Forecasting bifurcations is a significant challenge and an important need in several applications. Most of the existing forecasting approaches focus on bifurcations in nonoscillating systems. However, subcritical and supercritical flutter (Hopf) bifurcations are very common in a variety of systems, especially fluid–structural systems. This paper presents a unique approach to forecast (nonlinear) flutter based on observations of the system only in the prebifurcation regime. The proposed method is based on exploiting the phenomenon of critical slowing down (CSD) in oscillating systems near certain bifurcations. Techniques are introduced to enhance the prediction accuracy for cases of low-frequency oscillations and large-dimensional dynamical systems. The method is applied to an aeroelastic system responding to gust loads. Numerical results are provided to demonstrate the performance of the method in predicting the postbifurcation regime accurately in both supercritical and subcritical cases.

References

References
1.
Veraart
,
A. J.
,
Faassen
,
E. J.
,
Dakos
,
V.
,
van Nes
,
E. H.
,
Lürling
,
M.
, and
Scheffer
,
M.
,
2012
, “
Recovery Rates Reflect Distance to a Tipping Point in a Living System
,”
Nature
,
481
(
7381
), pp.
357
359
.
2.
Jeffries
,
C.
, and
Wiesenfeld
,
K.
,
1985
, “
Observation of Noisy Precursors of Dynamical Instabilities
,”
Phys. Rev. A
,
31
(
2
), pp.
1077
1084
.
3.
Guttal
,
V.
, and
Jayaprakash
,
C.
,
2008
, “
Changing Skewness: An Early Warning Signal of Regime Shifts in Ecosystems
,”
Ecol. Lett.
,
11
(
5
), pp.
450
460
.
4.
Horsthemke
,
W.
,
1984
,
Noise Induced Transitions
,
Springer
,
Berlin
.
5.
Lim
,
J.
, and
Epureanu
,
B. I.
,
2011
, “
Forecasting a Class of Bifurcations: Theory and Experiment
,”
Phys. Rev. E
83
(
1
), p.
016203
.
6.
Lim
,
J.
, and
Epureanu
,
B. I.
,
2012
, “
Forecasting Bifurcation Morphing: Application to Cantilever-Based Sensing
,”
Nonlinear Dyn.
,
67
(
3
), pp.
2291
2298
.
7.
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
.
8.
Lee
,
B. H. K.
,
Jiang
,
L. Y.
, and
Wong
,
Y. S.
,
1999
, “
Flutter of an Airfoil With a Cubic Restoring Force
,”
J. Fluids Struct.
,
13
(
1
), pp.
75
101
.
9.
Lee
,
B. H. K.
,
Price
,
S. J.
, and
Wong
,
Y. S.
,
1999
, “
Nonlinear Aeroelastic Analysis of Airfoils: Bifurcation and Chaos
,”
Progr. Aerosp. Sci.
,
35
(
3
), pp.
205
334
.
10.
Liu
,
L.
,
Wong
,
Y. S.
, and
Lee
,
B. H. K.
,
2000
, “
Application of the Center Manifold Theory in Non-Linear Aeroelasticity
,”
J. Sound Vib.
,
234
(
4
), pp.
641
659
.
11.
Kalmár-Nagy
,
T.
,
Stépán
,
G.
, and
Moon
,
F. C.
,
2001
, “
Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations
,”
Nonlinear Dyn.
,
26
(
2
), pp.
121
142
.
12.
Jafri
,
F. A.
,
Shukla
,
A.
, and
Thompson
,
D. F.
,
2007
, “
A Numerical Bifurcation Study of Friction Effects in a Slip-Controlled Torque Converter Clutch
,”
Nonlinear Dyn.
,
50
(
3
), pp.
627
638
.
13.
Song
,
Y.
, and
Wei
,
J.
,
2005
, “
Local Hopf Bifurcation and Global Periodic Solutions in a Delayed Predator–Prey System
,”
J. Math. Anal. Appl.
,
301
(
1
), pp.
1
21
.
14.
Mees
,
A.
, and
Chua
,
L. O.
,
1979
, “
The Hopf Bifurcation Theorem and Its Applications to Nonlinear Oscillations in Circuits and Systems
,”
IEEE Trans. Circuits Syst.
,
26
(
4
), pp.
235
254
.
15.
Luo
,
G. W.
, and
Xie
,
J. H.
,
1998
, “
Hopf Bifurcation of a Two-Degree-Of-Freedom Vibro-Impact System
,”
J. Sound Vib.
,
213
(
3
), pp.
391
408
.
16.
Ghadami
,
A.
, and
Epureanu
,
B. I.
,
2015
, “
Forecasting Subcritical and Supercritical Flutter Using Gust Responses
,”
ASME
Paper No. IMECE2015-53105.
17.
D'Souza
,
K.
,
Epureanu
,
B. I.
, and
Pascual
,
M.
,
2015
, “
Forecasting Bifurcations From Large Perturbation Recoveries in Feedback Ecosystems
,”
PloS One
,
10
(
9
), p.
e0137779
.
18.
Hannan
,
E. J.
, and
Quinn
,
B. G.
,
1979
, “
The Determination of the Order of an Autoregression
,”
J. R. Stat. Soc. Ser. B (Methodol.)
,
41
(
2
), pp.
190
195
.
19.
Juang
,
J. N.
,
1987
, “
Mathematical Correlation of Modal-Parameter-Identification Methods Via System-Realization Theory
,”
NASA TM
, Hampton, VA, NASA Technical Memorandum 87720.
20.
Juang
,
J. N.
, and
Pappa
,
R. S.
,
1985
, “
An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction
,”
J. Guid., Control, Dyn.
,
8
(
5
), pp.
620
627
.
21.
Pappa
,
R. S.
,
1994
, “
Eigensystem Realization Algorithm User's Guide for VAX/VMS Computers
,” NASA TM, Hampton, VA, Report No. 109066, p.
1994
.
22.
Lee
,
B. H. K.
,
Gong
,
L.
, and
Wong
,
Y. S.
,
1997
, “
Analysis and Computation of Nonlinear Dynamic Response of a Two-Degree-Of-Freedom System and Its Application in Aeroelasticity
,”
J. Fluids Struct.
,
11
(
3
), pp.
225
246
.
23.
Fung
,
Y. C.
,
2002
,
An Introduction to the Theory of Aeroelasticity
,
Dover Publications
, New York.
24.
Dessi
,
D.
, and
Mastroddi
,
F.
,
2008
, “
A Nonlinear Analysis of Stability and Gust Response of Aeroelastic Systems
,”
J. Fluids Struct.
,
24
(
3
), pp.
436
445
.
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