentical Josephson junction (JJ) circuit is explored and a new control strategy is applied in order to smoothen the unstable regimes. In such a system, periodic, quasiperiodic and chaotic behaviors have been encountered worldwidely. In addition, the existence of the hyperchaos has also been declared by Kurt and his coworker. It is understood that the dynamic features strictly depend on the coupling resistance R cp . Indeed the decrease in R cp causes a sudden increase in Lyapunov exponents, which prove the chaoticity of the system. Since a wide chaotic region is encountered, the control of the system becomes important for a number of engineering applications. A new control system approach has been applied in order to solve this problem.

The dynamics of a resistively coupled nonidentical Josephson junction (JJ) circuit is explored and a new control strategy is applied in order to smoothen the unstable regimes. In such a system, periodic, quasiperiodic and chaotic behaviors have been encountered worldwidely. In addition, the existence of the hyperchaos has also been declared by Kurt and his coworker. It is understood that the dynamic features strictly depend on the coupling resistance Rcp. Indeed the decrease in Rcp causes a sudden increase in Lyapunov exponents, which prove the chaoticity of the system. Since a wide chaotic region is encountered, the control of the system becomes important for a number of engineering applications. A new control system approach has been applied in order to solve this problem.

Instant chaos is the onset of chaotic behaviour as a local bifurcation directly from a trivial steady state. We describe a systematic method for constructing examples of instant chaos, by scaling spatial variables and time. In this way we generalize properties of examples previously studied by other authors. We show that whenever a chaotic attractor of limited amplitude is obtained using a scaling property then it appears in slow motionᎏfor any set S transverse to the vector field, the return time to S tends to infinity as we approach the bifurcation point. When instant chaos appears for a family of vector fields with a nontrivial scaling property, if it is not in slow motion then the amplitude of the chaotic attractor becomes arbitrarily large around the bifurcation point. We use this method to obtain the Lorenz attractor in a bifurcation directly from an asymptotically stable equilibrium.

The logistic map with a delayed feedback is studied as a generic model. The stability of the model and its bifurcation scheme is analyzed as a function of the feedback amplitude and of the delay. Stability analysis is performed semianalytically. A relation between the delay and the periodicity of the orbit, which explains why some terms used in chaos control are ineffective, was found. The consequences for chaos control are discussed. The structure of bifurcations is found to depend strongly on the parity and on the length of the delay. Boundary crisis, the tangent, the Neimark, as well as the period-doubling bifurcations occur in this system. The effective dimension of the model is also discussed.

An application of the path-integral approach to an analysis of the fluctuations in complex dynamical systems is discussed. It is shown that essentially the same ideas underly recent progress in the solutions of a number of long-standing problems in complex dynamics. In particular, we consider the problems of prediction, control and inference of chaotic dynamics perturbed by noise in the framework of path-integral approach.

Continuous and pulsed forms of control of a multistable system are compared directly, both theoretically and numerically, taking as an example the switching of a periodically-driven class-B laser between its stable and unstable pulsing regimes. It is shown that continuous control is the more energy-efficient. This result is illuminated by making use of the close correspondence that exists between the problems of energy-optimal control and the stability of a steady state. Ensuring the stability and effective control of a lasing mode represents an important problem in the applied theory of lasers [1]. It can be mapped onto analyses of spiking behavior in population dynamics [2] and neurons [3] and, in a more general context, appears as a fundamental problem in the theory of nonlinear dynamical systems [4, 5]. It is potentially of relevance wherever switching takes place between distinct regimes of behaviour, e.g. in cardiac and cortical systems. While the topic is thus of broad interdisciplinary interest, laser systems can provide especially reliable and convincing tests of the new theoretical concepts. In general, the problem can be analyzed within either one of two distinct theoretical and experimental frameworks: using either continuous or discrete time, with corresponding descriptions of the system dynamics in terms of either continuous flows or maps. Both methods have been extensively tested in application to laser systems. For example, a special protocol has been developed for the feedback control of Nd lasers [6]. To reduce uncertainty in switching, methods based on stochastic resonance have been proposed [7, 8], with addition of a weak periodic modulation. The targeting of stable and saddle orbits has been discussed [9] and achieved experimentally by the use of a single impulse in a lossmodulated CO 2 laser [10, 11] paying special attention to minimization of the duration of the transient processes. Optimization of switch-on properties in semi-conductor lasers has been considered by exploring phase space [12] and via the minimal-time control problem [13]. That no direct comparison between these two general approaches for control has yet been made is perhaps surprising, given its broad interdisciplinary implications. In this Letter we consider, both theoretically and numerically, a direct comparison between continuous and discrete time approaches to control the lasing mode in class-B lasers. In particular, we show that the continuous method is the more energy-efficient: the activation "energies" and the energies of the control functions differ by an order of magnitude. We use the duality of the control and stability problems discussed previously [14, 15] to provide insight into the origin of this difference.

Fluctuational transitions between two co-existing chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the mechanism for such transitions is determined by a hierarchy of homoclinic points. The most probable escape path from the chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.