VOLUME 77, NUMBER 11
PHYSICAL REVIEW LETTERS
9 SEPTEMBER 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 received considerable attention recently [1]; these systems
which can generally be described by the “particle-in≠Usxd
potential” paradigm, xÙ ­ 2 ≠x 1 Sstd 1 Nstd, 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, Sstd and Nstd
denote a deterministic signal (usually taken to be time periodic) and noise (usually taken to be Gaussian). The potential function Usxd 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 manifestations 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 x0 into the signal Sstd or, equivalently, a term xx0 into Usxd. 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 spectral 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 treatment 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 parameter, 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” stochastic resonance [1,3]. We start with a purely deterministic phenomenological theory that shows the occurrence of
multiple maxima in the spectral amplitudes in a generic
asymmetric system; we then introduce characteristic stochastic 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 resonance behavior in the spectral amplitudes of the harmonics when the noise is turned on. Finally we present theory
[to Osk ­ 2d] and numerical simulations on the rf SQUID
loop [to Osk ­ 4d] to buttress our results.
Consider a periodic signal A sin vt applied to a bistable
potential Usxd. We are concerned only with a dichotomous output fstd 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 fstd as
fstd ­ 1s0 # t # Qd, fstd ­ 21sQ # t # T d. Clearly
Q, the residence time in one of the two wells, depends on
the degree of asymmetry. We now Fourier analyze fstd.
When the potential is symmetric (i.e., Q ­ Ty2), the
Fourier series contains only odd harmonics. For the general case, we can, from the Fourier coefficients, determine
kvQ
2
the spectral amplitude Mk at kv as Mk ­ kp sin 2 ,
which has extrema when kvQ ­ np (n odd). We are
concerned only with the interval 0 # Q # T y2 and readily see that the fundamental (k ­ 1) has a single maximum for Q ­ Q1 ­ T y2 corresponding to the symmetric case, the first harmonic (k ­ 2) has a single maximum
for Q ­ Q2 ­ T y4, the k ­ 3 harmonic has maxima at
Q ­ Q3 ­ Ty6, Ty2, the k ­ 4 harmonic at Q ­ Q4 ­
T y8, 3T y8, and so on.
The extension to the noisy case is achieved by introducing the mean residence times ktl l and ktr l in the left
and right states of the potential (the left well has the shallower minimum). For convenience, these may be computed in the absence of the periodic signal; the presence of
the signal affects these mean times only slightly (Fig. 3)
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© 1996 The American Physical Society
0031-9007y96y77(11)y2162(4)$10.00
VOLUME 77, NUMBER 11
PHYSICAL REVIEW LETTERS
for weak signal amplitudes. We then postulate that to
achieve a maximum in a given spectral amplitude Mk (assuming the output to be approximately periodic), we must
achieve ktl l ­ Qk and ktr l ­ T 2 ktl l. For the first few
harmonics this yields immediately ktl l ­ T y2 ­ ktr l for
k ­ 1 (this is the classical frequency-matching condition
for stochastic resonance), ktl l ­ T y4 and ktr l ­ 3T y4 for
k ­ 2, etc. In fact, we find a precise matching of stochastic and deterministic time scales for every harmonic
k whenever the spectral amplitude Mk possesses a maximum. For harmonic k we may write the general conditions for these “resonances” as
ktl l ­
n T
,
k 2
ktr l ­ T 2 ktl l,
ny2k
ktl l
­
,
ktr l
1 2 ny2k
(1)
where n is odd and 1 # n # k. This leads to an elegant pattern of numbers (shown in Table I for k ­ 1 9)
which exposes a precise matching of stochastic (the mean
residence times) and deterministic (the signal period) time
scales that must exist to obtain the (multiple) resonances
(as a function of asymmetry) in the harmonics k when the
system is noisy. We now explore the resonance behavior
in a specific system, the rf SQUID loop.
The rf SQUID loop is a superconducting loop shorted
by a single Josephson junction [4]. The state variable
xstd is now the magnetic flux f (normalized by the
flux quantum f0 ) through the loop; it evolves temporally
according to the dynamics
tL xÙ ­ 2
≠Usxd
1 hstd 1 ystd ,
≠x
9 SEPTEMBER 1996
mal noise which is taken to be negligible for the purposes
of this Letter, any externally applied noise will usually
have a bandwidth far smaller than the SQUID bandwidth
tL21 . Hence we take ystd to be zero-mean Gaussian exponentially correlated noise having variance k y 2 l ­ Dyt;
it evolves via a white-noise-driven Ornstein-Uhlenbeck
(OU) process: yÙ ­ 2t 21 y 1 sFstd, where Fstd is zeromean white noise of one-sided spectral densityp2D, t is
the correlation time of the noise ystd, and s ; 2Dyt.
Stochastic resonance (defined in the conventional way
via the output SNR at the fundamental in the PSD) in the
rf SQUID loop has been demonstrated in laboratory experiments [5,6]. For convenience we prebias the SQUID
so that it possesses (for moderate values of the nonlinearity parameter b) a well-defined central bistable structure, possibly with outlying metastable states. This is
accomplished [5] by building in a dc bias of my2 (m
odd) in the potential Usxd: we replace x0 by x0 1 my2.
m21
x 1h11y2
The central minima located at x1 ø 2 1 0 11b and
m11
x 1h21y2
x2 ø 2 1 0 11b
are separated by a maximum at
m
x0 1h
xu ø 2 1 12b . We also make the standard adiabatic
assumption [1] of having the signal frequency lower than
the relaxation rate of the system, so that we may incorporate hstd into the potential Usxd as a quasiconstant term.
Finally, we assume that A ø U0 , the height of the central
maximum of the potential, so that there is no deterministic
switching.
Figure 1 shows the resonant behavior in the fundamental and first three harmonics (k ­ 1 4) of the simulated
SQUID output. We plot the power at each harmonic as a
function of the asymmetry-producing normalized dc mag-
(2)
b
1
where the potential Usxd ­ 2 sx 2 x0 d2 2 4p 2 cos 2px,
which is multistable when the nonlinearity parameter
b exceeds some critical value. The symmetry-breaking
(normalized) dc flux is x0 and hstd ­ A sinsvt 1 ud,
with u being a (often assumed random) phase factor. The
time constant tL is the ratio of the inductance to the normal state resistance of the SQUID loop. Typically tL ø
10212 so that, with the exception of the (internal) therTABLE I. Mean residence time ratios which maximize the
spectral amplitude at frequency kv.
kt1 lyktr l
k
1
2
3
4
5
6
7
8
9
1
1
3
3
5
5
7
7
9
9
1
3
3
5
5
7
7
9
1
5
3
7
5
9
7
11
1
7
3
9
5
11
1
9
3
11
5
13
1
11
3
13
1
13
3
15
1
15
1
17
FIG. 1. SQUID output power (in dB) at the fundamental
and the first three harmonics (k ­ 1 4) of the driving signal. System parameters: t ­ 0.01, v ­ 10, b ­ 5, A ­ 0.1,
m ­ 1.
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VOLUME 77, NUMBER 11
PHYSICAL REVIEW LETTERS
netic flux x0 and the noise variance s 2 ­ 2Dt 22 . This
is computed by simulating the effect of the noise ystd on
the dynamic potential underpinning the SQUID dynamics. The simulations (carried out on an Intel PARAGON
using an integration time step of 0.000 613 59 and averaging over 131 072 fast Fourier transforms of 32 768
points each) lead to kxstdl, the normalized output flux in
the loop, averaged over noise. The magnitude squared of
the Fourier transform of kxstdl, computed at a given frequency kv, yields the power at that harmonic for a given
noise variance and asymmetry. We note that the weights
of the delta spikes in the output PSD decrease very rapidly
at higher harmonics and their resolution above the noise
level becomes increasingly difficult. The contour plots
demonstrate the multiple maxima at higher harmonics k
as predicted by the simple phenomenological theory. The
time-scale matching conditions (1) that underpin the occurrence and locations of the maxima have indeed been
verified, within the simulation tolerances, for k ­ 1 4.
We reiterate that (1) represents a matching of purely stochastic time scales (the mean residence times ktl l and
ktr l) with a deterministic scale (the signal period T); it
is truly a manifestation of the bona fide resonance character of stochastic resonance (SR) [7] at higher orders with
a new parameter, the asymmetry-inducing signal x0 , controlling the resonance behavior through its effect on the
mean residence times ktl l and ktr l. For small x0 it is
easily demonstrable that the critical noise variance s 2 at
which the harmonics have peak power is the same variance that yields the (single) maximum, at x0 ­ 0, in the
k ­ 1 case, corresponding to conventional SR [1,3]. The
theory breaks down in the x0 ! 1y2 limit in which the
central bistable structure of the potential Usxd disappears.
In general, the maxima locations depend very weakly on
the signal frequency. The simulation results are identical
for any odd m, and are reflected about the vertical axis for
21y2 # x0 # 0. Any other x0 may be mapped into the
range 21y2 # x0 # 1y2 by modifying m.
We now present a very concise account of an approximate theoretical calculation of the first two spectral
amplitudes. Since tL ø t, the SQUID may be assumed
to remain in its (nonequilibrium) steady state, making transitions (accompanied by the emission of a single flux quantum) only when the noise causes the currently occupied
m21
minimum to vanish at a point of inflection xi1 ­ 2 1
1
m11
1
21
or xi2 ­ 2 2 2p arccoss2b 21 d.
2p arccoss2b d
Thus, the noise must achieve the values yc1,2 ­
m
b
xi1,2 2 x0 2 2 2 hstd 1 2p sin 2pxi1,2 to accomplish switching. Therefore we model the SQUID as a
two-state system with a hysteretic input-output characteristic having state probabilities p1,2 std and master equations
pÙ1 ­ W21 p2 2 W12 p1 ,
pÙ2 ­ W12 p1 2 W21 p2 ,
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(3)
9 SEPTEMBER 1996
where p1 1 p2 ­ 1 and Wik denotes the transition rate
from state i to state k. These rates are the approximate
inverses of the mean passage times ktl l and ktr l introduced
earlier. We compute the transition rates by solving the first
passage problem [8] for the OU process (underpinning the
noise) between the values yc1 and yc2 :
21
ø ktl l ­
W12
2
s2
Z
yc1
ez
2
ty2D
Z
z
e2z
02
ty2D
dz 0
2`
yc2
p Z
­ 2t r
dz
uc1
2
eu Fsud du ;
uc2
a corresponding expression exists for W21 . We have de1
fined Fsud ; 2 f1 1 erf sudg and uc1,2 ; s2Dd21y2 yc1,2 .
To solve (3) we define uc1,20 ; uc1,2 jhstd­0 and set
uc1,2 ­ upc1,20 2 h 0 std, h 0 std ; A0 sinsvt 1 ud, where
A0 ; Ay 2k y 2 l is a natural (and convenient) perturbation
expansion parameter; the theory is valid for A0 ø 1. We
now expand the transition rates as
W12 ø a0 1 a1 h 0 std 1 a2 h 02 std,
W21 ø b0 1 b1 h 0 std 1 b2 h 02 std ,
(4)
the expansion coefficients being obtained through a
straightforward expansion of the transition rates. Substituting these expansions into (3), we obtain the solution
up to OsA2 d, which should yield very good results up
through the first harmonic (k ­ 2). Then we can write
down for the mean value of the SQUID response
Z
kxstdl ­
xPsx, td dx ­ x10 p1 std 1 x20 p2 std , (5)
where x1,20 ; x12 jA­0 , and we have used the property appropriate to the SQUID two-state dynamics:
Psx, td ø p1 stddsx 2 x10 d 1 p2 stddsx 2 x20 d, Psx, td
being the global probability density function that characterizes the system.
Using the solution of (3) in (5), we directly express kxstdl as a Fourier expansion, kxstdl ­ M0 1
M1 cossvt 1 f1 d 1 M2 coss2vt 1 f2 d. The phaseaveraged autocorrelation function Kssd ; kxstdxst 1
sdlt can be readily computed to obtain Kssd ­ M02 1
FIG. 2. SQUID output power at the fundamental and the
first harmonic (k ­ 1, 2) of the driving signal. Solid curves:
theoretical prediction. Dashed curves: numerical simulation
(two-state filtered). Dots: numerical simulation (unfiltered).
s 2 ­ 12.619. Other parameters as in Fig. 1.
VOLUME 77, NUMBER 11
PHYSICAL REVIEW LETTERS
FIG. 3. Mean residence time ratios. Solid curves: theoretical
prediction, A ­ 0. Dashed curves: numerical simulation, A ­
0. Dots: numerical simulation, A ­ 0.1. Other parameters as
in Fig. 2.
M12
2
M2
cos vs 1 22 cos 2vs, so that the output powers in the
fundamental and the first harmonic are M12 y2 and M22 y2,
respectively.
Figure 2 shows the power in the fundamental and
in the first harmonic vs dc signal x0 . We find very
good agreement between the theory and simulations.
In fact, even for the case of the signal frequency v
lying somewhat outside the noise bandwidth t 21 we
get very good qualitative agreement; in particular, the
theory correctly captures the location of the maximum in
M22 , with the disagreement appearing as a vertical scale
factor in this case. The location of the maximum in
M22 also depends sensitively on the noise variance k y 2 l.
The time-scale matching conditions (Table I) between the
stochastic passage times ktl l, ktr l and the signal period
T that are expected (via the phenomenological theory
described earlier) to hold at the maxima, do indeed hold
(Fig. 3), within the limits of validity of the theoretical
formulation, and within the simulation tolerances. We
have already pointed out, in connection with Fig. 1, that
these “resonance” criteria have been shown, numerically,
to be satisfied at least up to fourth order (k ­ 4). Note
that in Fig. 2 we have plotted the results of simulations
of the system that take into account the intrawell motion
of the state point xstd as well as the “filtered” results
in which the intrawell motion is neglected; as expected,
the latter case agrees very well with the two-state theory,
while the former agrees well with the theory at moderateto-low asymmetry. At high asymmetry we see the effect
of intrawell motion manifested as an increased power
at the fundamental, k ­ 1, over that seen in the twostate theory and filtered cases. Intrawell motion does
not significantly affect the harmonics because they arise
from nonlinear response, e.g., well hopping; intrawell
motion is an approximately linear response. Finally, for a
fixed dc signal x0 , individual powers Mk2 y2 pass through
maxima at critical noise variances; this is an extension
of the conventional stochastic resonance effect at higher
orders. Of course, for a fixed x0 , one may suppress some
maxima in the higher-order spectral amplitudes (k . 1)
by suitably selecting the noise parameters D and k so
that the time-scale matching conditions for the occurrence
9 SEPTEMBER 1996
of the maxima cannot be met; this will be detailed in a
forthcoming publication.
The results of this Letter are applicable to generic
bistable and (in special situations such as described
here in connection with the SQUID) multistable systems
with broken symmetry. Many nonlinear detectors suffer
from significant low-frequency noise limitations (the noise
may be internal, e.g., 1yf, or external). By carefully
selecting the frequency v of the known bias signal,
the detection may be shifted to a more acceptable part
of the frequency spectrum. Then, in a detector that
has an a priori symmetric potential, the appearance of
the even multiples of v in the output PSD, together
with the change in the spectral amplitudes Mk in the
presence of the symmetry-breaking signal (which may be
dc, or have a single frequency in which case one looks
at the properties of beats in the output PSD), may be
used to detect or estimate the weak target signal. This
idea was, in fact, demonstrated in laboratory experiments
carried out with a specially designed “SR-SQUID” [6]
assuming only internal white (i.e., thermal) noise, as well
as a conventional rf SQUID [9] using externally applied
correlated noise. Our results (Fig. 2) mirror, qualitatively,
the experimental observations of Refs. [6,9].
A. R. B. acknowledges support from the Office of Naval
Research through Grant No. N0001496AF00002 as well
as the Internal Research program at NCCOSC. M. E. I.
was supported in part by a grant of HPC time from the
DoD Major Shared Resource Center at WPAFB on the
Intel PARAGON. We acknowledge helpful discussions
with F. Marchesoni and L. Kiss.
*Electronic address: [email protected]
†
Electronic address: [email protected]
‡
Electronic address: [email protected]
[1] For a good overview, see P. Jung, Phys. Rep. 234, 175
(1994).
[2] R. Bartussek, P. Jung, and P. Hanggi, Phys. Rev. E 49,
3930 (1994); P. Jung and R. Bartussek, in Fluctuations
and Order: The New Synthesis, edited by M. Millonas
(Springer-Verlag, New York, 1996).
[3] For good overviews, see K. Wiesenfeld and F. Moss,
Nature (London) 373, 33 (1995); A. Bulsara and L.
Gammaitoni, Phys. Today 49, 39 (1996), No. 3.
[4] A. Barone and G. Paterno, Physics and Applications of the
Josephson Effect (J. Wiley, New York, 1982).
[5] A. Hibbs, A. Singsaas, E. Jacobs, A. Bulsara, J.
Bekkedahl, and F. Moss, J. Appl. Phys. 77, 2582 (1995).
[6] R. Rouse, S. Han, and J. Lukens, Appl. Phys. Lett. 66,
108 (1995).
[7] L. Gammaitoni, F. Marchesoni, and S. Santucci, Phys.
Rev. Lett. 74, 1052 (1995).
[8] See, e.g., C. Gardiner, Handbook of Stochastic Methods
(Springer-Verlag, Berlin, 1983).
[9] New Type of Magnetic Flux Sensor Based on Stochastic
Resonance in a Flux Quantized Superconducting Loop,
Quantum Magnetics Inc., San Diego, CA, 1995.
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