VOLUME 77, NUMBER 11 PHYSICAL REVIEW LETTERS 9SEPTEMBER 1996
Noise-Controlled Resonance Behavior in Nonlinear Dynamical Systems with Broken Symmetry
A. R. Bulsara* and M. E. Inchiosa
†
Naval Command, Control and Ocean Surveillance Center, RDT & E Division, Code 364, San Diego, California 92152-5000
L. Gammaitoni
‡
Dipartimento di Fisica, Universita di Perugia, I-06100 Perugia, Italy
and Istituto Nazionale de Fisica Nucleare, Virgo Project, I-06100, Perugia, Italy
(Received 12 June 1996)
We study noise-controlled resonant behavior in periodically modulated, overdamped bistable dynamic
elements subject to a symmetry-breaking dc signal. The spectral amplitudes of the harmonics of the
modulation frequency are found to exhibit multiple maxima whose occurrence depends on matchings of
deterministic and stochastic time scales; in turn, these times depend on the noise statistics and the degree
of asymmetry. We demonstrate the phenomenological results via analytical and numerical computations
on an rf SQUID loop, and propose this technique to detect weak signals using a “frequency hopping”
mechanism to circumvent detector noise limitations. [S0031-9007(96)01154-4]
PACS numbers: 05.40.+j, 02.50.Fz, 87.10.+e
Periodically modulated stochastic systems have re-
ceived considerable attention recently [1]; these systems
which can generally be described by the “particle-in-
potential” paradigm, x 2
≠Ux
≠x
1 St 1 N t , exhibit
a richness of noise-mediated resonance behavior in the
spectral measures (e.g., the output signal-to-noise ratio,
SNR) of the response. In these systems, St and N t
denote a deterministic signal (usually taken to be time pe-
riodic) and noise (usually taken to be Gaussian). The po-
tential function Ux is even (often bistable), resulting in
an output power spectral density (PSD) consisting of odd
multiples of the signal frequency v superimposed on a
Lorentzian noise background. However, real-world man-
ifestations of these systems are often asymmetric, with the
dynamics containing even and odd functions of the state
variable. The simplest route to asymmetry in the above
dynamics is to incorporate a small dc term x
0
into the sig-
nal St or, equivalently, a term xx
0
into Ux. The output
PSD of asymmetric systems contains all the harmonics of
the periodic signal frequency; hence, the appearance and
magnitudes of the even multiples of v could be taken as
quantifying measures of the asymmetry-producing signal.
Asymmetric dynamic systems of the above form have
been studied [2] with Gaussian white noise. The spec-
tral amplitudes of the harmonics of the periodic signal,
in the output PSD, pass through maxima as a function of
noise variance. In this work we present a systematic treat-
ment of the resonant behavior of the spectral amplitudes
of the fundamental and harmonics at kv (k 1, 2, 3, . . .).
The resonant behavior depends on a new control parame-
ter, the degree of asymmetry, and can be interpreted at all
orders k, via a matching of deterministic and stochastic
time scales in the same manner as the “standard” stochas-
tic resonance [1,3]. We start with a purely determinis-
tic phenomenological theory that shows the occurrence of
multiple maxima in the spectral amplitudes in a generic
asymmetric system; we then introduce characteristic sto-
chastic time scales (these are critically dependent on the
asymmetry as well as the spectral characteristics of the
noise) and argue that a precise and elegant matching of
these time scales must occur for all k for there to be reso-
nance behavior in the spectral amplitudes of the harmon-
ics when the noise is turned on. Finally we present theory
[to Ok 2] and numerical simulations on the rf SQUID
loop [to Ok 4] to buttress our results.
Consider a periodic signal A sin vt applied to a bistable
potential Ux. We are concerned only with a dichoto-
mous output f t over a single period T of the signal;
i.e., we ignore the details of intrawell motion and assume
the signal to be just capable of achieving deterministic
switching between the potential wells. We define f t as
f t 10 # t #Q, f t 21Q# t # T . Clearly
Q, the residence time in one of the two wells, depends on
the degree of asymmetry. We now Fourier analyze f t .
When the potential is symmetric (i.e., Q T 2), the
Fourier series contains only odd harmonics. For the gen-
eral case, we can, from the Fourier coefficients, determine
the spectral amplitude M
k
at kv as M
k
2
kp
sin
kvQ
2
,
which has extrema when kvQ np (n odd). We are
concerned only with the interval 0 #Q# T 2 and read-
ily see that the fundamental (k 1) has a single maxi-
mum for Q Q
1
T 2 corresponding to the symmet-
ric case, the first harmonic (k 2) has a single maximum
for Q Q
2
T 4, the k 3 harmonic has maxima at
Q Q
3
T 6, T 2, the k 4 harmonic at Q Q
4
T 8, 3T 8, and so on.
The extension to the noisy case is achieved by intro-
ducing the mean residence times t
l
and t
r
in the left
and right states of the potential (the left well has the shal-
lower minimum). For convenience, these may be com-
puted in the absence of the periodic signal; the presence of
the signal affects these mean times only slightly (Fig. 3)
2162 0031-9007 96 77(11) 2162(4)$10.00 © 1996 The American Physical Society