%This is a Revtex file.
\documentstyle[aps,prl,twocolumn]{revtex}
%\documentstyle[preprint,aps]{revtex}
\begin{document}
\draft
\title{
Controlling Complexity}
\author{Leon Poon and Celso Grebogi}
\address{Institute for Plasma Research, University of Maryland, College Park, MD 20742}
\maketitle
\newpage
\begin{abstract}
Complex systems have the property that many competing behaviors are
possible and the system tends to alternate among them.
In fact, the ability of a complex system to access many different states,
combined with its sensitivity, offers great flexibility in manipulating the
system's dynamics to select a desired behavior. By understanding dynamically
how some of the complex features arise, we show that it is indeed possible
to control a complex system's behavior. This is illustrated by using
the noisy double rotor map as a paradigm.
\end{abstract}
\pacs{05.45.+b}
%
% Introduction
%
Scientists have come to the realization that many naturally occurring
systems are neither completely ordered and predictable nor completely random
and unpredictable. In fact, the behavior of many systems in nature fall
between these two opposite ends. The term ``complexity'' has been
coined to denote the study of ``complex systems'', that is, systems with
complicated and intricate features having both elements of order and
elements of randomness. They arise in fields as diverse as biology,
chemistry, computer science, geology, physics, and fluid mechanics. Some systems
that exhibit apparent complex behavior are Rayleigh-B\'{e}rnard convection
\cite{BD}, an extended optical system \cite{A}, neuronal activity \cite{spike},
and fluidized beds \cite{beds}.
But, what is a complex system? While there is no general
consensus on what constitutes a good quantitative measure of complexity,
the following general traits are some that most agree a complex system
often exhibits \cite{complex}: (i) A complex system is
composed of many parts that are interrelated in a complicated manner.
Usually, these intricate mutual relations result in some form of
coherent structure. (ii) A complex system possesses both ``ordered''
and ``random'' behaviors. (iii) A complex
system often exhibits a hierarchy of structures, that is, nontrivial
structures exist over a wide range of time and/or length scales.
These complex features are generally, but not necessarily, found in
systems with many degrees of freedom. Instead of evolving towards one
dominating attracting set, which is common in lower dimensional systems,
the interactions among the large number of attracting and unstable sets
often result in rich and varied dynamics where many
competing behaviors are possible. As a result, the dynamics of complex
systems tend to alternate among these different behaviors and which of
them are observed at a given time are often sensitive to minor perturbations.
These two key attributes, accessibility to many states and sensitivity, present
us with an opportunity to influence and manipulate a complex system's
dynamics.
In this paper, we show how complexity can be
realized and how its dynamics can be manipulated using small
perturbations. Specifically, we use the double rotor system as a
paradigm of a relatively low-dimensional dynamical system
(as opposed to an extended system) which exhibits many characteristics
typical of complex systems when it is subjected to random external noise.
We argue that the three traits mentioned above are apparent in many of
its behaviors. Furthermore, we also show how to use small amplitude
feedback control to influence and manipulate the behavior of the system
such that its trajectories, which formerly traversed the various attracting
and unstable chaotic sets, will now be confined to a neighborhood of one
of the attracting states of our choice.
The double rotor map is a rich dynamical system with
many complex features \cite{rotornote}.
For a wide range of parameters, the double rotor map has a
multitude of periodic attractors
(the coexistence of tiny chaotic attractors is also possible for some
parameters). Moreover, only a small portion of the basins of
attraction for these periodic attractors have ``smooth'' structure,
that is, only initial conditions within small balls centered at
each component of the periodic attractor unambiguously
asymptote to their respective attractors.
With the exception of these small open neighborhoods about the periodic attractors,
the majority of phase space is occupied by fractal basin boundaries whose
dimension ($\approx 3.998$ for the parameters we study) is very close to the dimension
of the phase space.
%
% Feature (i & ii)
%
The complicated basin structures of the double rotor system plays an
important role in the system's complex dynamics. The presence of
unstable invariant sets embedded in the fractal basin boundaries
has two appreciable effects on the system: long chaotic transient behavior
and final state sensitivity \cite{Nusse}. Typical trajectories with initial conditions
near the boundaries (which is highly probable since most of the phase space
is dominated by basin boundaries) will undergo chaotic motion for long times
before settling on one of the periodic attractors. The dynamics is
then characterized by a large number of periodic attractors ``embedded''
in a sea of transient chaos. The fine scale intermingling among the
various basins makes it very difficult to predict the future state
of trajectories for arbitrary initial conditions \cite{Nusse}.
Thus, the system is extremely sensitive to perturbations.
Because fractal basin boundaries permeate most of phase
space, the addition of small amplitude noise prevents the trajectories
from settling into any of the stable periodic behavior. What happens instead
is that a trajectory will come close to one of the periodic attractors and
stay in its neighborhood for some time. For this period of time, the
trajectory's behavior is governed by the periodic attractor and it is,
thereby, ordered. If one takes a large number of trajectories near this
periodic attractor and then a snapshot is taken after the noisy system has
evolved for some time, one would see coherent structures in phase space
in the vicinity of the periodic attractor as shown in Fig. \ref{structure}.
(Coherent structures are colored with darker shades. What the picture
represents and how it is generated will be discussed shortly).
However, this ordered behavior, for a particular trajectory, is transitory,
and noise will eventually move the trajectory out of this state into the
fractal boundary region. The trajectory will then spend some amount of time
within the massive basin boundary region executing an apparently chaotic motion
before approaching the same or another periodic attractor. The period
of time in the fractal basin boundaries corresponds to the trajectory's
``random'' behavior. For an ensemble of trajectories, these random structures
are represented by the lighter shades in Fig. \ref{structure}. Hence, a
typical noisy trajectory alternates between intervals of chaotic motion
and intervals of nearly periodic behavior as shown in Fig. \ref{noisy}.
A physical system that suggests the above behavior was observed by
Berg\'{e} and Dubois in a Rayleigh-Bernard convection experiment
\cite{BD}.
The dynamics we have just described correspond to traits (i) and
(ii) expected in complex systems. The fractal basin boundaries
with the embedded chaotic sets provide the link among the various
attractors. The presence of both ``ordered'' and ``random''
behaviors mentioned in (ii) is evidenced by the trajectories of our
system cascading from the random structures down to the coherent structures.
Noise, on the other hand, displaces trajectories from the coherent
structures back to the random structures. Therefore, dynamically there
is such an interplay between random and coherent structures such that
the system is neither completely predictable nor completely random.
The coherent structures mentioned above and shown in Fig. \ref{structure}
are found by examining the dynamics of a group of trajectories ($10^5$) near
one of the periodic orbits. We follow the evolution of an ensemble of initial
conditions in physical space (varying only $\theta_1$ and $\theta_2$) near
one of the periodic attractors while the system is subjected to
random noise with uniform distribution. In ordet to get an indication of
whether a trajectory is evolving chaotically or has settled into some sort of
periodic behavior after a certain number of iterates, say $n$, we
compute the largest eigenvalue of the Jacobian matrix for each of the
trajectory in the ensemble at the next iterate, i.e., $n+1$.
The largest eigenvalue is a measure of the local instability and hence
it indicates the ``jump'' the trajectory will make from its current location.
We then assign a color to a point corresponding to the trajectory's initial
condition according to how large a trajectory ``jumped'' at the $(n+1)$th iterate,
where the lighter the shade the bigger the jumps. Thus, a trajectory point with largest
eigenvalue less than 1 would be various shades of brown while a trajectory point with
largest eigenvalue more than 1 would be various shades of white.
In Fig. \ref{structure}, we see an intricate picture exhibiting
coherence and randomness. Thus, the combination of deterministic chaos
and stochastic noise has resulted in the emergence of coherent structures.
%
% Feature (iib)
%
We can quantify order and randomness by encoding the dynamics of the
system into symbolic sequences. We choose to encode the trajectory by
the sequence in which it visits the neighborhood of the attractors.
Of course, the trajectory visits
the different attracting sets by traversing through the chaotic sets in the
boundaries. These chaotic excursions are implicit in our choice of the
alphabet, each symbol corresponding to an attractor.
For the parameters we study, 8 symbols are needed in the alphabet
\cite{note}. We compute now the ``transition'' probabilities among the
various attracting sets. The complex dynamics can now be characterized by the
Kolmogorov-Sinai (KS) entropy \cite{KS}
\[
h = \lim_{n \rightarrow \infty} \frac{H_n}{n} = \lim_{n \rightarrow \infty}
\frac{1}{n}\left( -\sum_{|S| = n}p(S) \ln p(S) \right),
\]
where $S = s_1s_2\dots s_n$ denotes a finite symbol sequence;
$p(S)$, the relative frequency of $S$;
and $H_n$, block entropy of block length $n$.
Since $s_i \in \{1,\dots,8\}$, sequences where the letters are completely
uncorrelated would have a KS entropy value of $\ln 8 \simeq 2.08$, while
a periodic sequence would yield a value of 0. Our computation shows
that $H_n/n$ converges fairly rapidly to a value of $h \approx 1.42$.
KS entropy can be thought of as a measure of the coherence of
a trajectory as it evolves, so an intermediate value of $h$ means
not only that the symbolic sequence is unpredictable (since $h > 0$), but
there is also structure in the set of all possible symbol sequences.
It should be noted that the encoding scheme we have implemented does
not take into account the amount of time the trajectory spends
in the ordered as well as the chaotic regions.
%
% Feature (iii)
%
A consequence of the complex interplay between the coherent and
random structures, and the irregular switchings among them, is the
appearance of nontrivial length and/or time scalings in the noisy double
rotor system, a quality [trait (iii)] which we expect in complex systems.
First, there is the length of the chaotic
transients which is a measure of how long a trajectory spends
in the vicinity of each chaotic saddle embedded in the fractal
basin boundary.
It is known that the average length of a chaotic transient is
related to the dimension and the Lyapunov exponents of the chaotic
saddle \cite{Hsu}. Each chaotic saddle, in general, contributes a
distinct time scale, and the overall chaotic transient $\langle\tau\rangle$ would
then be a conglomeration of all these different time scales. Another
relevant measure of time is the mean escape time $\langle T\rangle$ for a trajectory
to leave the neighborhood of an attracting set, and it will in general
be different for the different attractors as can be seen in Fig. \ref{timescales}.
%
% Control
%
After establishing the complex features of the double rotor system, we proceed
to show how this knowledge can help us in manipulating and controlling the
behavior of this complex system. Unlike low-dimensional chaotic systems that
are commonly controlled using the ideas
introduced in Ref. \cite{OGY}, complex systems are not characterized by the
existence of one large chaotic attractor but by the coexistence of
many attractors. While the existence of a large chaotic attractor
is critical to those control schemes \cite{OGY}, we argue that
for a complex system, the unstable chaotic sets in the boundaries provide
us with the necessary sensitivity and flexibility to gear the dynamics toward
a specific periodic behavior using small perturbations. We can elect to
stabilize an unstable periodic orbit embedded within a chaotic saddle in
the boundary \cite{KST} or stabilize one of the (metastable) attracting
sets as we show next.
In controlling a metastable state in the complex double rotor map,
we employ a simple feedback scheme.
We denote the noiseless double rotor map as
${\bf x} \mapsto F({\bf x})$ where ${\bf x}$ is the four-dimensional phase space
coordinate, and the noisy double rotor map as
$\tilde{F}({\bf x}) \equiv F({\bf x}) + \bbox{\delta}$,
where $\bbox{\delta}$ is the noise vector whose norm is bounded
by $\delta$. For simplicity, we assume the periodic orbit to
be controlled is a fixed point (generalization to higher periodic
orbit is fairly straightforward). If we labeled this fixed point as
${\bf x_\star}$, then in a neighborhood of it, we have the following
linearization
$F({\bf x_\star} + \bbox{\varepsilon}) = {\bf x_\star} +
DF({\bf x_\star})\bbox{\varepsilon}$,
where the eigenvalues of $DF({\bf x_\star})$ are inside the unit circle (since
${\bf x_\star}$ is stable without noise).
Suppose now that on the $i$th iterate,
the trajectory lands in a neighborhood of this fixed point, so
${\bf x}_i = {\bf x_\star} + \bbox{\varepsilon}$. Without control,
${\bf x}_i \mapsto {\bf x}_{i+1} = \tilde{F}({\bf x}_i)$. However, assuming the
linearization
holds approximately for the noisy map near ${\bf x_\star}$, we can stabilize
the fixed point with the addition of a controlling term, or,
${\bf \hat{x}}_{i+1} = {\bf x}_{i+1} - DF({\bf x_\star})\bbox{\varepsilon}
= \tilde{F}({\bf x}_i) - DF({\bf x_\star})({\bf x}_i - {\bf x}_{\star}).$
Thus, the noisy trajectory with the above perturbation approaches the
fixed point in due course. Since we want to achieve control using only
small perturbations, the correction $|DF({\bf x_\star})\bbox{\varepsilon}|$
is scaled when necessary so it will not exceed some predetermined upper
bound of our choice.
Applying the correction to the noisy double rotor map, we
control the dynamics of the system. In Fig.\ \ref{control}, we follow a
typical noisy trajectory until it lands in a neighborhood of the
desired metastable attractor we wish to control, then we turn on the
control and let the system evolve for another thousand iterates
in the neighborhood of the desired attractor. Control is then turned
off and we let the trajectory wanders until it falls near the next
desired metastable attractor. In this fashion, we stabilize, say, eight
of the metastable attractors in the order we desire as demonstrated
in Fig.\ \ref{control}. Furthermore, if the trajectory is caught
in the vicinity of an attracting set that is undesirable, we can
destabilize it by applying a small amount of noise. In fact, this was
done in a brain experiment to control epilepsy \cite{Schiff} and
a fluidized bed experiment to control slugging \cite{Daw}.
These two experiments demonstrate the general result that small
perturbations, if chosen judiciously, not only can affect a desire
outcome in these systems, but it also validates and extends
these ideas to other systems.
In conclusion, we see that the myriad of possible behaviors in a complex
system is of great utility if we are able to harness it. In fact, the
ability of a complex system to access many different states, combined with
its sensitivity, offers great flexibility in manipulating and controlling
its dynamics. In general, many complex systems' behavior can be
modified to suit our needs using only small perturbation strategies
provided we are able to exploit their sensitivity.
This research was supported by a grant from DOE
(Office of Scientific Computing). The numerical computations
reported in this paper were supported in part by a grant from
the W. M. Keck Foundation.
\begin{references}
\bibitem{BD}
P. Berg\'{e} and M. Dubois, Phys. Lett. A {\bf 93}, 365 (1983).
\bibitem{A}
F.T. Arecchi, G. Giacomelli, P.L. Ramazza, and S. Residori,
. Rev. Lett. {\bf 65}, 2531 (1990).
\bibitem{spike}
P.E. Rapp, I.D. Zimmerman, E.P. Vining, N. Cohen, A.M. Albano,
and M.A. Jim\'{e}nez-Monta\~{n}o,
{\it et al.},
J. Neuroscience {\bf 14}, 4731 (1994).
\bibitem{beds}
C.S. Daw,
C.E.A. Finney, M. Vasudevan, N.A. van Goor, K. Nguyen, D.D. Bruns, E.J. Kostelich,
C. Grebogi, E. Ott, and J.A. Yorke,
Phys. Rev. Lett. {\bf 75}, 2308 (1995).
\bibitem{complex}
R. Badii and A. Politi, {\it Complexity: Hierarchical Structures and Scaling
in Physics}, (to be published);
{\it Measures of Complexity}, ed. by L. Peliti and A. Vulpiani (Springer-Verlag, Berlin, 1988);
P. Grassberger in {\it Fifth Mexican School on Statistical Mechanics},
ed. by F. Ramos-G\'{o}mez (World Scientific, Singapore, 1991), p. 57;
R. Badii in {\it Chaotic Dynamics: Theory and Practice}, ed. by T. Bountis
(Plenum Press, New York, 1992), p. 1;
J.P. Crutchfield and K. Young, Phys. Rev. Lett. {\bf 63}, 105 (1989);
J. Horgan, Sci. Am. {\bf 272}, No. 6, 104 (1995).
\bibitem{rotornote}
The kicked double rotor consists of two connected massless rods
(length $l_1$ and $l_2$) where one end of the first
rod pivots about a fixed point while the other end is attached
to and pivots about the middle of the second rod. A mass $m_1$
is placed at the end of the first rod and two masses $m_2/2$
are placed at each end of the second rod. The coefficients of
friction at the pivots are denoted $\nu_1$ and
$\nu_2$. Kicks of constant strength $\rho$ is applied to one
end of the second rod at periodic intervals. The state of the
system after each kick is described by the angular displacements
and velocities of the rods $(\theta_1,\theta_2,\dot{\theta}_1,\dot{\theta}_2)$.
The parameters values used in the study are:
$m_1 = m_2 = 1$, $\nu_1 = \nu_2 = 0.2$,
$l_1 = 1/\sqrt{2}$, $l_2 = 1$, and $\rho = 6.5$. See the following
references for more details:
C. Grebogi, E. Kostelich, E. Ott, and J.A. Yorke, Physica D {\bf 25}, 347
(1987);
F. Romeiras, C. Grebogi, E. Ott, and W.P. Dayawansa,
Physica D {\bf 58}, 165 (1992);
S. Dawson, C. Grebogi, T. Sauer, and J.A. Yorke,
Phys. Rev. Lett. {\bf 73}, 1927 (1994).
\bibitem{Nusse}
H.E. Nusse, C. Grebogi, E. Ott, and J.A. Yorke, in {\it Lect. Notes in
Math.}, Vol. 1342, ed. by J. C. Alexander (Springer-Verlag, New York, 1988),
p. 220;
C. Grebogi, S.W. McDonald, E. Ott, and J.A. Yorke, Phys. Lett. A, {\bf 99}, 415
(1983).
\bibitem{note}
Actually there are 16 attractors for the parameters chosen, but,
because of a simple symmetry in the double rotor system (Romeiras {\it et al.} in \cite{rotornote}),
we can group the attractors into 8 distinct pairs.
\bibitem{KS}
A.N. Kolmogorov, Dokl. Acad. Nauk SSSR {\bf 119}, 861 (1958);
Ya.G. Sinai, Dokl. Acad. Nauk SSSR {\bf 124}, 768 (1959).
\bibitem{Hsu}
H. Kantz and P. Grassberger, Physica D {\bf 17}, 75 (1985);
Hsu, G.-H., E. Ott, and C. Grebogi, Phys. Lett. A {\bf 127}, 199 (1988).
\bibitem{OGY}
E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. {\bf 64}, 1196 (1990);
T. Shinbrot, C. Grebogi, E. Ott, and J.A. Yorke, Nature {\bf 363}, 3
(1993) and references therein.
\bibitem{KST}
Z. Kov\'{a}cs, K. G. Szab\'{o}, and T. T\'{e}l, in
{\it Nonlinearity and chaos in engineering dynamics}, ed. by
J.M.T. Thompson and S.R. Bishop (Wiley, Chichester, 1994).
\bibitem{Schiff}
S.J. Schiff, K. Jerger, D.H. Duong, T. Chang, M.L. Spano, and W.L. Ditto
{\it et al.},
Nature {\bf 370}, 615 (1994).
\bibitem{Daw}
C.S. Daw, C.E.A. Finney, M. Vasudevan, in {\it Proceedings of the Third
Experimental Chaos Conference}, ed. by R. Harrison {\it et al.}
(World Scientific, in press).
\end{references}
\begin{figure}
\caption{
A blowup of a region which exhibit a combination of coherent and
random structures. The picture is generated after 100 iterations
of the noisy map. The apparent sharp vertical boundaries in the
picture is an artifact of projecting a four dimensional
figure onto two dimensional space, namely $(\theta_1,\theta_2,2.24,-3.65)$.
}
\label{structure}
\end{figure}
\begin{figure}
\caption{
Time series of a typical noisy trajectory.
}
\label{noisy}
\end{figure}
\begin{figure}
\caption{
The mean escape times $\langle T_i\rangle$ for some of the attractors and the average length of
the chaotic transient $\langle\tau\rangle$ as a function of noise amplitude $\delta$.}
\label{timescales}
\end{figure}
\begin{figure}
\caption{
Time series showing the result of applying the simple feedback control scheme
to successively control 8 different metastable states.}
\label{control}
\end{figure}
\end{document}