Controlling transient chaos

Management implications of long transients in ecological systems

The underlying biological processes that govern many ecological systems can create very long periods of transient dynamics. It is often difficult or impossible to distinguish this transient behaviour from similar dynamics that would persist indefinitely. In some cases, a shift from the transient to the long-term, stable dynamics may occur in the absence of any exogenous forces. Recognizing the possibility that the state of an ecosystem may be less stable than it appears is crucial to the long-term success of management strategies in systems with long transient periods. Here we demonstrate the importance of considering the potential of transient system behaviour for management actions across a range of ecosystem organizational scales and natural system types. Developing mechanistic models that capture essential system dynamics will be crucial for promoting system resilience and avoiding system collapses.

Terminating transient chaos in spatially extended systems

In many real-life systems, transient chaotic dynamics plays a major role. For instance, the chaotic spiral or scroll wave dynamics of electrical excitation waves during life-threatening cardiac arrhythmias can terminate by itself. Epileptic seizures have recently been related to the collapse of transient chimera states. Controlling chaotic transients, either by maintaining the chaotic dynamics or by terminating it as quickly as possible, is often desired and sometimes even vital (as in the case of cardiac arrhythmias). We discuss in this study that the difference of the underlying structures in state space between a chaotic attractor (persistent chaos) and a chaotic saddle (transient chaos) may have significant implications for efficient control strategies in real life systems. In particular, we demonstrate that in the latter case, chaotic dynamics in spatially extended systems can be terminated via a relatively low number of (spatially and temporally) localized perturbations. We demonstrate as a proof of principle that control and targeting of high-dimensional systems exhibiting transient chaos can be achieved with exceptionally small interactions with the system. This insight may impact future control strategies in real-life systems like cardiac arrhythmias.

Avoiding healthy cells extinction in a cancer model

We consider a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. For certain parameter choice, the dynamical system displays chaotic motion and by decreasing the response of the immune system to the tumor cells, a boundary crisis leading to transient chaotic dynamics is observed. This means that the system behaves chaotically for a finite amount of time until the unavoidable extinction of the healthy and immune cell populations occurs. Our main goal here is to apply a control method to avoid extinction. For that purpose, we apply the partial control method, which aims to control transient chaotic dynamics in the presence of external disturbances. As a result, we have succeeded to avoid the uncontrolled growth of tumor cells and the extinction of healthy tissue. The possibility of using this method compared to the frequently used therapies is discussed.

Exploring partial control of chaotic systems

In this paper we make a thorough exploration of the technique of partial control of chaotic systems. This control technique allows one to keep the trajectories of a dynamical system close to a chaotic saddle even if the control applied is smaller than the effects of environmental noise in the system, provided that the chaotic saddle is due to the existence of a horseshoelike mapping in phase space. We state this here in a mathematically precise way using the Conley-Moser conditions, and we prove that they imply that our partial control strategy can be applied. We also give an upper bound of the control-noise ratio needed to achieve this goal, and we describe how this technique can be applied for large noise values. Finally, we study in detail the effect of imperfect targeting in our control technique. All these results are illustrated numerically with the paradigmatic Hénon map.

Partial control of chaotic systems

In a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exist what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with control smaller than noise. We call this type of control partial control of chaos.

Long chaotic transients in complex networks

We show that long chaotic transients dominate the dynamics of randomly diluted networks of pulse-coupled oscillators. This contrasts with the rapid convergence towards limit cycle attractors found in networks of globally coupled units. The lengths of the transients strongly depend on the network connectivity and vary by several orders of magnitude, with maximum transient lengths at intermediate connectivities. The dynamics of the transients exhibit a novel form of robust synchronization. An approximation to the largest Lyapunov exponent characterizing the chaotic nature of the transient dynamics is calculated analytically.

Transient characteristics of chaos synchronization in a semiconductor laser subject to optical feedback

We have investigated the transient characteristics of two types of chaos synchronization in a semiconductor laser subject to optical feedback: complete synchronization and strong injection locking-type synchronization. We have calculated the statistical distribution of the transient response time of synchronization when the initial position in the starting attractor is varied. For complete synchronization, the distribution of the transient response time has much larger average and variance than the average period of the chaotic oscillations. Conversely, a short transient response time is obtained for strong injection locking-type synchronization. We found that the transient response time is dependent upon the maximum Lyapunov exponent of the chaotic temporal waveform for complete synchronization, whereas it is almost constant for strong injection locking-type synchronization.

Controlling chaotic transients: Yorke's game of survival

We consider the tent map as the prototype of a chaotic system with escapes. We show analytically that a small, bounded, but carefully chosen perturbation added to the system can trap forever an orbit close to the chaotic saddle, even in presence of noise of larger, although bounded, amplitude. This problem is focused as a two-person, mathematical game between two players called "the protagonist" and "the adversary." The protagonist's goal is to survive. He can lose but cannot win; the best he can do is survive to play another round, struggling ad infinitum. In the absence of actions by either player, the dynamics diverge, leaving a relatively safe region, and we say the protagonist loses. What makes survival difficult is that the adversary is allowed stronger "actions" than the protagonist. What makes survival possible is (i) the background dynamics (the tent map here) are chaotic and (ii) the protagonist knows the action of the adversary in choosing his response and is permitted to choose the initial point x(0) of the game. We use the "slope 3" tent map in an example of this problem. We show that it is possible for the protagonist to survive.

Control of transient chaos in tent maps near crisis. I. Fixed point targeting

Combinatorial techniques are applied to the symbolic dynamics representing transient chaotic behavior in tent maps in order to solve the problem of Ott-Grebogi-Yorke control to the nontrivial fixed point occurring in such maps. This approach allows "preimage overlap" to be treated exactly. Closed forms for both the probability of control being achieved and the average number of iterations to control are derived. The results are discussed in relation to the work of Tel and shed new light on the transition to the control of permanent chaos.

Control of transient chaos in tent maps near crisis. II. Periodic orbit targeting

Recent work on a symbolic approach to the calculation of probability distributions arising in the application of the Ott-Grebogi-Yorke strategy to transiently chaotic tent maps is extended to the case of control to a nontrivial periodic orbit. Closed forms are derived for the probability of control being achieved and the average number of iterations to control when it occurs. Both single-component and multiple-component targeting are considered, and illustrative examples of the results obtained are presented.