Accurately predicting the onset of large behavioral deviations associated with saddle-node bifurcations is imperative in a broad range of sciences and for a wide variety of purposes, including ecological assessment, signal amplification, and microscale mass sensing. In many such practices, noise and non-stationarity are unavoidable and ever-present influences. As a result, it is critical to simultaneously account for these two factors toward the estimation of parameters that may induce sudden bifurcations. Here, a new analytical formulation is presented to accurately determine the probable time at which a system undergoes an escape event as governing parameters are swept toward a saddle-node bifurcation point in the presence of noise. The double-well Duffing oscillator serves as the archetype system of interest since it possesses a dynamic saddle-node bifurcation. The stochastic normal form of the saddle-node bifurcation is derived from the governing equation of this oscillator to formulate the probability distribution of escape events. Non-stationarity is accounted for using a time-dependent bifurcation parameter in the stochastic normal form. Then, the mean escape time is approximated from the probability density function (PDF) to yield a straightforward means to estimate the point of bifurcation. Experiments conducted using a double-well Duffing analog circuit verifies that the analytical approximations provide faithful estimation of the critical parameters that lead to the non-stationary and noise-activated saddle-node bifurcation.

References

References
1.
Hanggi
,
P.
,
1986
, “
Escape From a Metastable State
,”
J. Stat. Phys.
,
42
(
1–2
), pp.
105
148
.
2.
Meunier
,
C.
, and
Verga
,
A. D.
,
1988
, “
Noise and Bifurcations
,”
J. Stat. Phys.
,
50
(
1–2
), pp.
345
375
.
3.
Dykman
,
M. I.
,
Mori
,
E.
,
Ross
,
J.
, and
Hunt
,
P. M.
,
1994
, “
Large Fluctuations and Optimal Paths in Chemical Kinetics
,”
J. Chem. Phys.
,
100
(
8
), pp.
5735
5750
.
4.
Arnold
,
L.
,
1998
,
Random Dynamical Systems
,
Springer
,
Berlin, Germany
.
5.
Dykman
,
M. I.
,
Schwartz
,
I. B.
, and
Shapiro
,
M.
,
2005
, “
Scaling in Activated Escape of Underdamped Systems
,”
Phys. Rev. E
,
72
(
2
), p.
021102
.
6.
Fulton
,
T. A.
, and
Dunkleberger
,
L. N.
,
1974
, “
Lifetime of the Zero-Voltage State in Josephson Tunnel Junctions
,”
Phys. Rev. B
,
9
(
11
), pp.
4760
4769
.
7.
Büttiker
,
M.
,
Harris
,
E. P.
, and
Landauer
,
R.
,
1983
, “
Thermal Activation in Extremely Underdamped Josephson-Junction Circuits
,”
Phys. Rev. B
,
28
(
3
), pp.
1268
1275
.
8.
Devoret
,
M. H.
,
Esteve
,
D.
,
Martinis
,
J. M.
,
Cleland
,
A.
, and
Clarke
,
J.
,
1987
, “
Resonant Activation of a Brownian Particle Out of a Potential Well: Microwave-Enhanced Escape From the Zero-Voltage State of a Josephson Junction
,”
Phys. Rev. B
,
36
(
1
), pp.
58
73
.
9.
Vijay
,
R.
,
Devoret
,
M. H.
, and
Siddiqi
,
I.
,
2009
, “
Invited Review Article: The Josephson Bifurcation Amplifier
,”
Rev. Sci. Instrum.
,
80
(
11
), p.
111101
.
10.
Zorin
,
A. B.
,
1996
, “
Quantum-Limited Electrometer Based on Single Cooper Pair Tunneling
,”
Phys. Rev. Lett.
,
76
(
23
), pp.
4408
4411
.
11.
Aldridge
,
J. S.
, and
Cleland
,
A. N.
,
2005
, “
Noise-Enabled Precision Measurements of a Duffing Nanomechanical Resonator
,”
Phys. Rev. Lett.
,
94
(
15
), p.
156403
.
12.
Badzey
,
R. L.
,
Zolfagharkhani
,
G.
,
Gaidarzhy
,
A.
, and
Mohanty
,
P.
,
2005
, “
Temperature Dependence of a Nanomechanical Switch
,”
Appl. Phys. Lett.
,
86
(
2
), p.
023106
.
13.
Stambaugh
,
C.
, and
Chan
,
H. B.
,
2006
, “
Noise-Activated Switching in a Driven Nonlinear Micromechanical Oscillator
,”
Phys. Rev. B
,
73
(
17
), p.
172302
.
14.
Longtin
,
A.
,
1993
, “
Stochastic Resonance in Neuron Models
,”
J. Stat. Phys.
,
70
(
1–2
), pp.
309
327
.
15.
Lindner
,
B.
,
Longtin
,
A.
, and
Bulsara
,
A.
,
2003
, “
Analytic Expressions for Rate and CV of a Type I Neuron Driven by White Gaussian Noise
,”
Neural Comput.
,
15
(
8
), pp.
1761
1788
.
16.
Lindner
,
B.
,
García-Ojalvo
,
J.
,
Neiman
,
A.
, and
Schimansky-Geier
,
L.
,
2004
, “
Effects of Noise in Excitable Systems
,”
Phys. Rep.
,
392
(
6
), pp.
321
424
.
17.
Guttal
,
V.
, and
Jayaprakash
,
C.
,
2008
, “
Changing Skewness: An Early Warning Signal of Regime Shifts in Ecosystems
,”
Ecol. Lett.
,
11
(
5
), pp.
450
460
.
18.
Dakos
,
V.
,
van Nes
,
E. H.
,
D'Odorico
,
P.
, and
Scheffer
,
M.
,
2012
, “
Robustness of Variance and Autocorrelation as Indicators of Critical Slowing Down
,”
Ecology
,
93
(
2
), pp.
264
271
.
19.
Scheffer
,
M.
,
Carpenter
,
S. R.
,
Lenton
,
T. M.
,
Bascompte
,
J.
,
Brock
,
W.
,
Dakos
,
V.
,
van de Koppel
,
J.
,
van de Leemput
,
I. A.
,
Levin
,
S. A.
,
van Nes
,
E. H.
,
Pascual
,
M.
, and
Vandermeer
,
J.
,
2012
, “
Anticipating Critical Transitions
,”
Science
,
338
(
6105
), pp.
344
348
.
20.
Lenton
,
T. M.
,
2011
, “
Early Warning of Climate Tipping Points
,”
Nat. Clim. Change
,
1
(
4
), pp.
201
209
.
21.
Thompson
,
J. M. T.
, and
Sieber
,
J.
,
2011
, “
Predicting Climate Tipping as a Noisy Bifurcation: A Review
,”
Int. J. Bifurcation Chaos
,
21
(
2
), pp.
399
423
.
22.
Scheffer
,
M.
,
Bascompte
,
J.
,
Brock
,
W. A.
,
Brovkin
,
V.
,
Carpenter
,
S. R.
,
Dakos
,
V.
,
Held
,
H.
,
van Nes
,
E. H.
,
Rietkerk
,
M.
, and
Sugihara
,
G.
,
2009
, “
Early-Warning Signals for Critical Transitions
,”
Nature
,
461
(
7260
), pp.
53
59
.
23.
Lu
,
C.-H.
, and
Evan-Iwanowski
,
R. M.
,
1994
, “
The Nonstationary Effects on a Softening Duffing Oscillator
,”
Mech. Res. Commun.
,
21
(
6
), pp.
555
564
.
24.
Haberman
,
R.
,
1979
, “
Slowly Varying Jump and Transition Phenomena Associated With Algebraic Bifurcation Problems
,”
SIAM J. Appl. Math.
,
37
(
1
), pp.
69
106
.
25.
Berglund
,
N.
, and
Kunz
,
H.
,
1999
, “
Memory Effects and Scaling Laws in Slowly Driven Systems
,”
J. Phys. A
,
32
(
1
), pp.
15
39
.
26.
Kogan
,
O.
,
2007
, “
Controlling Transitions in a Duffing Oscillator by Sweeping Parameters in Time
,”
Phys. Rev. E
,
76
(
3
), p.
037203
.
27.
Mandel
,
P.
, and
Erneux
,
T.
,
1987
, “
The Slow Passage Through a Steady Bifurcation: Delay and Memory Effects
,”
J. Stat. Phys.
,
48
(
5–6
), pp.
1059
1070
.
28.
Baer
,
S. M.
,
Erneux
,
T.
, and
Rinzel
,
J.
,
1989
, “
The Slow Passage Through a Hopf Bifurcation: Delay, Memory Effects, and Resonance
,”
SIAM J. Appl. Math.
,
49
(
1
), pp.
55
71
.
29.
Mandel
,
P.
, and
Erneux
,
T.
,
1984
, “
Laser Lorenz Equations With a Time-Dependent Parameter
,”
Phys. Rev. Lett.
,
53
(
19
), pp.
1818
1820
.
30.
Zeghlache
,
H.
,
Mandel
,
P.
, and
Van den Broeck
,
C.
,
1989
, “
Influence of Noise on Delayed Bifurcations
,”
Phys. Rev. A
,
40
(
1
), pp.
286
294
.
31.
Erneux
,
T.
, and
Laplante
,
J. P.
,
1989
, “
Jump Transition Due to a Time-Dependent Bifurcation Parameter in the Bistable Ioadate–Arsenous Acid Reaction
,”
J. Chem. Phys.
,
90
(
11
), pp.
6129
6134
.
32.
Koper
,
M. T. M.
, and
Aguda
,
B. D.
,
1996
, “
Experimental Demonstration of Delay and Memory Effects in the Bifurcations of Nickel Electrodissolution
,”
Phys. Rev. E
,
54
(
1
), pp.
960
963
.
33.
Koper
,
M. T. M.
,
1998
, “
Non-Linear Phenomena in Electrochemical Systems
,”
J. Chem. Soc., Faraday Trans.
,
94
(
10
), pp.
1369
1378
.
34.
Elmer
,
F. J.
,
1997
, “
Nonlinear Dynamics of Dry Friction
,”
J. Phys. A
,
30
(
17
), pp.
6057
6063
.
35.
Sang
,
Y.
,
Dubé
,
M.
, and
Grant
,
M.
,
2001
, “
Thermal Effects on Atomic Friction
,”
Phys. Rev. Lett.
,
87
(
17
), p.
174301
.
36.
Conley
,
W. G.
,
Krousgrill
,
C. M.
, and
Raman
,
A.
,
2008
, “
Stick-Slip Motions in the Friction Force Microscope: Effects of Tip Compliance
,”
Tribol. Lett.
,
29
(
1
), pp.
23
32
.
37.
Siddiqi
,
I.
,
Vijay
,
R.
,
Pierre
,
F.
,
Wilson
,
C. M.
,
Metcalfe
,
M.
,
Rigetti
,
C.
,
Frunzio
,
L.
, and
Devoret
,
M. H.
,
2004
, “
RF-Driven Josephson Bifurcation Amplifier for Quantum Measurement
,”
Phys. Rev. Lett.
,
93
(
20
), p.
207002
.
38.
Zhang
,
W.
, and
Turner
,
K. L.
,
2005
, “
Application of Parametric Resonance Amplification in a Single-Crystal Silicon Micro-Oscillator Based Mass Sensor
,”
Sens. Actuators, A
,
122
(
1
), pp.
23
30
.
39.
Yie
,
Z.
,
Zielke
,
M. A.
,
Burgner
,
C. B.
, and
Turner
,
K. L.
,
2011
, “
Comparison of Parametric and Linear Mass Detection in the Presence of Detection Noise
,”
J. Micromech. Microeng.
,
21
(
2
), p.
025027
.
40.
Burgner
,
C. B.
,
Miller
,
N. J.
,
Shaw
,
S. W.
, and
Turner
,
K. L.
,
2010
, “
Parameter Sweep Strategies for Sensing Using Bifurcations in MEMS
,”
Solid-State Sensor, Actuator, and Microsystems Workshop
, Hilton Head Island, SC, June 6–10, 2010, p.
130
.
41.
Younis
,
M. I.
, and
Alsaleem
,
F.
,
2009
, “
Exploration of New Concepts for Mass Detection in Electrostatically-Actuated Structures Based on Nonlinear Phenomena
,”
ASME J. Comput. Nonlinear Dyn.
,
4
(
2
), p.
021010
.
42.
Kumar
,
V.
,
Boley
,
J. W.
,
Yang
,
Y.
,
Ekowaluyo
,
H.
,
Miller
,
J. K.
,
Chiu
,
G. T.-C.
, and
Rhoads
,
J. F.
,
2011
, “
Bifurcation-Based Mass Sensing Using Piezoelectrically-Actuated Microcantilevers
,”
Appl. Phys. Lett.
,
98
(
15
), p.
153510
.
43.
Harne
,
R. L.
, and
Wang
,
K. W.
,
2014
, “
A Bifurcation-Based Coupled Linear-Bistable System for Microscale Mass Sensing
,”
J. Sound Vib.
,
333
(
8
), pp.
2241
2252
.
44.
Khater
,
M. E.
,
Al-Ghamdi
,
M.
,
Park
,
S.
,
Stewart
,
K. M. E.
,
Abdel-Rahman
,
E. M.
,
Penlidis
,
A.
,
Nayfeh
,
A. H.
,
Abdel-Aziz
,
A. K. S.
, and
Basha
,
M.
,
2014
, “
Binary MEMS Gas Sensors
,”
J. Micromech. Microeng.
,
24
(
6
), p.
065007
.
45.
Harne
,
R. L.
, and
Wang
,
K. W.
,
2013
, “
Robust Sensing Methodology for Detecting Change With Bistable Circuitry Dynamics Tailoring
,”
Appl. Phys. Lett.
,
102
(
20
), p.
203506
.
46.
Kim
,
J.
,
Harne
,
R. L.
, and
Wang
,
K. W.
,
2015
, “
Enhancing Structural Damage Identification Robustness to Noise and Damping With Integrated Bistable and Adaptive Piezoelectric Circuitry
,”
ASME J. Vib. Acoust.
,
137
(
1
), p.
011003
.
47.
Guckenheimer
,
J.
, and
Holmes
,
P.
,
1983
,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
,
Springer-Verlag
,
New York
.
48.
Kovacic
,
I.
, and
Brennan
,
M. J.
,
2011
,
The Duffing Equation: Nonlinear Oscillators and Their Behaviour
,
Wiley
,
Hoboken, NJ
.
49.
Virgin
,
L. N.
,
2000
,
Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration
,
Cambridge University Press
,
Cambridge, UK
.
50.
Breban
,
R.
,
Nusse
,
H. E.
, and
Ott
,
E.
,
2003
, “
Scaling Properties of Saddle-Node Bifurcations on Fractal Basin Boundaries
,”
Phys. Rev. E
,
68
(
6
), p.
066213
.
51.
Berglund
,
N.
, and
Gentz
,
B.
,
2006
,
Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach
,
Springer Science & Business Media
,
New York
.
52.
Kuehn
,
C.
,
2015
,
Multiple Time Scale Dynamics
,
Springer
,
New York
.
53.
Nicolis
,
C.
, and
Nicolis
,
G.
,
2014
, “
Dynamical Responses to Time-Dependent Control Parameters in the Presence of Noise: A Normal Form Approach
,”
Phys. Rev. E
,
89
(
2
), p.
022903
.
54.
Miller
,
N. J.
, and
Shaw
,
S. W.
,
2012
, “
Escape Statistics for Parameter Sweeps Through Bifurcations
,”
Phys. Rev. E
,
85
(
4
), p.
046202
.
55.
Cao
,
Q.
,
Wiercigroch
,
M.
,
Pavlovskaia
,
E. E.
,
Thompson
,
J. M. T.
, and
Grebogi
,
C.
,
2008
, “
Piecewise Linear Approach to an Archetypal Oscillator for Smooth and Discontinuous Dynamics
,”
Philos. Trans. R. Soc., A
,
366
(
1865
), pp.
635
652
.
56.
Zou
,
K.
, and
Nagarajaiah
,
S.
,
2015
, “
Study of a Piecewise Linear Dynamic System With Negative and Positive Stiffness
,”
Commun. Nonlinear Sci. Numer. Simul.
,
22
(
1
), pp.
1084
1101
.
57.
Miller
,
N.
,
Burgner
,
C.
,
Dykman
,
M.
,
Shaw
,
S.
, and
Turner
,
K.
,
2010
, “
Fast Estimation of Bifurcation Conditions Using Noisy Response Data
,”
SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring
, San Diego, CA, Mar. 31, 2010, p.
764700
.
58.
Abramowitz
,
M.
, and
Stegun
,
I. A.
, eds.,
1972
,
Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables
,
Dover
,
New York
.
59.
Higham
,
D. J.
,
2001
, “
An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations
,”
SIAM Rev.
,
43
(
3
), pp.
525
546
.
60.
Mann
,
B. P.
,
Barton
,
D. A. W.
, and
Owens
,
B. A. M.
,
2012
, “
Uncertainty in Performance for Linear and Nonlinear Energy Harvesting Strategies
,”
J. Intell. Mater. Syst. Struct.
,
23
(
13
), pp.
1451
1460
.
61.
Stanton
,
S. C.
,
Owens
,
B. A. M.
, and
Mann
,
B. P.
,
2012
, “
Harmonic Balance Analysis of the Bistable Piezoelectric Inertial Generator
,”
J. Sound Vib.
,
331
(
15
), pp.
3617
3627
.
62.
Harne
,
R. L.
,
Thota
,
M.
, and
Wang
,
K. W.
,
2013
, “
Concise and High-Fidelity Predictive Criteria for Maximizing Performance and Robustness of Bistable Energy Harvesters
,”
Appl. Phys. Lett.
,
102
(
5
), p.
053903
.
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