VOLUME 85, NUMBER 10 PHYSICAL REVIEW LETTERS 4SEPTEMBER 2000
Fluctuations and the Energy-Optimal Control of Chaos
I. A. Khovanov,
1
D. G. Luchinsky,
2
R. Mannella,
3
and P. V. E. McClintock
2
1
Department of Physics, Saratov State University, Astrahanskaya 83, 410026, Saratov, Russia
2
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
3
Dipartimento di Fisica, Università di Pisa and INFM UdR Pisa, Piazza Torricelli 2, 56100 Pisa, Italy
(Received 28 March 2000)
The energy-optimal entraining of the dynamics of a periodically driven oscillator, moving it from a
chaotic attractor to a coexisting stable limit cycle, is investigated via analysis of fluctuational transitions
between the two states. The deterministic optimal control function is identified with the corresponding
optimal fluctuational force, which is found by numerical and analog simulations.
PACS numbers: 05.45.Gg, 02.50.–r, 05.20.–y, 05.40.–a
The stability of chaotic systems in the presence of noise
and methods for controlling these systems are of intrinsic
interdisciplinary interest and of obvious importance in re-
lation to applications. Methods already available [1] for
the control of chaos include entraining to a chosen “goal
dynamics,” which necessarily requires large modifications
of the system’s dynamics [2,3], and a variety of minimal
forms of interaction [4–6] which have hitherto been re-
stricted by the linear approximations adopted.
The energy-optimal implementation of deterministic
switching from the basin of attraction of a chaotic attractor
(CA) has remained an unsolved archetypal problem [7]
for a long time. Its solution must amount to an important
extension of the range of model-exploration objectives
(cf. [2] and [5]) achievable through minimal control tech-
niques. At the same time, the seemingly separate question
of noise-induced escape from the basin of attraction of a
CA has remained a major scientific challenge ever since
the first attempts to generalize the classical escape problem
to cover this case [8]. These two apparently quite different
problems are usually considered separately within the
distinct subfields of deterministic and stochastic nonlinear
dynamics. See, however, Ref. [9] for a discussion of the
interrelationship between stochastic and control problems,
and, in particular, [9](c) for the analogy between their
variational formulations.
In this Letter we show how the energy-optimal control
of chaos can be effected via an analogy between the varia-
tional formulations of both problems using a statistical
analysis of fluctuational trajectories. The main difficulty
in tackling these problems stems from the complexity of
the system dynamics near a CA and is related, in particular,
to delicate questions concerning the uniqueness of the so-
lution and the boundary conditions at a CA. The approach
proposed below is based on the analysis of an oscillator
interacting with a thermal bath. In the zero-noise intensity
limit, a consistent theoretical development [10,11] from the
microscopic to the macroscopic equations of motion leads
to descriptions of both its deterministic (dissipative) and
fluctuational dynamics within the framework of Hamil-
tonian formalism [12]. It can be shown both on physical
grounds and rigorously that the Wentzel-Freidlin Hamil-
tonian [12] arising in this approach is equivalent to the
Pontryagin Hamiltonian in the control problem [7] with
an additive linear unrestricted control; the corresponding
optimal control function is equivalent to the optimal fluc-
tuational force [9](c).
We illustrate the approach by analyzing the motion of a
periodically driven nonlinear oscillator
q
1
K
1
qt q
2
,
q
2
K
2
qt 1 ut (1)
22Gq
2
2v
2
0
q
1
2bq
2
1
2gq
3
1
2 h cosVt 1 ut .
Here ut is the control function. Parameters were chosen
such that the potential is monostable b
2
, 4gv
2
0
, the
dependence of the energy of oscillations on their frequency
is nonmonotonic
b
2
gv
2
0
.
9
10
, and the motion is under-
damped G ø V 2v
0
. This model is of interest in a
number of contexts in which theoretical analysis is pos-
sible for a wide range of parameter values [13]. It is a
system in which chaos can be observed at relatively small
values h 0.1 of the driving force amplitude.
For a given damping G 0.025 the amplitude and fre-
quency of the driving force were chosen so that the chaotic
attractor coexists with the stable limit cycle (SC in Fig. 1).
The chaotic state appears via period-doubling bifurcations
and thus corresponds to a nonhyperbolic attractor (NHA).
Its boundary of attraction ≠V is nonfractal and is formed
by the saddle cycle of period 1 (S1). For details about the
phase diagram, see Ref. [14].
We have considered the following energy-optimal con-
trol problem. The system (1) with unconstrained control
function ut is to be steered from the NHA to the stable
limit cycle in such a way that the energy (“cost”) func-
tional R is minimized, with t
1
unspecified,
R inf
u[U
1
2
Z t
1
t
0
u
2
t dt . (2)
Here the control set U consists of functions (control sig-
nals) which are able to move the system from the NHA to
the SC.
2100 0031-9007 00 85(10) 2100(4)$15.00 © 2000 The American Physical Society