The control of chaos: theory and applications

TLDR: In this paper, the Ott-Grebogi-Yorke (OGY) method and the adaptive method for chaotic control are discussed. But the authors focus on the targeting problem, i.e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions.

About: This article is published in Physics Reports.The article was published on 2000-05-01 and is currently open access. It has received 742 citations till now. The article focuses on the topics: Control of chaos & Synchronization of chaos.

Figures

Fig. 28. From Ref. [125]. (a) A hyperbolic chaotic saddle embedded in the HeH non attractor with gap size s"0.2. (b) The topological entropy h T (s) of the chaotic saddle as a function of the gap size s.

Fig. 41. Reprinted with permission from Science 257, p. 1230, copyright (1992) American Association for the Advancement of Science. (a}c) Interbeat interval I n versus the beat number during the chaotic phases. (d}f ) Corresponding PoincareH sections.

Fig. 12. From Ref. [66]. (a) An (x, y) projection of the unperturbed RoK ssler dynamics (dots) and path followed by the perturbed dynamics to reach the target (thick dashed line). In this case the perturbation acts only on the x variable of Eqs. (72). Again, the path is inside the chaotic attractor; thus it is compatible with the natural evolution of the system and it goes from higher to lower probability regions. (b) Temporal evolution of the perturbation during the targeting process. The range spanned by ; is less than 1% of the range spanned by the x dynamics. Initial conditions and control parameters as in the text.

Fig. 38. From Ref. [16]. Temporal traces of the output intensity, control signal and FFT corresponding to (a) no control, (b) period 1, (c) period 4 and (d) period 9.

Fig. 1. From Ref. [41]. The double rotor.

Fig. 26. Encoding the word `TIGERa into a trajectory on the chaotic repeller at r"4.1 for the logistic map. The binary representation of the word is shown on the top of the "gure. Shown is a time series where small control is initiated at n"1 and the "rst binary bit of the message is encoded into the trajectory at n"8. Time dependent perturbations are applied at subsequent iterations so that the entire message `TIGERa can be encoded into the trajectory. The magnitudes of the control perturbations required are shown in (b).

Frequently asked questions

Q1. What are the contributions mentioned in the paper "The control of chaos: theory and applications" ?

The authors review the major ideas involved in the control of chaos, and present in detail two methods: the Ott } Grebogi } Yorke ( OGY ) method and the adaptive method. The authors also discuss a series of relevant issues connected with chaos control, such as the targeting problem, i. e., how to bring a trajectory to a small neighborhood of a desired location in the chaotic attractor in both low and high dimensions, and point out applications for controlling fractal basin boundaries. In short, the authors describe procedures for stabilizing desired chaotic orbits embedded in a chaotic attractor and discuss the issues of communicating with chaos by controlling symbolic sequences and of synchronizing chaotic systems. Finally, the authors give a review of relevant experimental applications of these ideas and techniques. 

Q2. How do the authors encode a binary message into a trajectory that lives on the chaotic repeller?

To encode an arbitrary binary message into a trajectory that lives on the chaotic repeller, it is necessary to use small perturbations to an accessible system parameter or a dynamical variable. 

Q3. What is the simplest way to formulate an applicable control law?

The simplest way to formulate an applicable control law is to make use of the fact that the dynamics of any smooth nonlinear system is approximately linear in a small e-neighborhood of a "xed point. 

Q4. Why is the trajectory not able to reach the desired "xed point?

due to the nonlinearity not included in the linear expansion (10), the control may not be able to bring the trajectory to the desired "xed point. 

Q5. What makes the hope of synchronizing two chaotic systems even more remote?

in practical applications the existence of noise (both external and internal) and system imperfect identi"cation makes the hope of synchronizing two chaotic systems even more remote. 

Q6. What is the fundamental requirement that a chaotic system can be de"ned?

The fundamental requirement that quali"es a chaotic system for communication is whether a good symbolic dynamics can be de"ned which faithfully represents the dynamics in the phase space. 

Q7. What is the way to represent the transitions between the allowed n-bit words?

A way to represent the transitions between the allowed n-bit words is to use the directed-graph method in Ref. [55] which was originally discussed for one-dimensional noninvertible chaotic maps (with an in"nite shift space). 

Q8. What is the relevant consequence of the properties of a chaotic system?

A relevant consequence of these properties is that a chaotic dynamics can be seen as shadowing some periodic behavior at a given time, and erratically jumping from one to another periodic orbit. 

Q9. What is the probability that a trajectory will enter the neighborhood of the periodic orbit?

In the presence of external noise, a controlled trajectory will occasionally be &kicked' out of the neighborhood of the periodic orbit. 

Q10. What is the probability for driving an arbitrary initial condition to the desirable attractor?

In this case, the probability for driving an arbitrary initial condition to the desirable attractor is proportional to the fraction of uncertain initial conditions, which scales with the perturbation as ea. 

Q11. How many iterations did the authors have to do to stabilize the period one orbit?

Within the parameter range of Fig. 35, the authors were able to stabilize the period one orbit for more than 200 000 iterations (about 64 h), using a maximum perturbation of 9% of the unperturbed dynamics. 

Q12. How can one encode a binary sequence into a dynamical trajectory?

Together with the coding function which also needs to be determined beforehand, one can in principle encode any binary sequences into a dynamical trajectory on the chaotic repeller. 

Q13. What is the probability that a trajectory enters the neighborhood of the target periodic orbit?

For one-dimensional maps, the probability that a trajectory enters the neighborhood of a particular component (component i) of the periodic orbit is given byP(e)"P x(i)`ex(i)~e o[x(i)] dx+2eo[x(i)] , (5)where o is the frequency that a chaotic trajectory visits a small neighborhood of the point x on the attractor. 

Q14. How can one reconstruct the attractor from a time series of measurements?

To do this, one can make use of the time delay embedding technique [26], allowing to reconstruct the attractor from a time series of measurements of a single variable, say x(t), from Eqs. (72). 

Q15. Why is the trajectory not kept in the vicinity of the "xed point?

Even then, because of nonlinearity not included in the linearized Eq. (37), the control may not be able to keep the trajectory in the vicinity of the "xed point. 

Q16. What is the largest possible value of the topological entropy for a symbolic map?

for the logistic map, whose dynamical behaviors are seen in a large class of deterministic chaotic systems, the largest possible value of the topological entropy, or the channel capacity, is achieved in a parameter regime of transient chaos where the invariant sets are chaotic repellers. 

Q17. What is the probability that the two already synchronized trajectories will lose ?

Under the in#uence of external noise, there is a "nite probability that the two already synchronized trajectories may lose synchronization. 

Q18. What is the probability of a chaotic attractor falling into the control parallelogram?

If one distributes a large number of initial conditions on the chaotic attractor according to the natural measure and then follows the trajectories resulting from these initial conditions, this probability k(P# ) gives the rate at which these orbits fall into the control parallelogram.