Nonlinear Dynamic Analysis and Chaotic Behavior in Atomic Force Microscopy

2005 ◽  
Author(s):  
Hossein Nejat Pishkenari ◽  
Nader Jalili ◽  
Aria Alasty ◽  
Ali Meghdari
Keyword(s):  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Nonlinear Dynamic Analysis ◽  
Melnikov Method ◽  
Single Mode ◽  
Intermolecular Forces ◽  
Invariant Sets ◽  
Nonlinear Dynamical ◽  
Atomic Force ◽  
Irregular Motion

The atomic force microscope (AFM) system has evolved into a useful tool for direct measurements of intermolecular forces with atomic-resolution characterization that can be employed in a broad spectrum of applications. In this paper, the nonlinear dynamical behavior of the AFM is studied. This is achieved by modeling the microcantilever as a single mode approximation (lumped-parameters model) and considering the interaction between the sample and cantilever in the form of van der Waals potential. The resultant nonlinear system is then analyzed using Melnikov method, which predicts the regions in which only periodic and quasi-periodic motions exist, and also predicts the regions that chaotic motion is possible. Numerical simulations are used to verify the presence of such chaotic invariant sets determined by Melnikov theory. Finally, the amplitude of vibration in which chaos is appeared is investigated and such irregular motion is proven by several methods including Poincare maps, Fourier transform, autocorrelation function and Lyapunov exponents.

Complex Dynamics in a Harmonically Excited Lennard-Jones Oscillator: Microcantilever-Sample Interaction in Scanning Probe Microscopes1

1998 ◽  
Vol 122 (1) ◽  
pp. 240-245 ◽  
Author(s):  
M. Basso ◽  
L. Giarre´ ◽  
M. Dahleh ◽  
I. Mezic´
Keyword(s):  
Parameter Space ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Operating Conditions ◽  
Invariant Sets ◽  
Jones Potential ◽  
Lennard Jones ◽  
Atomic Force

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]

scholarly journals Bistable dynamics beyond rotating wave approximation

2014 ◽  
Vol 23 (02) ◽  
pp. 1450019 ◽  
Author(s):  
Y. A. Sharaby ◽  
S. Lynch ◽  
A. Joshi ◽  
S. S. Hassan
Keyword(s):  
Ring Cavity ◽  
Dynamical Behavior ◽  
Single Mode ◽  
Unstable State ◽  
Nonlinear Dynamical ◽  
Rotating Wave Approximation ◽  
Wave Approximation ◽  
Atomic Medium ◽  
Bistable Dynamics ◽  
Rotating Wave

In this paper, we investigate the nonlinear dynamical behavior of dispersive optical bistability (OB) for a homogeneously broadened two-level atomic medium interacting with a single mode of the ring cavity without invoking the rotating wave approximation (RWA). The periodic oscillations (self-pulsing) and chaos of the unstable state of the OB curve is affected by the counter rotating terms through the appearance of spikes during its periods. Further, the bifurcation with atomic detuning, within and outside the RWA, shows that the OB system can be converted from a chaotic system to self-pulsing system and vice-versa.

scholarly journals Standing Swells Surveyed Showing Surprisingly Stable Solutions for the Lorenz '96 Model

2014 ◽  
Vol 24 (10) ◽  
pp. 1430027 ◽  
Author(s):  
Morgan R. Frank ◽  
Lewis Mitchell ◽  
Peter Sheridan Dodds ◽  
Christopher M. Danforth
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Standing Waves ◽  
Nonlinear Dynamical ◽  
Hidden Order ◽  
Stable Solutions ◽  
Lorenz 96

The Lorenz '96 model is an adjustable dimension system of ODEs exhibiting chaotic behavior representative of the dynamics observed in the Earth's atmosphere. In the present study, we characterize statistical properties of the chaotic dynamics while varying the degrees of freedom and the forcing. Tuning the dimensionality of the system, we find regions of parameter space with surprising stability in the form of standing waves traveling amongst the slow oscillators. The boundaries of these stable regions fluctuate regularly with the number of slow oscillators. These results demonstrate hidden order in the Lorenz '96 system, strengthening the evidence for its role as a hallmark representative of nonlinear dynamical behavior.

A Fresh Insight Into the Microcantilever-Sample Interaction Problem in Non-Contact Atomic Force Microscopy

2004 ◽  
Vol 126 (2) ◽  
pp. 327-335 ◽  
Author(s):  
Nader Jalili ◽  
Mohsen Dadfarnia ◽  
Darren M. Dawson
Keyword(s):  
Imaging Techniques ◽  
Interaction Force ◽  
Intermolecular Forces ◽  
Scanning Tunneling ◽  
Atomic Interaction ◽  
Force Estimation ◽  
Contact Mode ◽  
Distributed Parameters ◽  
Atomic Force ◽  
Insight Into

The atomic force microscope (AFM) system has evolved into a useful tool for direct measurements of intermolecular forces with atomic-resolution characterization that can be employed in a broad spectrum of applications. The non-contact AFM offers unique advantages over other contemporary scanning probe techniques such as contact AFM and scanning tunneling microscopy, especially when utilized for reliable measurements of soft samples (e.g., biological species). Current AFM imaging techniques are often based on a lumped-parameters model and ordinary differential equation (ODE) representation of the micro-cantilevers coupled with an adhoc method for atomic interaction force estimation (especially in non-contact mode). Since the magnitude of the interaction force lies within the range of nano-Newtons to pica-Newtons, precise estimation of the atomic force is crucial for accurate topographical imaging. In contrast to the previously utilized lumped modeling methods, this paper aims at improving current AFM measurement technique through developing a general distributed-parameters base modeling approach that reveals greater insight into the fundamental characteristics of the microcantilever-sample interaction. For this, the governing equations of motion are derived in the global coordinates via the Hamilton’s Extended Principle. An interaction force identification scheme is then designed based on the original infinite dimensional distributed-parameters system which, in turn, reveals the unmeasurable distance between AFM tip and sample surface. Numerical simulations are provided to support these claims.

Research on Chaotic Behavior of Concrete under External Load Based on Discontinuous Medium Mechanical Model

2009 ◽  
Vol 79-82 ◽  
pp. 1145-1148
Author(s):  
Hao Qin ◽  
Shu Cai Li
Keyword(s):  
External Load ◽  
Mechanical Model ◽  
Dynamical Equation ◽  
Chaotic Behavior ◽  
Concrete Material ◽  
Nonlinear Dynamical ◽  
Load Change ◽  
Nonlinear Science ◽  
Set Up ◽  
Discontinuous Medium

Outburst and random are typical characters of concrete when under external load. Traditional mechanics methods are difficult to be applied in. Depend on nonlinear science to set up the nonlinear dynamical equation that is fit for its characteristics. The subsystem dynamical equation of concrete is set up based on discontinuous medium mechanical model, and find that the concrete dynamical equation under external load is a Duffin equation. Then analyze and discuss the effect of external load on concrete by math analysis soft of maple. Results show that the concrete material take on complicated response to external load change.

Second Order Averaging Study of an Autoparametric System

1993 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj ◽  
Patricia Davies
Keyword(s):  
Dynamical Behavior ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Pitchfork Bifurcation ◽  
Second Order ◽  
System Response ◽  
Response Curves ◽  
Vibratory System ◽  
First Order

Abstract The autoparametric vibratory system consisting of a primary spring-mass-dashpot system coupled with a damped simple pendulum serves as an useful example of two degree-of-freedom nonlinear systems that exhibit complex dynamic behavior. It exhibits 1:2 internal resonance and amplitude modulated chaos under harmonic forcing conditions. First-order averaging studies of this system using AUTO and KAOS have yielded useful information about the amplitude dynamics of this system. Response curves of the system indicate saturation and the pitchfork bifurcation sets are found to be symmetric. The period-doubling route to chaotic solutions is observed. However questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid cannot be answered by a first-order study. Some observed dynamical behavior, like saturation, may not persist when higher-order nonlinear effects are taken into account. Second-order averaging of the system, using Mathematica (Maeder, 1991; Wolfram, 1991) is undertaken to address these questions. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. The dynamics of the system is found to be very sensitive to damping, with extremely complicated behavior arising for low values of damping. For large ε second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response.

scholarly journals Chaos and Control in Coronary Artery System

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yanxiang Shi
Keyword(s):  
Coronary Artery ◽  
Feedback Control ◽  
Numerical Simulations ◽  
Melnikov Method ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical ◽  
The Road ◽  
Two Systems ◽  
And Control

Two types of coronary artery system N-type and S-type, are investigated. The threshold conditions for the occurrence of Smale horseshoe chaos are obtained by using Melnikov method. Numerical simulations including phase portraits, potential diagram, homoclinic bifurcation curve diagrams, bifurcation diagrams, and Poincaré maps not only prove the correctness of theoretical analysis but also show the interesting bifurcation diagrams and the more new complex dynamical behaviors. Numerical simulations are used to investigate the nonlinear dynamical characteristics and complexity of the two systems, revealing bifurcation forms and the road leading to chaotic motion. Finally the chaotic states of the two systems are effectively controlled by two control methods: variable feedback control and coupled feedback control.

A Review of Recent Developments in Atomic Force Microscopy Systems With Application to Manufacturing and Biological Processes

2003 ◽  
Author(s):  
Karthik Laxminarayana ◽  
Nader Jalili
Keyword(s):  
Resonance Frequency ◽  
Intermolecular Forces ◽  
Minimal Amount ◽  
Contact Mode ◽  
Tapping Mode ◽  
Atomic Force ◽  
Natural Resonance ◽  
Target Sample ◽  
Recent Developments ◽  
Natural Resonance Frequency

The atomic force microscope (AFM) system has evolved into a useful tool for direct measurements of microstructural parameters and intermolecular forces at nanoscale level with atomic-resolution characterization. Typically, these microcantilever systems are operated in three open-loop modes; non-contact mode, contact mode, and tapping mode. In order to probe electric, magnetic, and/or atomic forces of a selected sample, the non-contact mode is utilized by moving the cantilever slightly away from the sample surface and oscillating the cantilever at or near its natural resonance frequency. Alternatively, the contact mode acquires sample attributes by monitoring interaction forces while the cantilever tip remains in contact with the target sample. The tapping mode of operation combines qualities of both the contact and non-contact modes by gleaning sample data and oscillating the cantilever tip at or near its natural resonance frequency while allowing the cantilever tip to impact the target sample for a minimal amount of time. Recent research on AFM systems has focused on many fabrication and manufacturing processes at molecular levels due to its tremendous surface microscopic capabilities. This paper provides a review of such recent developments in AFM imaging systems with emphasis on operational modes, microcantilever dynamic modeling and control. Due to the important contributions of AFM systems to manufacturing, this paper also provides a comprehensive review of recent applications of different AFM systems in these important areas.

Stable and Unstable Quasiperiodic Oscillations in Robot Dynamics

1993 ◽  
Author(s):  
G. Stépán ◽  
G. Haller
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Time Lag ◽  
Nonlinear Behavior ◽  
Degree Of Freedom ◽  
Robot Dynamics ◽  
Robot Arm ◽  
Delayed Systems ◽  
Nonlinear Dynamical ◽  
Quasiperiodic Oscillations

Abstract Delays in robot control may result in unexpectedly sophisticated nonlinear dynamical behavior. Experiments on force controlled robots frequently show periodic and quasiperiodic oscillations which cannot be explained without including the time lag and/or the sampling time of the system in our models. Delayed systems, even of low degree of freedom, can produce phenomena which are already well understood in the theory of nonlinear dynamical systems but hardly ever occur in simple mechanical models. To illustrate this, we analyze the delayed positioning of a single degree of freedom robot arm. The analytical results show typical nonlinear behavior in the system which may go through a codimension two Hopf bifurcation for an infinite set of parameter values, leading to the creation of two-tori in the phase space. These results give a qualitative explanation for the existence of self-excited quasiperiodic oscillations in the dynamics of force controlled robots.

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