In this paper we model the microcantilever-sample interaction in an atomic force microscope (AFM) via a Lennard-Jones potential and consider the dynamical behavior of a harmonically forced system. Using nonlinear analysis techniques on attracting limit sets, we numerically verify the presence of chaotic invariant sets. The chaotic behavior appears to be generated via a cascade of period doubling, whose occurrence has been studied as a function of the system parameters. As expected, the chaotic attractors are obtained for values of parameters predicted by Melnikov theory. Moreover, the numerical analysis can be fruitfully employed to analyze the region of the parameter space where no theoretical information on the presence of a chaotic invariant set is available. In addition to explaining the experimentally observed chaotic behavior, this analysis can be useful in finding a controller that stabilizes the system on a nonchaotic trajectory. The analysis can also be used to change the AFM operating conditions to a region of the parameter space where regular motion is ensured. [S0022-0434(00)01401-5]

Issue Section:
Technical Briefs
Keywords:
atomic force microscopy, scanning probe microscopy, Lennard-Jones potential, chaos, control systems, intermolecular forces
Topics:
Atomic force microscopy, Attractors, Dynamics (Mechanics), Scanning probe microscopy, Numerical analysis, Lennard-Jones potential, Dimensions, Chaos
1.
Burnham
,
N. A.
,
Kulik
,
A.
,
Gremaud
,
G.
, and
Briggs
,
G.
,
1995
, “
Nanosubarmonics: The Dynamics of Small Nonlinear Contacts
,”
Phys. Rev.
,
74
, pp.
5092
–
5095
.
2.
Ashhab, M., Salapaka, M., Dahleh, M., and Mezic´, I., 1997, “Control of Chaos in Atomic Force Microscopes,” ACC, Albuquerque, NM.
3.
Ashhab
,
M.
,
Salapaka
,
M. V.
,
Dahleh
,
M.
, and
Mezic
,
I.
, “
Melnikov-based Dynamical Analysis of Microcantilevers in Scanning Probe Microscopy
,”
J. Nonlinear Dynam.
,
20
, pp.
197
–
220
.
4.
Israelachvili, J. N., 1985, Intermolecular and Surface Forces, Academic Press, New York.
5.
Wiggins, S., 1990, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York.
6.
Nayfeh, A., and Balachandran, B., 1995, Applied Nonlinear Dynamics, Wiley, New York.
7.
Osedelec
,
V. I.
, “
A Multiplicative Ergodic Theorem: Lyapunov Characteristic Numbers for Dynamical System
,”
Trans. Mosc. Math. Soc.
,
19
, p.
197
197
.
8.
Parker, T. S., and Chua, L. O., 1989, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York.
9.
Ott, E., 1993, Chaos in Dynamical Systems, Cambridge University Press, New York.
10.
Ruelle, D., 1987, Chaotic Evolution and Strange Attractors, Cambridge University Press, New York.
11.
Doedel
,
E. J.
,
Keller
,
H. B.
, and
Kernevez
,
J. P.
,
1991
, “
Numerical Analysis and Control of Bifurcation Problems I: Bifurcations in Finite Dimension
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
,
1
, pp.
493
–
520
.
12.
Ashhab
,
M.
,
Salapaka
,
M. V.
,
Dahleh
,
M.
, and
Mezic
,
I.
, “
Dynamical Analysis and Control of Microcantilevers
,”
Automatica
,
35
, pp.
1663
–
1670
.
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