We present an experimental procedure to track periodic orbits through a fold (saddle-node) bifurcation and demonstrate it with a parametrically excited pendulum experiment where the tracking parameter is the amplitude of the excitation. Specifically, we track the initially stable period-one rotation of the pendulum through its fold bifurcation and along the unstable branch. The fold bifurcation itself corresponds to the minimal amplitude that supports sustained rotation. Our scheme is based on a modification of time-delayed feedback in a continuation setting and we show for an idealized model that it converges with the same efficiency as classical proportional-plus-derivative control.

1.
Doedel
,
E.
, 2007, “
Lecture Notes on Numerical Analysis of Nonlinear Equations
,”
Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems
,
B.
Krauskopf
,
H.
Osinga
, and
J.
Galán-Vioque
, eds.,
Springer-Verlag
,
Dordrecht
, pp.
1
49
.
2.
Kuznetsov
,
Y. A.
, 2004, “
Elements of Applied Bifurcation Theory
,”
Applied Mathematical Sciences
,
3rd ed.
,
Springer-Verlag
,
New York
, Vol.
12
.
3.
Dhooge
,
A.
,
Govaerts
,
W.
, and
Kuznetsov
,
Y.
, 2003, “
MatCont: A Matlab Package for Numerical Bifurcation Analysis of ODEs
,”
ACM Trans. Math. Softw.
0098-3500,
29
(
2
), pp.
141
164
.
4.
Engelborghs
,
K.
,
Luzyanina
,
T.
, and
Samaey
,
G.
, 2001, “
DDE-BIFTOOL v.2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations
,” Report No. TW 330, Katholieke Universiteit Leuven.
5.
Lust
,
K.
,
Roose
,
D.
,
Spence
,
A.
, and
Champneys
,
A.
, 1998, “
An Adaptive Newton-Picard Algorithm With Subspace Iteration for Computing Periodic Solutions
,”
SIAM J. Sci. Comput. (USA)
1064-8275,
19
(
4
), pp.
1188
1209
.
6.
Kevrekidis
,
I.
,
Gear
,
C.
, and
Hummer
,
G.
, 2004, “
Equation-Free: The Computer-Aided Analysis of Complex Multiscale Systems
,”
AIChE J.
0001-1541,
50
(
7
), pp.
1346
1355
.
7.
Sieber
,
J.
, and
Krauskopf
,
B.
, 2008, “
Control Based Bifurcation Analysis for Experiments
,”
Nonlinear Dyn.
0924-090X,
51
(
3
), pp.
365
377
.
8.
Blakeborough
,
A.
,
Williams
,
M.
,
Darby
,
A.
, and
Williams
,
D.
, 2001, “
The Development of Real-Time Substructure Testing
,”
Philos. Trans. R. Soc. London, Ser. A
0962-8428,
359
, pp.
1869
1891
.
9.
Gonzalez-Buelga
,
A.
,
Wagg
,
D.
, and
Neild
,
S.
, 2007, “
Parametric Variation of a Coupled Pendulum-Oscillator System Using Real-Time Dynamic Substructuring
,”
Struct. Control Health Monit.
1545-2255,
14
(
7
), pp.
991
1012
.
10.
Sieber
,
J.
,
Gonzalez-Buelga
,
A.
,
Neild
,
S.
,
Wagg
,
D.
, and
Krauskopf
,
B.
, 2008, “
Experimental Continuation of Periodic Orbits Through a Fold
,”
Phys. Rev. Lett.
0031-9007,
100
, p.
244101
.
11.
Barton
,
D.
, and
Burrow
,
S.
, 2009, “
Numerical Continuation in a Physical Experiment: Investigation of a Nonlinear Energy Harvester
,”
ASME
Paper No. DETC2009-87318.
12.
Szemplińska-Stupnicka
,
W.
,
Tyrkiel
,
E.
, and
Zubrzycki
,
A.
, 2000, “
The Global Bifurcations That Lead to Transient Tumbling Chaos in a Parametrically Driven Pendulum
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274,
10
(
9
), pp.
2161
2175
.
13.
Hirsch
,
M.
,
Pugh
,
C.
, and
Shub
,
M.
, 1977, “
Invariant Manifolds
,”
Lecture Notes in Mathematics
,
Springer-Verlag
,
Berlin
, Vol.
583
.
14.
Eyert
,
V.
, 1996, “
A Comparative Study on Methods for Convergence Acceleration of Iterative Vector Sequences
,”
J. Comput. Phys.
0021-9991,
124
(
2
), pp.
271
285
.
15.
Pyragas
,
K.
, 1992, “
Continuous Control of Chaos by Self-Controlling Feedback
,”
Phys. Lett. A
0375-9601,
170
, pp.
421
428
.
16.
Guckenheimer
,
J.
, and
Holmes
,
P.
, 1990, “
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
,”
Applied Mathematical Sciences
,
Springer-Verlag
,
New York
, Vol.
42
.
17.
Lehman
,
B.
, and
Weibel
,
S.
, 1999, “
Fundamental Theorems of Averaging for Functional Differential Equations
,”
J. Differ. Equations
0022-0396,
152
, pp.
160
190
.
18.
Bates
,
P.
,
Lu
,
K.
, and
Zeng
,
C.
, 1999, “
Persistence of Overowing Manifolds for Semi-Flow
,”
Commun. Pure Appl. Math.
0010-3640,
52
(
8
), pp.
893
1046
.
19.
Roose
,
D.
, and
Szalai
,
R.
, 2007, “
Continuation and Bifurcation Analysis of Delay Differential Equations
,”
Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems
,
B.
Krauskopf
,
H.
Osinga
, and
J.
Galán-Vioque
, eds.,
Springer-Verlag
,
Dordrecht
, pp.
51
75
.
20.
Nam
,
K.
, and
Arapostathis
,
A.
, 1992, “
A Sufficient Condition for Local Controllability of Nonlinear Systems Along Closed Orbits
,”
IEEE Trans. Autom. Control
0018-9286,
37
(
3
), pp.
378
380
.
You do not currently have access to this content.